causal discovery from observational data
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Causal Discovery from Observational Data Brady Neal - PowerPoint PPT Presentation

Causal Discovery from Observational Data Brady Neal causalcourse.com What if we dont have the causal graph? Brady Neal 2 / 45 What if we dont have the causal graph? Causal discovery: data causal graph Brady Neal 2 / 45


  1. Faithfulness Assumption Recall the Markov assumption: ⊥ G Y | Z = ⊥ P Y | Z <latexit sha1_base64="zlPcHd38t40hUna9e0GNDw+9jeg=">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</latexit> X ⊥ ⇒ X ⊥ Causal graph Data Causal graph Data Faithfulness: ⊥ G Y | Z ⇐ ⊥ P Y | Z <latexit sha1_base64="hcvMBKxaM6ynzjkNMhCc6VBJEg=">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</latexit> X ⊥ = X ⊥ Brady Neal Assumptions 5 / 45

  2. Violation of Faithfulness Brady Neal Assumptions 6 / 45

  3. Violation of Faithfulness Faithfulness: ⊥ G Y | Z ⇐ ⊥ P Y | Z <latexit sha1_base64="hcvMBKxaM6ynzjkNMhCc6VBJEg=">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</latexit> X ⊥ = X ⊥ Brady Neal Assumptions 6 / 45

  4. <latexit sha1_base64="TZPsAeWD3LFNlhH1rFZ4zmsmB1o=">AV93icrVjZbtGFB27W+IuWQoDLfrCxg6QALIquy0SIDCQWE5QFE2RAE6c1DYCihxJhLiVHEV1CL30R/pW9KlAP6ef0L/ovWeGIqmNchARIod37r7NDux76Wq1fp3bf29z/48KNLlzc+/uTz65cvXb9eRoNE0c+cyI/Sl507FT6XifKU/58kWcSDvo+PK4M2jz/PFrmaReFB6p81ieBXYv9LqeYysCvbq29vtpR/a8MFPe4E3sOWqYyPGZX43rZ8jV6aT9OQXk9SZSt5Zt16cNuylXWr1WjdtrLtB9vjewsQD25bJx3pRyPLl1l7Ud68EZkRwsJmlPSBKv1y/RtBfTHM6hOWCaw5ymZNhDt1c2LZVHxZJgrPG9xbOtWfnDvK5w9m59vTcqQzdirtfXd1qNVv4WbODXTPYEub3JLp2+W9xKlwRCUcMRSCkCIWisS9skdJ1InZFS8QEOxMZwRIaeZiXYiw2iHZIWJIwbIO6N6jtxMDemdeagdkiKT/+EKC1x0+C4NO4Cqp8s3yrhLpKRgTfreE7PjuEZEFSJPkHr6HLM1ek6dAU1Viuy4S6s9ciSGBD2g1OxuUtPn94VWcj3c8KUNHKJKqGRQzCfoBrCMhJ6as+zb/qIhA08SPWepney7Tm+YiuAfGyaZxCU9bVEo9MXEJIltCVcXzEdDH38hCrd8yXt2JDR7irq1g3COCDMQbGlU5L5NZzq7FWMpwHs+RxfgKGCFhpwSJkPke+c8jGquOsSTZdiIZA+2xMigpuH8I+bzbI4pnjvQJ0XULOSbBzmxqaxC71xPn54d8E6IPqO5PuSwLxqkjwTvBNmbe7EB7ITeRnhzoKMzBW+iMjmeDbKJs6tBfCOa8Sa8tCe0P3ONtHYZwSxz7YifIF9S3BvwCud1g2gjaKizOIWs0MR+n/oK+yWgO0PZszmkARkB9OgjWwYEK+RpTizvG7IiRf34hmtGTx9ei8GlAensnxHGAWziA4he0RjF3KY2qJe1xR3jB7jilT2jYfuNCvVQv5x7erOC2V6UJUqmWych+SWuI7YzFHgPUue80x3VD35BTWjae06kD/omtwjlb1Y6oiPtVYucjGPqiqfNmL8+2cZ8se6qSBWLve4SxD6qugaeozdhI2ijVX9tUSYR4OHgGJid0vbO0dRMRvDAeMonPF3nvCbxzK/E0ZOncJ75flHB8a+nw14I6fAbqc7jkq0qb0CibzCyn9zCS8I8ecQ1vxi+jmHL15O/VZpZlfe74RfOcOSMVYbTqtxd5MZ8y4/EFvWCLcqj850gxJsDuJabw9wdwmvow7gJSQIFzh2WR+XBuDQ4KMjce6mMvrsNDBQ+dzgXlqMk9j6IpSUxhc1culcl7OyxsNX07rTvK9Sqvhy2kZ8hq56Zl+kILXixo6hTgwRVDK0qMaqghrgoZrnV+uoF+ItXZo8iIxGh7XUAboMdqyFTl41r9FLpgYnzCNL+8hQdfo/+NsQO6qB8LWnUhbxZ052/l04J+dEHPFpTBf1bUL6poZRYG7wJjGme1tYUYz2pxdL5VLcmSFpN8v2LrYG7h3yIq/ZrulMlvOCv06riPHtBYqT/Prjtcybu1dTxE9+2gDyWQ7C6pnWfvjFsRweEK9qXI9+J8uUp9zaNbPW/aiMO87pjP6B1IsQvJTw9paX3ywL84WTYBY7MSpbvB3eQATqGP1AvfEx7rDEkfUv7u126P6qseqty5XPW0PTLMt894nyH1qtDrGsX5xub09BO5cxUlfGQJOT/sTldFti+OQMs4nQRufM56xzZ84i05L0brNPWPoEdl7Dscj+fyWx3c+Z/2lJMIJoDiHvQvO5TPmavyKnFn8RaDuO8Xi03NMzy7dfXT01OAeIut8dHZJPUaf8PXuqT3zrYLzd4+qgnM3u3F9Sl6V/Tpt4fT65AwChx9ilGg1rAyx0Xn6Bg0yuwWU5w5i5N1s3JS1vymdUtLeRUYfvumH0mzWmy8urq1O/31bXbwfK+5+32z9XRv6/5d82XukvhK3BC3yE93xH2qzCfkV2ftv/Ur61+sf7l5vnH5p+bf2nU9TVD87mo/Db/+R9yz1D</latexit> Violation of Faithfulness Faithfulness: ⊥ G Y | Z ⇐ ⊥ P Y | Z <latexit sha1_base64="hcvMBKxaM6ynzjkNMhCc6VBJEg=">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</latexit> X ⊥ = X ⊥ A B C D Brady Neal Assumptions 6 / 45

