SLIDE 1 E-step: update M-step: update
loglikelihood
θ0 L
Expectation-Maximization (EM)
initial parameter guess
φ
<latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit>
θ
<latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit><latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit><latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit><latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit>
θk = arg max
θ
Z p(z|φk) log p(x, z|θ)dz
<latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit><latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit><latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit><latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit>
q(z|φk) = p(z|x, θk−1)
<latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit><latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit><latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit><latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit>
F(φ1, θ)
<latexit sha1_base64="/B0l5qsDzm8kMWYxhk9fNHO0okY=">AB+XicdVDJSgNBEO2JW4zbqEcvjUGIENPEmO8BQXxGMEskITQ0+kTXoWumsCYcifePGgiFf/xJt/Y2cRVPRBweO9KqrqeZEUGgj5sFIrq2vrG+nNzNb2zu6evX9Q12GsGK+xUIaq6VHNpQh4DQRI3owUp74necMbXc/8xpgrLcLgHiYR7/h0EIi+YBSM1LXtm1w7Goque9aGIQd62rWzxCmTUiFfxMTJF8n5ZcEQUnZLpIxdh8yRUtUu/Z7uxey2OcBMEm1brkgk5CFQgm+TjWPKBvRAW8ZGlCf604yv3yKT4zSw/1QmQoAz9XvEwn1tZ74nun0KQz1b28m/uW1YuiXO4kIoh4wBaL+rHEOJZDLgnFGcgJ4ZQpoS5FbMhVZSBCStjQvj6FP9P6nHJY57V8xWrpZxpNEROkY5KILVEG3qIpqiKExekBP6NlKrEfrxXpdtKas5cwh+gHr7ROGtZLy</latexit><latexit sha1_base64="/B0l5qsDzm8kMWYxhk9fNHO0okY=">AB+XicdVDJSgNBEO2JW4zbqEcvjUGIENPEmO8BQXxGMEskITQ0+kTXoWumsCYcifePGgiFf/xJt/Y2cRVPRBweO9KqrqeZEUGgj5sFIrq2vrG+nNzNb2zu6evX9Q12GsGK+xUIaq6VHNpQh4DQRI3owUp74necMbXc/8xpgrLcLgHiYR7/h0EIi+YBSM1LXtm1w7Goque9aGIQd62rWzxCmTUiFfxMTJF8n5ZcEQUnZLpIxdh8yRUtUu/Z7uxey2OcBMEm1brkgk5CFQgm+TjWPKBvRAW8ZGlCf604yv3yKT4zSw/1QmQoAz9XvEwn1tZ74nun0KQz1b28m/uW1YuiXO4kIoh4wBaL+rHEOJZDLgnFGcgJ4ZQpoS5FbMhVZSBCStjQvj6FP9P6nHJY57V8xWrpZxpNEROkY5KILVEG3qIpqiKExekBP6NlKrEfrxXpdtKas5cwh+gHr7ROGtZLy</latexit><latexit sha1_base64="/B0l5qsDzm8kMWYxhk9fNHO0okY=">AB+XicdVDJSgNBEO2JW4zbqEcvjUGIENPEmO8BQXxGMEskITQ0+kTXoWumsCYcifePGgiFf/xJt/Y2cRVPRBweO9KqrqeZEUGgj5sFIrq2vrG+nNzNb2zu6evX9Q12GsGK+xUIaq6VHNpQh4DQRI3owUp74necMbXc/8xpgrLcLgHiYR7/h0EIi+YBSM1LXtm1w7Goque9aGIQd62rWzxCmTUiFfxMTJF8n5ZcEQUnZLpIxdh8yRUtUu/Z7uxey2OcBMEm1brkgk5CFQgm+TjWPKBvRAW8ZGlCf604yv3yKT4zSw/1QmQoAz9XvEwn1tZ74nun0KQz1b28m/uW1YuiXO4kIoh4wBaL+rHEOJZDLgnFGcgJ4ZQpoS5FbMhVZSBCStjQvj6FP9P6nHJY57V8xWrpZxpNEROkY5KILVEG3qIpqiKExekBP6NlKrEfrxXpdtKas5cwh+gHr7ROGtZLy</latexit><latexit sha1_base64="/B0l5qsDzm8kMWYxhk9fNHO0okY=">AB+XicdVDJSgNBEO2JW4zbqEcvjUGIENPEmO8BQXxGMEskITQ0+kTXoWumsCYcifePGgiFf/xJt/Y2cRVPRBweO9KqrqeZEUGgj5sFIrq2vrG+nNzNb2zu6evX9Q12GsGK+xUIaq6VHNpQh4DQRI3owUp74necMbXc/8xpgrLcLgHiYR7/h0EIi+YBSM1LXtm1w7Goque9aGIQd62rWzxCmTUiFfxMTJF8n5ZcEQUnZLpIxdh8yRUtUu/Z7uxey2OcBMEm1brkgk5CFQgm+TjWPKBvRAW8ZGlCf604yv3yKT4zSw/1QmQoAz9XvEwn1tZ74nun0KQz1b28m/uW1YuiXO4kIoh4wBaL+rHEOJZDLgnFGcgJ4ZQpoS5FbMhVZSBCStjQvj6FP9P6nHJY57V8xWrpZxpNEROkY5KILVEG3qIpqiKExekBP6NlKrEfrxXpdtKas5cwh+gHr7ROGtZLy</latexit>
SLIDE 2
E-step: update M-step: update
Expectation-Maximization (EM)
φ
<latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit>
θ
<latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit><latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit><latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit><latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit>
θk = arg max
θ
Z p(z|φk) log p(x, z|θ)dz
<latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit><latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit><latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit><latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit>
q(z|φk) = p(z|x, θk−1)
<latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit><latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit><latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit><latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit>
loglikelihood
θ0 L θ1
SLIDE 3
E-step: update M-step: update
Expectation-Maximization (EM)
φ
<latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit>
θ
<latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit><latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit><latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit><latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit>
θk = arg max
θ
Z p(z|φk) log p(x, z|θ)dz
<latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit><latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit><latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit><latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit>
q(z|φk) = p(z|x, θk−1)
<latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit><latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit><latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit><latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit>
loglikelihood
θ0 L θ1
F(φ2, θ)
<latexit sha1_base64="2Y3k+nc3PE0XFuROsFLP9JeOYDk=">AB+XicdVDJSgNBEO2JW4zbqEcvjUGIENPEmO8BQXxGMEskITQ0+kTXoWumsCYcifePGgiFf/xJt/Y2cRVPRBweO9KqrqeZEUGgj5sFIrq2vrG+nNzNb2zu6evX9Q12GsGK+xUIaq6VHNpQh4DQRI3owUp74necMbXc/8xpgrLcLgHiYR7/h0EIi+YBSM1LXtm1w7Gopu/qwNQw70tGtniVMmpUK+iImTL5Lzy4IhpOyWSBm7Dpkji5aodu3di9ksc8DYJq3XJBJ2EKhBM8mHWseUTaiA94yNKA+151kfvkUnxilh/uhMhUAnqvfJxLqaz3xPdPpUxjq395M/MtrxdAvdxIRDHwgC0W9WOJIcSzGHBPKM5ATgyhTAlzK2ZDqigDE1bGhPD1Kf6f1POSxz3rpitXC3jSKMjdIxyEUXqIJuURXVENj9ICe0LOVWI/Wi/W6aE1Zy5lD9APW2yeIQZLz</latexit><latexit sha1_base64="2Y3k+nc3PE0XFuROsFLP9JeOYDk=">AB+XicdVDJSgNBEO2JW4zbqEcvjUGIENPEmO8BQXxGMEskITQ0+kTXoWumsCYcifePGgiFf/xJt/Y2cRVPRBweO9KqrqeZEUGgj5sFIrq2vrG+nNzNb2zu6evX9Q12GsGK+xUIaq6VHNpQh4DQRI3owUp74necMbXc/8xpgrLcLgHiYR7/h0EIi+YBSM1LXtm1w7Gopu/qwNQw70tGtniVMmpUK+iImTL5Lzy4IhpOyWSBm7Dpkji5aodu3di9ksc8DYJq3XJBJ2EKhBM8mHWseUTaiA94yNKA+151kfvkUnxilh/uhMhUAnqvfJxLqaz3xPdPpUxjq395M/MtrxdAvdxIRDHwgC0W9WOJIcSzGHBPKM5ATgyhTAlzK2ZDqigDE1bGhPD1Kf6f1POSxz3rpitXC3jSKMjdIxyEUXqIJuURXVENj9ICe0LOVWI/Wi/W6aE1Zy5lD9APW2yeIQZLz</latexit><latexit sha1_base64="2Y3k+nc3PE0XFuROsFLP9JeOYDk=">AB+XicdVDJSgNBEO2JW4zbqEcvjUGIENPEmO8BQXxGMEskITQ0+kTXoWumsCYcifePGgiFf/xJt/Y2cRVPRBweO9KqrqeZEUGgj5sFIrq2vrG+nNzNb2zu6evX9Q12GsGK+xUIaq6VHNpQh4DQRI3owUp74necMbXc/8xpgrLcLgHiYR7/h0EIi+YBSM1LXtm1w7Gopu/qwNQw70tGtniVMmpUK+iImTL5Lzy4IhpOyWSBm7Dpkji5aodu3di9ksc8DYJq3XJBJ2EKhBM8mHWseUTaiA94yNKA+151kfvkUnxilh/uhMhUAnqvfJxLqaz3xPdPpUxjq395M/MtrxdAvdxIRDHwgC0W9WOJIcSzGHBPKM5ATgyhTAlzK2ZDqigDE1bGhPD1Kf6f1POSxz3rpitXC3jSKMjdIxyEUXqIJuURXVENj9ICe0LOVWI/Wi/W6aE1Zy5lD9APW2yeIQZLz</latexit><latexit sha1_base64="2Y3k+nc3PE0XFuROsFLP9JeOYDk=">AB+XicdVDJSgNBEO2JW4zbqEcvjUGIENPEmO8BQXxGMEskITQ0+kTXoWumsCYcifePGgiFf/xJt/Y2cRVPRBweO9KqrqeZEUGgj5sFIrq2vrG+nNzNb2zu6evX9Q12GsGK+xUIaq6VHNpQh4DQRI3owUp74necMbXc/8xpgrLcLgHiYR7/h0EIi+YBSM1LXtm1w7Gopu/qwNQw70tGtniVMmpUK+iImTL5Lzy4IhpOyWSBm7Dpkji5aodu3di9ksc8DYJq3XJBJ2EKhBM8mHWseUTaiA94yNKA+151kfvkUnxilh/uhMhUAnqvfJxLqaz3xPdPpUxjq395M/MtrxdAvdxIRDHwgC0W9WOJIcSzGHBPKM5ATgyhTAlzK2ZDqigDE1bGhPD1Kf6f1POSxz3rpitXC3jSKMjdIxyEUXqIJuURXVENj9ICe0LOVWI/Wi/W6aE1Zy5lD9APW2yeIQZLz</latexit>
SLIDE 4
E-step: update M-step: update
Expectation-Maximization (EM)
φ
<latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit>
θ
<latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit><latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit><latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit><latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit>
θk = arg max
θ
Z p(z|φk) log p(x, z|θ)dz
<latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit><latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit><latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit><latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit>