  5. <latexit sha1_base64="TZPsAeWD3LFNlhH1rFZ4zmsmB1o=">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</latexit> Violation of Faithfulness Faithfulness: ⊥ G Y | Z ⇐ ⊥ P Y | Z <latexit sha1_base64="hcvMBKxaM6ynzjkNMhCc6VBJEg=">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</latexit> X ⊥ = X ⊥ A B C D Brady Neal Assumptions 6 / 45

  6. <latexit sha1_base64="TZPsAeWD3LFNlhH1rFZ4zmsmB1o=">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</latexit> Violation of Faithfulness Faithfulness: ⊥ G Y | Z ⇐ ⊥ P Y | Z <latexit sha1_base64="hcvMBKxaM6ynzjkNMhCc6VBJEg=">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</latexit> X ⊥ = X ⊥ <latexit sha1_base64="28pEVlnQhwmLIAdGi6E7jp1a97o=">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</latexit> A ⊥ ⊥ D A B C D Brady Neal Assumptions 6 / 45

  7. <latexit sha1_base64="TZPsAeWD3LFNlhH1rFZ4zmsmB1o=">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</latexit> Violation of Faithfulness Faithfulness: ⊥ G Y | Z ⇐ ⊥ P Y | Z <latexit sha1_base64="hcvMBKxaM6ynzjkNMhCc6VBJEg=">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</latexit> X ⊥ = X ⊥ <latexit sha1_base64="28pEVlnQhwmLIAdGi6E7jp1a97o=">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</latexit> A ⊥ ⊥ D A but A and D aren’t d-separated B C D Brady Neal Assumptions 6 / 45

  8. <latexit 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Violation of Faithfulness Faithfulness: ⊥ G Y | Z ⇐ ⊥ P Y | Z <latexit sha1_base64="hcvMBKxaM6ynzjkNMhCc6VBJEg=">AUs3icrVhLb9tGEN6kr8R9OalvbBxCvQgK5KbwLkYDWAnLYoGcAC/kigwJKyCPNVkrLqCPo5vfaP9Noe+m/6zbdLUdSLcmAJIlezM9/Mzmu57MS+l2aNxn+3bn/08Sefnbn7trnX3z51dfr9+4fp1E/sd0jO/Kj5LTl3fC92jzMt89zRO3HbQ8d2TzsWezJ9cuknqReFhdhW74L2eh1PbudgXS2/tOp1fJCx41dXMLs7GfrtdUKPMd6A3oAC1ync2VNMbXiMdfZ+maj3uDHmh0zWBTmc9BdO/uX6qlHBUpW/VoFwVqgxjX7Viu9b1VQNFYP2Tg1BSzDyO+qkVqDbB9cLjaoF7geo5/bw01xH/BTCltQ4uPXwJS31veByMu6Tqu+i3JngX6RgSW2y8wr1jMANQM9UDtUou51xdroNvULHqDGt4ytV6WElMivjBLq25i7uP/xlWKNcrcLoYOZBKMLJB80HVFNGR4K49L7pMRJt8rkYidXL7F5mtcxH+F4Aq41xSkvFVku9MHEJqdmlrcLjM6aLEf/ACrV9y7C64zV4jLtehfAegnKh3mNURl6mczK7FnNlBnk0R5fwZ+QIwZ2CEjHzPfjPA0c5V21gio42I3nOtcTMoLpB/pXzeTbHiOcW7UkZNYv5lFPbCqrsDu308e9Q+wE8kPM9ahHfFGDPS6xE2Zv7sUauRP8G/CfTRvtKXqdlSnxrGFNkl014EaY8cZY2hPan7lF2rohaJb5bqnfqN9F3Gv0iuR1DbIRLdRZnFJXaGK/i74ifglwFap4NqfUqCOgHT1mywVohT6NJPoeYRUp68c3qEPcfXotJkqN2sU/A4Drki2qfuAcYO9Yi0hV5XVzvGjlFJq/jGY3ea1Wox/6R2dWec1ipyISvVMlm5S0N9disWCIgdk96zTbdUPfklKsbTVnVof1F15AcLdsnUkV8yrFymI09SpVxYvz1zlvLduskxpjKb4/B8cupbqGnrI2Y6NpbaL+9kyVRIyHzXtgckLXu2gbTM0MQ+Mp3zw6TqXPUlmfgdmznVovcm5chPjfy+W4gHX9I6nK5Y1ZWTbBaDieWS7vceTSP3okEd4MX0dcy3fjX/WxMyq2DeDF84gSsZmBmlVdIe5MX/lh2oTvWATWVX2p0RaKAmfTmKMH45HwJXeC+oJQRFKnw4nh9VxmAflJHxWJdzeR0WNnjsfA45WybzNIeuqGyKQ6p6uVbJy3l5o+nLZ1xvpdlNX25rFAumZue6QcpsU4r5DLGQSCiSw9rJCKuCdourb59Qr2hdxr+yYvEmPhSYVkwB6jVxWZqnxZaV/GLpgYn4jMmw/w4CX734hPQNf1YyGbXcubhdzVB/m0kB9c07OFZHBN/xaS7yskXe4N3pgmMq8qa0q4Diq5dD5V7QkudpP8+UVXW43XDrwoe7ZjOpNl9hvJCr07rIH1Fbqz7P7jlRys7KO+y+HfahJqdJbVzdGNoRQT7K6wvZb4X58tV6mue3Op5s8c4zOuO+Yx+AimeQvLTQzqxP3nEL04WQz4FDsxOlj8PbjEDdAx/QS98iWesETX9iOe7Jq4vSrveqhyzuqbfjmJuw3kHexX+9zXro8bm9PQVunMVNbxHBry38icLgtu35wBFiFdR+98ZJ1zsxZFqTftrsgUufwK4qEIs8no+3PL7zkfWbkogngOIcdhPIk2fM1fCKnFn8RqDqPcXi03OMexdXnx09Nbz7zDqfnd1Fj9EnfP30tDfzrkLydxtVIbk7fHB/Sl8Zf/r0e87Tax8cBY8+xWSU1rRJxEXn6JgymXlaTHnmLE7W9dJWeN25ZO5FVg8HZNP3LNbrF2tr7ZnH7Njs43q43n9Qbrx5vPntq3szdUd+qB+oH+GlHPUNlHsCvtvpT/a3+Uf9uPNl4u9HZcDTr7VtG5htV+mwE/wPzuQuZ</latexit> X ⊥ = X ⊥ <latexit sha1_base64="28pEVlnQhwmLIAdGi6E7jp1a97o=">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</latexit> A ⊥ ⊥ D A but A and D aren’t d-separated γ α B C β δ D Brady Neal Assumptions 6 / 45