q(z|φk) = p(z|x, θk−1)
<latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit><latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit><latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit><latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit>
loglikelihood
θ0 L θ1
F(φ2, θ)
<latexit sha1_base64="2Y3k+nc3PE0XFuROsFLP9JeOYDk=">AB+XicdVDJSgNBEO2JW4zbqEcvjUGIENPEmO8BQXxGMEskITQ0+kTXoWumsCYcifePGgiFf/xJt/Y2cRVPRBweO9KqrqeZEUGgj5sFIrq2vrG+nNzNb2zu6evX9Q12GsGK+xUIaq6VHNpQh4DQRI3owUp74necMbXc/8xpgrLcLgHiYR7/h0EIi+YBSM1LXtm1w7Gopu/qwNQw70tGtniVMmpUK+iImTL5Lzy4IhpOyWSBm7Dpkji5aodu3di9ksc8DYJq3XJBJ2EKhBM8mHWseUTaiA94yNKA+151kfvkUnxilh/uhMhUAnqvfJxLqaz3xPdPpUxjq395M/MtrxdAvdxIRDHwgC0W9WOJIcSzGHBPKM5ATgyhTAlzK2ZDqigDE1bGhPD1Kf6f1POSxz3rpitXC3jSKMjdIxyEUXqIJuURXVENj9ICe0LOVWI/Wi/W6aE1Zy5lD9APW2yeIQZLz</latexit><latexit sha1_base64="2Y3k+nc3PE0XFuROsFLP9JeOYDk=">AB+XicdVDJSgNBEO2JW4zbqEcvjUGIENPEmO8BQXxGMEskITQ0+kTXoWumsCYcifePGgiFf/xJt/Y2cRVPRBweO9KqrqeZEUGgj5sFIrq2vrG+nNzNb2zu6evX9Q12GsGK+xUIaq6VHNpQh4DQRI3owUp74necMbXc/8xpgrLcLgHiYR7/h0EIi+YBSM1LXtm1w7Gopu/qwNQw70tGtniVMmpUK+iImTL5Lzy4IhpOyWSBm7Dpkji5aodu3di9ksc8DYJq3XJBJ2EKhBM8mHWseUTaiA94yNKA+151kfvkUnxilh/uhMhUAnqvfJxLqaz3xPdPpUxjq395M/MtrxdAvdxIRDHwgC0W9WOJIcSzGHBPKM5ATgyhTAlzK2ZDqigDE1bGhPD1Kf6f1POSxz3rpitXC3jSKMjdIxyEUXqIJuURXVENj9ICe0LOVWI/Wi/W6aE1Zy5lD9APW2yeIQZLz</latexit><latexit sha1_base64="2Y3k+nc3PE0XFuROsFLP9JeOYDk=">AB+XicdVDJSgNBEO2JW4zbqEcvjUGIENPEmO8BQXxGMEskITQ0+kTXoWumsCYcifePGgiFf/xJt/Y2cRVPRBweO9KqrqeZEUGgj5sFIrq2vrG+nNzNb2zu6evX9Q12GsGK+xUIaq6VHNpQh4DQRI3owUp74necMbXc/8xpgrLcLgHiYR7/h0EIi+YBSM1LXtm1w7Gopu/qwNQw70tGtniVMmpUK+iImTL5Lzy4IhpOyWSBm7Dpkji5aodu3di9ksc8DYJq3XJBJ2EKhBM8mHWseUTaiA94yNKA+151kfvkUnxilh/uhMhUAnqvfJxLqaz3xPdPpUxjq395M/MtrxdAvdxIRDHwgC0W9WOJIcSzGHBPKM5ATgyhTAlzK2ZDqigDE1bGhPD1Kf6f1POSxz3rpitXC3jSKMjdIxyEUXqIJuURXVENj9ICe0LOVWI/Wi/W6aE1Zy5lD9APW2yeIQZLz</latexit><latexit sha1_base64="2Y3k+nc3PE0XFuROsFLP9JeOYDk=">AB+XicdVDJSgNBEO2JW4zbqEcvjUGIENPEmO8BQXxGMEskITQ0+kTXoWumsCYcifePGgiFf/xJt/Y2cRVPRBweO9KqrqeZEUGgj5sFIrq2vrG+nNzNb2zu6evX9Q12GsGK+xUIaq6VHNpQh4DQRI3owUp74necMbXc/8xpgrLcLgHiYR7/h0EIi+YBSM1LXtm1w7Gopu/qwNQw70tGtniVMmpUK+iImTL5Lzy4IhpOyWSBm7Dpkji5aodu3di9ksc8DYJq3XJBJ2EKhBM8mHWseUTaiA94yNKA+151kfvkUnxilh/uhMhUAnqvfJxLqaz3xPdPpUxjq395M/MtrxdAvdxIRDHwgC0W9WOJIcSzGHBPKM5ATgyhTAlzK2ZDqigDE1bGhPD1Kf6f1POSxz3rpitXC3jSKMjdIxyEUXqIJuURXVENj9ICe0LOVWI/Wi/W6aE1Zy5lD9APW2yeIQZLz</latexit>
θ2
SLIDE 5
E-step: update M-step: update
Expectation-Maximization (EM)
φ
<latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit>
θ
<latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit><latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit><latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit><latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit>
θk = arg max
θ
Z p(z|φk) log p(x, z|θ)dz
<latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit><latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit><latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit><latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit>
q(z|φk) = p(z|x, θk−1)
<latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit><latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit><latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit><latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit>
loglikelihood
θ0 L θ2 θ1
F(φ3, θ)