  9. <latexit 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Violation of Faithfulness Faithfulness: ⊥ G Y | Z ⇐ ⊥ P Y | Z <latexit sha1_base64="hcvMBKxaM6ynzjkNMhCc6VBJEg=">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</latexit> X ⊥ = X ⊥ <latexit sha1_base64="GPZAItzBzV43Hq4LUOy6ep/upA=">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</latexit> B := α A <latexit sha1_base64="28pEVlnQhwmLIAdGi6E7jp1a97o=">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</latexit> A ⊥ ⊥ D A C := γ A but A and D aren’t d-separated γ α D := β B + δ C B C β δ D Brady Neal Assumptions 6 / 45

  10. <latexit 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Violation of Faithfulness Faithfulness: ⊥ G Y | Z ⇐ ⊥ P Y | Z <latexit sha1_base64="hcvMBKxaM6ynzjkNMhCc6VBJEg=">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</latexit> X ⊥ = X ⊥ <latexit sha1_base64="GPZAItzBzV43Hq4LUOy6ep/upA=">AUuHicrVhZb9tGEN6kV+JeTuq3vrBRWhSorEpuCgcBKSRExRFAziAj7R4FLkSiLEqyRl1RH0h/rQ39LX9Nd05tulSOogpcASRK5mZ76ZnWu57IWuEyfN5tsbN97/4MP7p1e+fjTz797PdO3fP4mAcWfLUCtwgetkzY+k6vjxNnMSVL8NIml7Ple9UYfnzy9lFDuBf5JchfK1Zw58p+9YZkKki92jJ8Y3j9pG13TDoWn8ZHS7OwZ9Oo6MD0vRz1S1J5MTOJ8Z3RtaVLw87Fbq3ZaOJjLA9aelAT+nMc3Ln9j+gKWwTCEmPhCSl8kdDYFaI6ftKtERThER7LaZEi2jkYF6Kmdgh2TFxSeIwiTqi64D+vdJUn/4zZgxpi7S49ItI0hBfax6bxn1Q1Z31GznedTqmwGYbr+je05geURMxJGqVXMq5uVyPvl7FqhNaw0Os1qGVhKCwH6zCmvt0d+l/Qivk6xVxShrZJBXRyCKaS1RFYR0R3ZXn2TdDRMIEn6QRW1md5nVPB/Qd0RYJo1jWMq2GuKZjosPzRK2Mo+LmK5H/ItWqOwrw+rP1+Ag7moVzHtClJF4Q6MicpnOfHat50o08myFLuZPwOETd0yUAJnvkP8c4ijmqkWYrMNEJAdYS4gMamjkXzCfZnNI8dyHPTGiZiDfHOgJdWVldqd2unTvATsi+SnNDaGHfVEneySwI2Rv6sU6uCP6N8E/CzZaC/QGKpPjWac1cXbVCTegGWeOpTyh/JlapKybEs3Q3xK/RLinsdXuG8rpNsAtVFsfQ5evYt6mvsF8ujKVPZtS6tDhwY4hsmVEtEyfQmJ939MqYtSPq1GndHfhtRAodWhn/0w9rAmjugYuic0tqGHpQ3qdQ1xqO2YFbSybx0p2WtBvKPa1d1xkWtLOejUg2dlW1oaoHesUcAbY7zVLd0PVk2OsbrZgVQ/2Z12Dc7RoH0tl8SnGykY2DiFVxGUvrl7nqrUcoE7qiCX7fkAcbUj1NT1GbYZa06u/jq6SgLEw8Ld0zmh6p21TRZmpkT3tKdc4lN1znsSz/xJmCZyqgv5efLEJ9q+XQ34I4/BbVc7gyVZSNaDSdz5TLOxhJ+EeNOIKL4SvQ6zlq/nPyM1sin09eP4SImdsopE2RbeRG6tXfiJq1AtqlFVFf3KkmRLh6Sk8f053CZd4RtPhE4QqfzudnlTE4IspMe6yPubQOMxscdD4bnF2deYpDVSywMFVXa6V83JV3ih6uaw9z/eirKXyzLlErnp6H4QA+tlhVyCOLCEl8vSkwqpAHuCoiubf9vAPh97VjnRaQtPK+Q9NBj1KoCXZXPK+1L0AUj7ROW+f0dPHiJ/jfDE9C2fsxk628mcldvZNPM/nJlp7NJL0t/ZtJvqmQlNgbnDmNZV5U1hRzHVdyqXyq2hMk7Sbp84uqtjquPfIi79m27kyG3m84K9Tu2EYPqG/Un5f3Ha7kVmUdj9F9e+hDETbJbVzem1oWQTHG6wvRr5n58tN6muV3OZ50EcVnXHdEY9gWRPIenpIc7tTw7ws5PFE+BE72Tpc+D+8gAFcOfqRc+p2esGT9QM93Lbo+K+x6m6LyOWus+2Ue94CQD2m/OsK+tj1uqE9D+4UzU1HU9KQ/mb6dJlxu/oMsA5pG72rkVXW2UtnkUVN6mlzSFzqBHZVgZjl8Wq8viuRlZvSgKcALJz2HUg58+Ym+FlObP+jUDVe4r1p+eQ7n26ujoseY9Qta56OySeow64aunp87SuwrO3wOqCs7d6b27C/qK+Iun3wFOr2PiyHjUKSaBtKLlEdedo0PIJPpMcaZMztZNwonZYW3aFucytP47V1P5J6t9i52K21Ft+LQ/ODhqtHxvNFw9qjx/qN3O3xJfinviW/HQoHlNlHpNfLfG3+Fe8Ff/tPdr7Y2+w5yjWmze0zBei8NmL/gcvtQa+</latexit> B := α A <latexit sha1_base64="28pEVlnQhwmLIAdGi6E7jp1a97o=">AUhHicrVjbtGEN2kbZq4Nyf1W1/YKAX6IKuSk8B5cZHCTlAUDeAvqSNA0MiKYsQbyUpq46gP+lr+9j/6d/0zNmlKOpGObAEkavZmTOzc1suO7HvpVmz+d+t2x9/MmdT+/e2/js8y+/Grz/oOTNBoktntsR36UvOm0U9f3Qvc48zLfRMnbjvo+O5p78v86eXbpJ6UXiUXcXu6B9EXpdz25nIJ1vbv5knXmh48YuLmFmHZxv1pqNJj/W/KBlBjVlPofR/Xv/qjPlqEjZaqAC5apQZRj7q1SfN+qlmqGLR3agRagpHeVeN1QZkB+BywdEGtY/rBf69NdQ/wUzpbQNLT5+CSQt9Z3hcTDukqrvot+a4l2mY0RsfEK947BDEDNVA/UKrmc325Dr5BxaozrOEZV+thJTEp4ge7tOYu7j7+Z1ihXK/A6WLkQCrByAbNB1VTREeCu/a8+KbHSLTJ52IkVq+ye5XVMh/h2wdWG+OUloqtlnp4hJSs0tbhcdnTJcj/okVavtWYXUna/AYd70K4T0Cpa/eY1RGXqVzOruWc2UGebxAl/Bn5AjBnYISMfM9+M8DRzlXbWCKjYjecG1xMyghkH+hfN5NseI5zbtSRk1i/nmU9sKquwO7fTx71D7ATyI8z1qEd8UYc9LrETZm/uxTq5E/wb8p9NG+0ZeoOVKfGsY02SXgRpjxJljaE9qfuUXauhFolvluq1+p30Xc6/SK5HUdshEt1FmcUldoYr+HviJ+CXAVqng2p9SpI6AdPWZLH7RCn0YSfT9gFSnrxzeoI9x9ei0mSp3axT9DjgOuSI6oO4hxg71iLSFXtdQu8aOcUmr+MZjd5rXajH/pHZ1Z5zVKnIhK9UyWblHTU31xKxYIiB2T3vNt1Q9+SUqxvPWNWh/UXkBwt2ydSRXzKsXKYjT1KlXHFi4vXuWgtO6yTOmMpvr8Ax6luoaesjZjo2ljqv72TZVEjIfNe2ByQte7aBvOzIxAD4ynfPDpOpc9SWb+AGabOXVG703Pr0J8YeTz3UA6/ojU1XInrKybILRaDKzWt7jyKV/9EgirvFi+jrmWr6d/KypmXWxbwYvnEOUjM0M0roDnNj8cqPVA29oIasKvtTIi2UhE8nMcaPJpyPgCu8fWoJQZEKH03mx5UxOABlbDzW5Vxeh4UNHjufQ84zk3maQ1dUNsMhVb1aq+TlorzR9NWyziTfy7KavlpWKJfMTc/0g5RYbyrkMsZBJIKpLD2qkIq4J2i6tvm3NewLudcOTF4kxsLTCsmAPUavKjJV+arSvoxdMDE+EZnfP8CDl+x/Yz4BXdePhWx2LW8Wclcf5NCfnhNzxaSwTX9W0i+r5B0uTd4E5rIvK6sKeE6rOTS+VS1J7jYTfLnF1tdV478KLs2Y7pTJbZbyQr9O64x5QX6s/z+87UsmtyjoesPt2IcSanZW1M7xjaEVERysb6U+V6cL9epr0Vy6+fNPuOwqDvmM/oJpHgKyU8P6dT+5BG/OFmM+BQ4NDtZ/jy4zQzQMfwZvfAVnrHG1PQYz3ctXF+Wdr1UeWcNTD9chp3B8i72K8OuK9dHzc2p6Ht0pmprOMFNOS/sTldFty+OQMsQ7qO3sXIOucubPIrCb9tNkDlz6BXVUgFnm8G91fBcj6zclEU8AxTnsJpCnz5jr4RU5s/yNQNV7iuWn5xj3Lq4+O3pqeA+YdT47u4seo0/4+ulpf+5dheTvDqpCcnf08MGMvjL+7On3gqfXATgKHn2KySitadOIy87RMWUy87SY8sxZnKwbpZOyxpu1LZ3Kq8Dg7Zl+5JrdYuN8s9afs2PzjZabSeNpqvn9SePzNv5u6qb9RD9T38tKueozIP4Vcbfesv9bf6Z+vOVn3r8dZTzXr7lpH5WpU+Wz/+D5z+SE=</latexit> A ⊥ ⊥ D A C := γ A but A and D aren’t d-separated γ α D := β B + δ C <latexit sha1_base64="IyMNvAxKZ/+ioFHKajhqN/Dpd94=">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</latexit> D = ( αβ + γδ ) A B C β δ D Brady Neal Assumptions 6 / 45

  11. <latexit 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Violation of Faithfulness Faithfulness: ⊥ G Y | Z ⇐ ⊥ P Y | Z <latexit sha1_base64="hcvMBKxaM6ynzjkNMhCc6VBJEg=">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</latexit> X ⊥ = X ⊥ <latexit 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B := α A <latexit 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A ⊥ ⊥ D A C := γ A but A and D aren’t d-separated γ α D := β B + δ C <latexit 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D = ( αβ + γδ ) A B C β δ Paths cancel if <latexit 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αβ = − γδ D Brady Neal Assumptions 6 / 45

  12. Causal Sufficiency and Acyclicity Brady Neal Assumptions 7 / 45

  13. Causal Sufficiency and Acyclicity Causal Sufficiency: there are no unobserved confounders of any of the variables in the graph. Brady Neal Assumptions 7 / 45

  14. Causal Sufficiency and Acyclicity Causal Sufficiency: there are no unobserved confounders of any of the variables in the graph. Acyclicity: still assuming there are no cycles in the graph. Brady Neal Assumptions 7 / 45

  15. Causal Sufficiency and Acyclicity Causal Sufficiency: there are no unobserved confounders of any of the variables in the graph. Acyclicity: still assuming there are no cycles in the graph. All assumptions: • Markov assumption • Faithfulness • Causal sufficiency • Acyclicity Brady Neal Assumptions 7 / 45

  16. Causal Sufficiency and Acyclicity Causal Sufficiency: there are no unobserved confounders of any of the variables in the graph. Acyclicity: still assuming there are no cycles in the graph. All assumptions: • Markov assumption • Faithfulness • Causal sufficiency • Acyclicity Brady Neal Assumptions 7 / 45

  17. Question: Why is the Markov assumption (plus causal sufficiency and acyclicity) not enough for learning causal graphs from data?