<latexit sha1_base64="ILOP13jgxfosZ/qGHmqVbujASQ=">AB+XicdVDJSgNBEO2JW4zbqEcvjUGIENPEmO8BQXxGMEskITQ0+kTXoWumsCYcifePGgiFf/xJt/Y2cRVPRBweO9KqrqeZEUGgj5sFIrq2vrG+nNzNb2zu6evX9Q12GsGK+xUIaq6VHNpQh4DQRI3owUp74necMbXc/8xpgrLcLgHiYR7/h0EIi+YBSM1LXtm1w7Gopu4awNQw70tGtniVMmpUK+iImTL5Lzy4IhpOyWSBm7Dpkji5aodu3di9ksc8DYJq3XJBJ2EKhBM8mHWseUTaiA94yNKA+151kfvkUnxilh/uhMhUAnqvfJxLqaz3xPdPpUxjq395M/MtrxdAvdxIRDHwgC0W9WOJIcSzGHBPKM5ATgyhTAlzK2ZDqigDE1bGhPD1Kf6f1POSxz3rpitXC3jSKMjdIxyEUXqIJuURXVENj9ICe0LOVWI/Wi/W6aE1Zy5lD9APW2yeJzZL0</latexit><latexit sha1_base64="ILOP13jgxfosZ/qGHmqVbujASQ=">AB+XicdVDJSgNBEO2JW4zbqEcvjUGIENPEmO8BQXxGMEskITQ0+kTXoWumsCYcifePGgiFf/xJt/Y2cRVPRBweO9KqrqeZEUGgj5sFIrq2vrG+nNzNb2zu6evX9Q12GsGK+xUIaq6VHNpQh4DQRI3owUp74necMbXc/8xpgrLcLgHiYR7/h0EIi+YBSM1LXtm1w7Gopu4awNQw70tGtniVMmpUK+iImTL5Lzy4IhpOyWSBm7Dpkji5aodu3di9ksc8DYJq3XJBJ2EKhBM8mHWseUTaiA94yNKA+151kfvkUnxilh/uhMhUAnqvfJxLqaz3xPdPpUxjq395M/MtrxdAvdxIRDHwgC0W9WOJIcSzGHBPKM5ATgyhTAlzK2ZDqigDE1bGhPD1Kf6f1POSxz3rpitXC3jSKMjdIxyEUXqIJuURXVENj9ICe0LOVWI/Wi/W6aE1Zy5lD9APW2yeJzZL0</latexit><latexit sha1_base64="ILOP13jgxfosZ/qGHmqVbujASQ=">AB+XicdVDJSgNBEO2JW4zbqEcvjUGIENPEmO8BQXxGMEskITQ0+kTXoWumsCYcifePGgiFf/xJt/Y2cRVPRBweO9KqrqeZEUGgj5sFIrq2vrG+nNzNb2zu6evX9Q12GsGK+xUIaq6VHNpQh4DQRI3owUp74necMbXc/8xpgrLcLgHiYR7/h0EIi+YBSM1LXtm1w7Gopu4awNQw70tGtniVMmpUK+iImTL5Lzy4IhpOyWSBm7Dpkji5aodu3di9ksc8DYJq3XJBJ2EKhBM8mHWseUTaiA94yNKA+151kfvkUnxilh/uhMhUAnqvfJxLqaz3xPdPpUxjq395M/MtrxdAvdxIRDHwgC0W9WOJIcSzGHBPKM5ATgyhTAlzK2ZDqigDE1bGhPD1Kf6f1POSxz3rpitXC3jSKMjdIxyEUXqIJuURXVENj9ICe0LOVWI/Wi/W6aE1Zy5lD9APW2yeJzZL0</latexit><latexit sha1_base64="ILOP13jgxfosZ/qGHmqVbujASQ=">AB+XicdVDJSgNBEO2JW4zbqEcvjUGIENPEmO8BQXxGMEskITQ0+kTXoWumsCYcifePGgiFf/xJt/Y2cRVPRBweO9KqrqeZEUGgj5sFIrq2vrG+nNzNb2zu6evX9Q12GsGK+xUIaq6VHNpQh4DQRI3owUp74necMbXc/8xpgrLcLgHiYR7/h0EIi+YBSM1LXtm1w7Gopu4awNQw70tGtniVMmpUK+iImTL5Lzy4IhpOyWSBm7Dpkji5aodu3di9ksc8DYJq3XJBJ2EKhBM8mHWseUTaiA94yNKA+151kfvkUnxilh/uhMhUAnqvfJxLqaz3xPdPpUxjq395M/MtrxdAvdxIRDHwgC0W9WOJIcSzGHBPKM5ATgyhTAlzK2ZDqigDE1bGhPD1Kf6f1POSxz3rpitXC3jSKMjdIxyEUXqIJuURXVENj9ICe0LOVWI/Wi/W6aE1Zy5lD9APW2yeJzZL0</latexit>
SLIDE 6
E-step: update M-step: update
Expectation-Maximization (EM)
φ
<latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit>
θ
<latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit><latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit><latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit><latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDbH7tlJ45pU8H2rhSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MBRThxh3Ah3K+UjZhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl8ckz/wPn8Ao/ePKA=</latexit>
θk = arg max
θ
Z p(z|φk) log p(x, z|θ)dz
<latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit><latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit><latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit><latexit sha1_base64="95FRklBUuUZ9Q82HUFePJbRBU/g=">ACK3icbZDLSgNBEV74ju+oi7dNAYhAQkzIuhGCLpxqWBUyAxDTaeTNOl50F0jSUb/x42/4kIXPnDrf9hJZqHGCw2XU1V01Q0SKTa9rtVmJmdm19YXCour6yurZc2Nq90nCrGyWsboJQHMpIt5AgZLfJIpDGEh+HfROR/XrW60iKNLHCTcC6ETibZgAb5pRMXuxzB79Fj6oLqUDeEvj+B1BUR0qQyvHOTrvB7VerKuGNAf8+gcUuVtoZ+qWzX7LHotHFyUya5zv3Ss9uKWRryCJkErZuOnaCXgULBJL8vuqnmCbAedHjT2AhCr1sfOs93TWkRduxMs9sN6Y/JzItR6EgekMAbv6b20E/6s1U2wfeZmIkhR5xCYftVNJMaj4GhLKM5QDowBpoTZlbIuKGBo4i2aEJy/J0+bq/2aY9eci4Ny/SPY5Fskx1SIQ45JHVyRs5JgzDyQJ7IK3mzHq0X68P6nLQWrHxmi/yS9fUN1mqmzw=</latexit>
q(z|φk) = p(z|x, θk−1)
<latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit><latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit><latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit><latexit sha1_base64="uzljJ/72tp76uLWhDFwf7GrzHZo=">ACDXicbVC7SgNBFJ2Nrxhfq5Y2g1FIQMOuCNoIQRvLCOYBybLMTibJkNmHM3fFuO4P2PgrNhaK2Nrb+TdOHoUmHrhw5px7mXuPFwmuwLK+jczc/MLiUnY5t7K6tr5hbm7VBhLyqo0FKFseEQxwQNWBQ6CNSLJiO8JVvf6F0O/fsuk4mFwDYOIOT7pBrzDKQEtuebeTeH+oRX1uJv0yI+w5F+3x20oMeAaO3QTos518xbJWsEPEvsCcmjCSqu+dVqhzT2WQBUEKWathWBkxAJnAqW5lqxYhGhfdJlTU0D4jPlJKNrUryvlTbuhFJXAHik/p5IiK/UwPd0p0+gp6a9ofif14yhc+okPIhiYAEdf9SJBYQD6PBbS4ZBTHQhFDJ9a6Y9ogkFHSAwxDs6ZNnSe2oZFsl+o4Xz6fxJFO2gXFZCNTlAZXaIKqiKHtEzekVvxpPxYrwbH+PWjDGZ2UZ/YHz+AKS5mp4=</latexit>
loglikelihood
θ0 L θ2 θ1
F(φ3, θ)