  18. Independence-Based Causal Discovery Assumptions Markov Equivalence and Main Theorem The PC Algorithm Can We Do Better? Semi-Parametric Causal Discovery No Identifiability Without Parametric Assumptions Linear Non-Gaussian Setting Nonlinear Additive Noise Setting Brady Neal Markov Equivalence and Main Theorem 9 / 45

  19. <latexit 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Chains and Forks Encode Same Independencies X 2 X 1 X 2 X 3 X 1 X 2 X 3 X 1 X 3 Brady Neal Markov Equivalence and Main Theorem 10 / 45

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Chains and Forks Encode Same Independencies X 2 X 1 X 2 X 3 X 1 X 2 X 3 X 1 X 3 Markov: ⊥ X 3 | X 2 <latexit 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X 1 ⊥ Brady Neal Markov Equivalence and Main Theorem 10 / 45

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Chains and Forks Encode Same Independencies X 2 X 1 X 2 X 3 X 1 X 2 X 3 X 1 X 3 Markov: ⊥ X 3 | X 2 <latexit 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X 1 ⊥ Minimality: and <latexit 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X 1 6 ? ? X 2 <latexit sha1_base64="v5lvLyKb57cohMlUhqh5fZzhM=">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</latexit> X 2 6 ? ? X 3 Brady Neal Markov Equivalence and Main Theorem 10 / 45

  22. <latexit 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<latexit 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<latexit 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Chains and Forks Encode Same Independencies X 2 X 1 X 2 X 3 X 1 X 2 X 3 X 1 X 3 Markov: ⊥ X 3 | X 2 <latexit 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X 1 ⊥ Minimality: and <latexit 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X 1 6 ? ? X 2 <latexit sha1_base64="v5lvLyKb57cohMlUhqh5fZzhM=">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</latexit> X 2 6 ? ? X 3 Faithfulness: <latexit sha1_base64="+cPdfwzFN3rkMtWA0GecChlBa9o=">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</latexit> X 1 6 ? ? X 3 Brady Neal Markov Equivalence and Main Theorem 10 / 45

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<latexit 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Chains and Forks Encode Same Independencies Markov equivalent (all in the same Markov equivalence class) X 2 X 1 X 2 X 3 X 1 X 2 X 3 X 1 X 3 Markov: ⊥ X 3 | X 2 <latexit 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X 1 ⊥ Minimality: and <latexit 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X 1 6 ? ? X 2 <latexit sha1_base64="v5lvLyKb57cohMlUhqh5fZzhM=">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</latexit> X 2 6 ? ? X 3 Faithfulness: <latexit sha1_base64="+cPdfwzFN3rkMtWA0GecChlBa9o=">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</latexit> X 1 6 ? ? X 3 Brady Neal Markov Equivalence and Main Theorem 10 / 45

  24. <latexit 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<latexit 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<latexit 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Immoralities are Special Markov equivalence class where and ⊥ X 3 | X 2 <latexit 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X 1 ⊥ <latexit 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X 1 6 ? ? X 3 X 1 X 2 X 3 X 1 X 2 X 3 X 2 X 1 X 3 Brady Neal Markov Equivalence and Main Theorem 11 / 45

  25. <latexit 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<latexit 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<latexit 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<latexit 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Immoralities are Special Markov equivalence class where and ⊥ X 3 | X 2 <latexit 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X 1 ⊥ <latexit 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X 1 6 ? ? X 3 X 1 X 2 X 3 X 1 X 2 X 3 X 1 X 3 X 2 X 2 X 1 X 3 Brady Neal Markov Equivalence and Main Theorem 11 / 45

  26. <latexit 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<latexit 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<latexit 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<latexit 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Immoralities are Special Markov equivalence class where and ⊥ X 3 | X 2 <latexit 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X 1 ⊥ <latexit 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X 1 6 ? ? 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<latexit 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Immoralities are Special Markov equivalence class where and ⊥ X 3 | X 2 <latexit 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X 1 ⊥ <latexit 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X 1 6 ? ? X 3 and ? 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<latexit 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Immoralities are Special Markov equivalence class where and ⊥ X 3 | X 2 <latexit 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X 1 ⊥ <latexit 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X 1 6 ? ? X 3 Markov equivalence class where and ? X 3 | X 2 <latexit sha1_base64="veYSvntNIRv+tzokfKT9VYCEtE=">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</latexit> X 1 6 ? <latexit sha1_base64="HzBDvUkE1Jqilm4iQatypo4JCIc=">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</latexit> X 1 ⊥ ⊥ X 3 X 1 X 2 X 3 X 1 X 2 X 3 X 1 X 3 X 2 X 2 X 1 X 3 Brady Neal Markov Equivalence and Main Theorem 11 / 45