<latexit sha1_base64="ILOP13jgxfosZ/qGHmqVbujASQ=">AB+XicdVDJSgNBEO2JW4zbqEcvjUGIENPEmO8BQXxGMEskITQ0+kTXoWumsCYcifePGgiFf/xJt/Y2cRVPRBweO9KqrqeZEUGgj5sFIrq2vrG+nNzNb2zu6evX9Q12GsGK+xUIaq6VHNpQh4DQRI3owUp74necMbXc/8xpgrLcLgHiYR7/h0EIi+YBSM1LXtm1w7Gopu4awNQw70tGtniVMmpUK+iImTL5Lzy4IhpOyWSBm7Dpkji5aodu3di9ksc8DYJq3XJBJ2EKhBM8mHWseUTaiA94yNKA+151kfvkUnxilh/uhMhUAnqvfJxLqaz3xPdPpUxjq395M/MtrxdAvdxIRDHwgC0W9WOJIcSzGHBPKM5ATgyhTAlzK2ZDqigDE1bGhPD1Kf6f1POSxz3rpitXC3jSKMjdIxyEUXqIJuURXVENj9ICe0LOVWI/Wi/W6aE1Zy5lD9APW2yeJzZL0</latexit><latexit sha1_base64="ILOP13jgxfosZ/qGHmqVbujASQ=">AB+XicdVDJSgNBEO2JW4zbqEcvjUGIENPEmO8BQXxGMEskITQ0+kTXoWumsCYcifePGgiFf/xJt/Y2cRVPRBweO9KqrqeZEUGgj5sFIrq2vrG+nNzNb2zu6evX9Q12GsGK+xUIaq6VHNpQh4DQRI3owUp74necMbXc/8xpgrLcLgHiYR7/h0EIi+YBSM1LXtm1w7Gopu4awNQw70tGtniVMmpUK+iImTL5Lzy4IhpOyWSBm7Dpkji5aodu3di9ksc8DYJq3XJBJ2EKhBM8mHWseUTaiA94yNKA+151kfvkUnxilh/uhMhUAnqvfJxLqaz3xPdPpUxjq395M/MtrxdAvdxIRDHwgC0W9WOJIcSzGHBPKM5ATgyhTAlzK2ZDqigDE1bGhPD1Kf6f1POSxz3rpitXC3jSKMjdIxyEUXqIJuURXVENj9ICe0LOVWI/Wi/W6aE1Zy5lD9APW2yeJzZL0</latexit><latexit sha1_base64="ILOP13jgxfosZ/qGHmqVbujASQ=">AB+XicdVDJSgNBEO2JW4zbqEcvjUGIENPEmO8BQXxGMEskITQ0+kTXoWumsCYcifePGgiFf/xJt/Y2cRVPRBweO9KqrqeZEUGgj5sFIrq2vrG+nNzNb2zu6evX9Q12GsGK+xUIaq6VHNpQh4DQRI3owUp74necMbXc/8xpgrLcLgHiYR7/h0EIi+YBSM1LXtm1w7Gopu4awNQw70tGtniVMmpUK+iImTL5Lzy4IhpOyWSBm7Dpkji5aodu3di9ksc8DYJq3XJBJ2EKhBM8mHWseUTaiA94yNKA+151kfvkUnxilh/uhMhUAnqvfJxLqaz3xPdPpUxjq395M/MtrxdAvdxIRDHwgC0W9WOJIcSzGHBPKM5ATgyhTAlzK2ZDqigDE1bGhPD1Kf6f1POSxz3rpitXC3jSKMjdIxyEUXqIJuURXVENj9ICe0LOVWI/Wi/W6aE1Zy5lD9APW2yeJzZL0</latexit><latexit sha1_base64="ILOP13jgxfosZ/qGHmqVbujASQ=">AB+XicdVDJSgNBEO2JW4zbqEcvjUGIENPEmO8BQXxGMEskITQ0+kTXoWumsCYcifePGgiFf/xJt/Y2cRVPRBweO9KqrqeZEUGgj5sFIrq2vrG+nNzNb2zu6evX9Q12GsGK+xUIaq6VHNpQh4DQRI3owUp74necMbXc/8xpgrLcLgHiYR7/h0EIi+YBSM1LXtm1w7Gopu4awNQw70tGtniVMmpUK+iImTL5Lzy4IhpOyWSBm7Dpkji5aodu3di9ksc8DYJq3XJBJ2EKhBM8mHWseUTaiA94yNKA+151kfvkUnxilh/uhMhUAnqvfJxLqaz3xPdPpUxjq395M/MtrxdAvdxIRDHwgC0W9WOJIcSzGHBPKM5ATgyhTAlzK2ZDqigDE1bGhPD1Kf6f1POSxz3rpitXC3jSKMjdIxyEUXqIJuURXVENj9ICe0LOVWI/Wi/W6aE1Zy5lD9APW2yeJzZL0</latexit>
θML
alternate until convergence
SLIDE 7
θ0
Expectation-Maximization (EM)
Alternative view: coordinate ascent of free energy: F(φ, θ)
<latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit><latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit><latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit><latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit>
φ
<latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit>
φ1
<latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit><latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit><latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit><latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit>
SLIDE 8
θ0
Expectation-Maximization (EM)
Alternative view: coordinate ascent of free energy:
θ1
F(φ, θ)
<latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit><latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit><latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit><latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit>
φ
<latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit>
φ1
<latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit><latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit><latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit><latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit>
SLIDE 9
θ0 θ1
Expectation-Maximization (EM)
Alternative view: coordinate ascent of free energy: F(φ, θ)
<latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit><latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit><latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit><latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit>
φ
<latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit>
φ1
<latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit><latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit><latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit><latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit>
φ2
<latexit sha1_base64="sIjD/slOEMP2/Io5OvKpdr1mOw=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHoxWME84BkCbOT2WTM7Mwy0yuEkH/w4kERr/6PN/GSbIHTSxoKq6e6KUiks+v63t7a+sbm1Xdgp7u7tHxyWjo6bVmeG8QbTUpt2RC2XQvEGCpS8nRpOk0jyVjS6nfmtJ26s0OoBxykPEzpQIhaMopOa3XQoetVeqexX/DnIKglyUoYc9V7pq9vXLEu4QiaptZ3ATzGcUIOCST4tdjPLU8pGdMA7jiqacBtO5tdOyblT+iTWxpVCMld/T0xoYu04iVxnQnFol72Z+J/XyTC+DidCpRlyxRaL4kwS1GT2OukLwxnKsSOUGeFuJWxIDWXoAiq6EILl1dJs1oJ/Epwf1mu3eRxFOAUzuACAriCGtxBHRrA4BGe4RXePO29eO/ex6J1zctnTuAPvM8fPMqO5A=</latexit><latexit sha1_base64="sIjD/slOEMP2/Io5OvKpdr1mOw=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHoxWME84BkCbOT2WTM7Mwy0yuEkH/w4kERr/6PN/GSbIHTSxoKq6e6KUiks+v63t7a+sbm1Xdgp7u7tHxyWjo6bVmeG8QbTUpt2RC2XQvEGCpS8nRpOk0jyVjS6nfmtJ26s0OoBxykPEzpQIhaMopOa3XQoetVeqexX/DnIKglyUoYc9V7pq9vXLEu4QiaptZ3ATzGcUIOCST4tdjPLU8pGdMA7jiqacBtO5tdOyblT+iTWxpVCMld/T0xoYu04iVxnQnFol72Z+J/XyTC+DidCpRlyxRaL4kwS1GT2OukLwxnKsSOUGeFuJWxIDWXoAiq6EILl1dJs1oJ/Epwf1mu3eRxFOAUzuACAriCGtxBHRrA4BGe4RXePO29eO/ex6J1zctnTuAPvM8fPMqO5A=</latexit><latexit sha1_base64="sIjD/slOEMP2/Io5OvKpdr1mOw=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHoxWME84BkCbOT2WTM7Mwy0yuEkH/w4kERr/6PN/GSbIHTSxoKq6e6KUiks+v63t7a+sbm1Xdgp7u7tHxyWjo6bVmeG8QbTUpt2RC2XQvEGCpS8nRpOk0jyVjS6nfmtJ26s0OoBxykPEzpQIhaMopOa3XQoetVeqexX/DnIKglyUoYc9V7pq9vXLEu4QiaptZ3ATzGcUIOCST4tdjPLU8pGdMA7jiqacBtO5tdOyblT+iTWxpVCMld/T0xoYu04iVxnQnFol72Z+J/XyTC+DidCpRlyxRaL4kwS1GT2OukLwxnKsSOUGeFuJWxIDWXoAiq6EILl1dJs1oJ/Epwf1mu3eRxFOAUzuACAriCGtxBHRrA4BGe4RXePO29eO/ex6J1zctnTuAPvM8fPMqO5A=</latexit><latexit sha1_base64="sIjD/slOEMP2/Io5OvKpdr1mOw=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHoxWME84BkCbOT2WTM7Mwy0yuEkH/w4kERr/6PN/GSbIHTSxoKq6e6KUiks+v63t7a+sbm1Xdgp7u7tHxyWjo6bVmeG8QbTUpt2RC2XQvEGCpS8nRpOk0jyVjS6nfmtJ26s0OoBxykPEzpQIhaMopOa3XQoetVeqexX/DnIKglyUoYc9V7pq9vXLEu4QiaptZ3ATzGcUIOCST4tdjPLU8pGdMA7jiqacBtO5tdOyblT+iTWxpVCMld/T0xoYu04iVxnQnFol72Z+J/XyTC+DidCpRlyxRaL4kwS1GT2OukLwxnKsSOUGeFuJWxIDWXoAiq6EILl1dJs1oJ/Epwf1mu3eRxFOAUzuACAriCGtxBHRrA4BGe4RXePO29eO/ex6J1zctnTuAPvM8fPMqO5A=</latexit>
SLIDE 10
θ0 θ1
Expectation-Maximization (EM)
Alternative view: coordinate ascent of free energy:
θ2
F(φ, θ)
<latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit><latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit><latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit><latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit>
φ
<latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit>
φ1
<latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit><latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit><latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit><latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit>
φ2
<latexit sha1_base64="sIjD/slOEMP2/Io5OvKpdr1mOw=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHoxWME84BkCbOT2WTM7Mwy0yuEkH/w4kERr/6PN/GSbIHTSxoKq6e6KUiks+v63t7a+sbm1Xdgp7u7tHxyWjo6bVmeG8QbTUpt2RC2XQvEGCpS8nRpOk0jyVjS6nfmtJ26s0OoBxykPEzpQIhaMopOa3XQoetVeqexX/DnIKglyUoYc9V7pq9vXLEu4QiaptZ3ATzGcUIOCST4tdjPLU8pGdMA7jiqacBtO5tdOyblT+iTWxpVCMld/T0xoYu04iVxnQnFol72Z+J/XyTC+DidCpRlyxRaL4kwS1GT2OukLwxnKsSOUGeFuJWxIDWXoAiq6EILl1dJs1oJ/Epwf1mu3eRxFOAUzuACAriCGtxBHRrA4BGe4RXePO29eO/ex6J1zctnTuAPvM8fPMqO5A=</latexit><latexit sha1_base64="sIjD/slOEMP2/Io5OvKpdr1mOw=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHoxWME84BkCbOT2WTM7Mwy0yuEkH/w4kERr/6PN/GSbIHTSxoKq6e6KUiks+v63t7a+sbm1Xdgp7u7tHxyWjo6bVmeG8QbTUpt2RC2XQvEGCpS8nRpOk0jyVjS6nfmtJ26s0OoBxykPEzpQIhaMopOa3XQoetVeqexX/DnIKglyUoYc9V7pq9vXLEu4QiaptZ3ATzGcUIOCST4tdjPLU8pGdMA7jiqacBtO5tdOyblT+iTWxpVCMld/T0xoYu04iVxnQnFol72Z+J/XyTC+DidCpRlyxRaL4kwS1GT2OukLwxnKsSOUGeFuJWxIDWXoAiq6EILl1dJs1oJ/Epwf1mu3eRxFOAUzuACAriCGtxBHRrA4BGe4RXePO29eO/ex6J1zctnTuAPvM8fPMqO5A=</latexit><latexit sha1_base64="sIjD/slOEMP2/Io5OvKpdr1mOw=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHoxWME84BkCbOT2WTM7Mwy0yuEkH/w4kERr/6PN/GSbIHTSxoKq6e6KUiks+v63t7a+sbm1Xdgp7u7tHxyWjo6bVmeG8QbTUpt2RC2XQvEGCpS8nRpOk0jyVjS6nfmtJ26s0OoBxykPEzpQIhaMopOa3XQoetVeqexX/DnIKglyUoYc9V7pq9vXLEu4QiaptZ3ATzGcUIOCST4tdjPLU8pGdMA7jiqacBtO5tdOyblT+iTWxpVCMld/T0xoYu04iVxnQnFol72Z+J/XyTC+DidCpRlyxRaL4kwS1GT2OukLwxnKsSOUGeFuJWxIDWXoAiq6EILl1dJs1oJ/Epwf1mu3eRxFOAUzuACAriCGtxBHRrA4BGe4RXePO29eO/ex6J1zctnTuAPvM8fPMqO5A=</latexit><latexit sha1_base64="sIjD/slOEMP2/Io5OvKpdr1mOw=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHoxWME84BkCbOT2WTM7Mwy0yuEkH/w4kERr/6PN/GSbIHTSxoKq6e6KUiks+v63t7a+sbm1Xdgp7u7tHxyWjo6bVmeG8QbTUpt2RC2XQvEGCpS8nRpOk0jyVjS6nfmtJ26s0OoBxykPEzpQIhaMopOa3XQoetVeqexX/DnIKglyUoYc9V7pq9vXLEu4QiaptZ3ATzGcUIOCST4tdjPLU8pGdMA7jiqacBtO5tdOyblT+iTWxpVCMld/T0xoYu04iVxnQnFol72Z+J/XyTC+DidCpRlyxRaL4kwS1GT2OukLwxnKsSOUGeFuJWxIDWXoAiq6EILl1dJs1oJ/Epwf1mu3eRxFOAUzuACAriCGtxBHRrA4BGe4RXePO29eO/ex6J1zctnTuAPvM8fPMqO5A=</latexit>
SLIDE 11
θ0 θ2 θ1
Expectation-Maximization (EM)
Alternative view: coordinate ascent of free energy: F(φ, θ)
<latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit><latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit><latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit><latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit>
φ
<latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit>
φ1
<latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit><latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit><latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit><latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit>
φ2
<latexit sha1_base64="sIjD/slOEMP2/Io5OvKpdr1mOw=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHoxWME84BkCbOT2WTM7Mwy0yuEkH/w4kERr/6PN/GSbIHTSxoKq6e6KUiks+v63t7a+sbm1Xdgp7u7tHxyWjo6bVmeG8QbTUpt2RC2XQvEGCpS8nRpOk0jyVjS6nfmtJ26s0OoBxykPEzpQIhaMopOa3XQoetVeqexX/DnIKglyUoYc9V7pq9vXLEu4QiaptZ3ATzGcUIOCST4tdjPLU8pGdMA7jiqacBtO5tdOyblT+iTWxpVCMld/T0xoYu04iVxnQnFol72Z+J/XyTC+DidCpRlyxRaL4kwS1GT2OukLwxnKsSOUGeFuJWxIDWXoAiq6EILl1dJs1oJ/Epwf1mu3eRxFOAUzuACAriCGtxBHRrA4BGe4RXePO29eO/ex6J1zctnTuAPvM8fPMqO5A=</latexit><latexit sha1_base64="sIjD/slOEMP2/Io5OvKpdr1mOw=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHoxWME84BkCbOT2WTM7Mwy0yuEkH/w4kERr/6PN/GSbIHTSxoKq6e6KUiks+v63t7a+sbm1Xdgp7u7tHxyWjo6bVmeG8QbTUpt2RC2XQvEGCpS8nRpOk0jyVjS6nfmtJ26s0OoBxykPEzpQIhaMopOa3XQoetVeqexX/DnIKglyUoYc9V7pq9vXLEu4QiaptZ3ATzGcUIOCST4tdjPLU8pGdMA7jiqacBtO5tdOyblT+iTWxpVCMld/T0xoYu04iVxnQnFol72Z+J/XyTC+DidCpRlyxRaL4kwS1GT2OukLwxnKsSOUGeFuJWxIDWXoAiq6EILl1dJs1oJ/Epwf1mu3eRxFOAUzuACAriCGtxBHRrA4BGe4RXePO29eO/ex6J1zctnTuAPvM8fPMqO5A=</latexit><latexit sha1_base64="sIjD/slOEMP2/Io5OvKpdr1mOw=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHoxWME84BkCbOT2WTM7Mwy0yuEkH/w4kERr/6PN/GSbIHTSxoKq6e6KUiks+v63t7a+sbm1Xdgp7u7tHxyWjo6bVmeG8QbTUpt2RC2XQvEGCpS8nRpOk0jyVjS6nfmtJ26s0OoBxykPEzpQIhaMopOa3XQoetVeqexX/DnIKglyUoYc9V7pq9vXLEu4QiaptZ3ATzGcUIOCST4tdjPLU8pGdMA7jiqacBtO5tdOyblT+iTWxpVCMld/T0xoYu04iVxnQnFol72Z+J/XyTC+DidCpRlyxRaL4kwS1GT2OukLwxnKsSOUGeFuJWxIDWXoAiq6EILl1dJs1oJ/Epwf1mu3eRxFOAUzuACAriCGtxBHRrA4BGe4RXePO29eO/ex6J1zctnTuAPvM8fPMqO5A=</latexit><latexit sha1_base64="sIjD/slOEMP2/Io5OvKpdr1mOw=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHoxWME84BkCbOT2WTM7Mwy0yuEkH/w4kERr/6PN/GSbIHTSxoKq6e6KUiks+v63t7a+sbm1Xdgp7u7tHxyWjo6bVmeG8QbTUpt2RC2XQvEGCpS8nRpOk0jyVjS6nfmtJ26s0OoBxykPEzpQIhaMopOa3XQoetVeqexX/DnIKglyUoYc9V7pq9vXLEu4QiaptZ3ATzGcUIOCST4tdjPLU8pGdMA7jiqacBtO5tdOyblT+iTWxpVCMld/T0xoYu04iVxnQnFol72Z+J/XyTC+DidCpRlyxRaL4kwS1GT2OukLwxnKsSOUGeFuJWxIDWXoAiq6EILl1dJs1oJ/Epwf1mu3eRxFOAUzuACAriCGtxBHRrA4BGe4RXePO29eO/ex6J1zctnTuAPvM8fPMqO5A=</latexit>
SLIDE 12
θ0 θ2 θ1
Expectation-Maximization (EM)
Alternative view: coordinate ascent of free energy:
θML
1D manifold where free energy and log-likehood touch
F(φ, θ)
<latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit><latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit><latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit><latexit sha1_base64="EAzN5J9tivpx73cITrA3mpoXfMo=">AB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgVQY9BQTxGMA/IrmF2MpsMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhslrFq+1RzKSLeQIGStxPFaehL3vKH1xO/9ciVFnF0j6OEeyHtRyIQjKRHm4qbjIQpy4ONKTbqlsV+0pyCJxclKGHPVu6cvtxSwNeYRMUq07jp2gl1GFgk+Lrqp5glQ9rnHUMjGnLtZdOrx+TYKD0SxMpUhGSq/p7IaKj1KPRNZ0hxoOe9ifif10kxuPQyESUp8ojNFgWpJBiTSQSkJxRnKEeGUKaEuZWwAVWUoQmqaEJw5l9eJM2zqmNXnbvzcu0qj6MAh3AEFXDgAmpwC3VoAMFz/AKb9aT9WK9Wx+z1iUrnzmAP7A+fwB9wpHY</latexit>
φ
<latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8m2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzqmiTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX35lXQvmr7X9O8vG62bMo4qnMApnIMPV9CO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIj8=</latexit>
φ1
<latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit><latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit><latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit><latexit sha1_base64="sro7AUF6X7sk3u57/oe2B5z/C+g=">AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOJsmY2ZlplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4N4w2mpTbtiFouheINFCh5OzGcxpHkrWh8O/NbT9xYodUDThIexnSoxEAwik5qdpOR6AW9csWv+nOQVRLkpAI56r3yV7evWRpzhUxSazuBn2CYUYOCST4tdVPLE8rGdMg7jioacxtm82un5MwpfTLQxpVCMld/T2Q0tnYSR64zpjiy95M/M/rpDi4DjOhkhS5YotFg1QS1GT2OukLwxnKiSOUGeFuJWxEDWXoAiq5EILl1dJ86Ia+NXg/rJSu8njKMIJnMI5BHAFNbiDOjSAwSM8wyu8edp78d69j0VrwctnjuEPvM8fO0aO4w=</latexit>
φ2
<latexit sha1_base64="sIjD/slOEMP2/Io5OvKpdr1mOw=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHoxWME84BkCbOT2WTM7Mwy0yuEkH/w4kERr/6PN/GSbIHTSxoKq6e6KUiks+v63t7a+sbm1Xdgp7u7tHxyWjo6bVmeG8QbTUpt2RC2XQvEGCpS8nRpOk0jyVjS6nfmtJ26s0OoBxykPEzpQIhaMopOa3XQoetVeqexX/DnIKglyUoYc9V7pq9vXLEu4QiaptZ3ATzGcUIOCST4tdjPLU8pGdMA7jiqacBtO5tdOyblT+iTWxpVCMld/T0xoYu04iVxnQnFol72Z+J/XyTC+DidCpRlyxRaL4kwS1GT2OukLwxnKsSOUGeFuJWxIDWXoAiq6EILl1dJs1oJ/Epwf1mu3eRxFOAUzuACAriCGtxBHRrA4BGe4RXePO29eO/ex6J1zctnTuAPvM8fPMqO5A=</latexit><latexit sha1_base64="sIjD/slOEMP2/Io5OvKpdr1mOw=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHoxWME84BkCbOT2WTM7Mwy0yuEkH/w4kERr/6PN/GSbIHTSxoKq6e6KUiks+v63t7a+sbm1Xdgp7u7tHxyWjo6bVmeG8QbTUpt2RC2XQvEGCpS8nRpOk0jyVjS6nfmtJ26s0OoBxykPEzpQIhaMopOa3XQoetVeqexX/DnIKglyUoYc9V7pq9vXLEu4QiaptZ3ATzGcUIOCST4tdjPLU8pGdMA7jiqacBtO5tdOyblT+iTWxpVCMld/T0xoYu04iVxnQnFol72Z+J/XyTC+DidCpRlyxRaL4kwS1GT2OukLwxnKsSOUGeFuJWxIDWXoAiq6EILl1dJs1oJ/Epwf1mu3eRxFOAUzuACAriCGtxBHRrA4BGe4RXePO29eO/ex6J1zctnTuAPvM8fPMqO5A=</latexit><latexit sha1_base64="sIjD/slOEMP2/Io5OvKpdr1mOw=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHoxWME84BkCbOT2WTM7Mwy0yuEkH/w4kERr/6PN/GSbIHTSxoKq6e6KUiks+v63t7a+sbm1Xdgp7u7tHxyWjo6bVmeG8QbTUpt2RC2XQvEGCpS8nRpOk0jyVjS6nfmtJ26s0OoBxykPEzpQIhaMopOa3XQoetVeqexX/DnIKglyUoYc9V7pq9vXLEu4QiaptZ3ATzGcUIOCST4tdjPLU8pGdMA7jiqacBtO5tdOyblT+iTWxpVCMld/T0xoYu04iVxnQnFol72Z+J/XyTC+DidCpRlyxRaL4kwS1GT2OukLwxnKsSOUGeFuJWxIDWXoAiq6EILl1dJs1oJ/Epwf1mu3eRxFOAUzuACAriCGtxBHRrA4BGe4RXePO29eO/ex6J1zctnTuAPvM8fPMqO5A=</latexit><latexit sha1_base64="sIjD/slOEMP2/Io5OvKpdr1mOw=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHoxWME84BkCbOT2WTM7Mwy0yuEkH/w4kERr/6PN/GSbIHTSxoKq6e6KUiks+v63t7a+sbm1Xdgp7u7tHxyWjo6bVmeG8QbTUpt2RC2XQvEGCpS8nRpOk0jyVjS6nfmtJ26s0OoBxykPEzpQIhaMopOa3XQoetVeqexX/DnIKglyUoYc9V7pq9vXLEu4QiaptZ3ATzGcUIOCST4tdjPLU8pGdMA7jiqacBtO5tdOyblT+iTWxpVCMld/T0xoYu04iVxnQnFol72Z+J/XyTC+DidCpRlyxRaL4kwS1GT2OukLwxnKsSOUGeFuJWxIDWXoAiq6EILl1dJs1oJ/Epwf1mu3eRxFOAUzuACAriCGtxBHRrA4BGe4RXePO29eO/ex6J1zctnTuAPvM8fPMqO5A=</latexit>