  29. <latexit 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<latexit 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<latexit 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Skeletons ⊥ X 3 | X 2 <latexit 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X 1 ⊥ X 1 X 2 X 3 X 1 X 2 X 3 X 2 X 1 X 3 Brady Neal Markov Equivalence and Main Theorem 12 / 45

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Skeletons X 1 X 2 X 3 ⊥ X 3 | X 2 <latexit 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X 1 ⊥ X 1 X 2 X 3 X 1 X 2 X 3 X 2 X 1 X 3 Brady Neal Markov Equivalence and Main Theorem 12 / 45

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Skeletons X 1 X 2 X 3 ⊥ X 3 | X 2 <latexit 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X 1 ⊥ X 1 X 2 X 3 X 1 X 2 X 3 X 1 X 2 X 3 X 2 X 1 X 3 Brady Neal Markov Equivalence and Main Theorem 12 / 45

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Skeletons X 1 X 2 X 3 ⊥ X 3 | X 2 <latexit 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X 1 ⊥ ? X 3 | X 2 <latexit sha1_base64="veYSvntNIRv+tzokfKT9VYCEtE=">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</latexit> X 1 6 ? X 1 X 2 X 3 X 1 X 2 X 3 X 1 X 2 X 3 X 2 X 1 X 3 Brady Neal Markov Equivalence and Main Theorem 12 / 45

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Skeletons X 1 X 2 X 3 X 1 X 2 X 3 ⊥ X 3 | X 2 <latexit 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X 1 ⊥ ? X 3 | X 2 <latexit sha1_base64="veYSvntNIRv+tzokfKT9VYCEtE=">AVBnicrVjLbtGFJ2kr8R9Oal3bBxCnQhq5KTItkYCGAnKIoGcAC/2jgwJKyCfNVkorCNoW/Ziu7Tb/ka/oL/Rc8MRVIPUg4sQeTozr3n3rmv4bAf+16adTr/3rj53vsfPjRrdsrH3/y6Wefr965e5BGw8R29+3Ij5Kjfi91fS909zMv892jOHF7Qd93D/vn2zJ/+NpNUi8K97L2H0V9E5Db+DZvQyk1Xr6KRrHYdR5oWOG7u4hBloD6zjwHMw2DxZXe+0O/xYs4OuGawr89mN7tz+Tx0rR0XKVkMVKFeFKsPYVz2V4vtSdVHxaC9UiPQEow8zrtqrFYgOwSXC4eqOe4nuLfS0MN8V8wU0rb0OLjl0DSUl8bHgfjAan6LvqtEu8iHSNi42XuPcNZgBqps5AbZLOZeX6+MbNKw6wxoec7UeVhKTIn6wK2se4O7jf4YVyvUSnC5GDqQSjGzQfFA1RXQkuGvPi2/OGIke+VyMxOo6u+uslvkI3Ng9TBOanYaqlnJi4hNbu0VXh8xnQx4q9YobavDmswWYPHuOtVCO8eKOfqDUZV5Dqd5exazJUZ5PEcXcKfkSMEdwpKxMz34D8PHNVctYEpOnqM5CnXEjOD2gb5B87n2Rwjnhu0J2XULOabRz2xqazC7txOH/c+sRPIjzB3Rj3ixbscYmdMHtzL7bIneDfBf/ZtNGeordZmRLPFtYk2dUCboQZb4KlPaH9mVukrRuBZpnvhvqR+l3EvUWvSF63IBvRQp3FKXWFJvZb6CvilwBXoYpnc0qLOgLacZsOQet0KeRN+3WEXK+vEN6gh3n16LidKidvHPBcB1yQRHVL3BcYO9Yi0hV7XVo+MHeOKVvGNx+40q9Vi/knt6s4rVXkQlaqZbJyi5o6qFZsURA7C57zTbdUPfklKsbT1nVp/1F15AcrdonUkV8qrFymI1nlKrihfnr3PeWjZJy3GUnx/Co4tSg0MPWVtxkbTSqn+tk2VRIyHzXtgckLXu2i7mJoZgR4YT/ng03Uue5LM/ALMHnPqmN4rz9chPjXy+W4gHX9Ear3cASurKptgNJrM1Mt7HLn0jx5JxDVeTF/HXMtXk59VmlkW+3rwhlEydjMIC2L7jA35q98T62jF6wjq6r+lEgLJeHTSYzx/QnfeAK7zm1hKBIhY8m8+PGOyAMjYeG3Aur8PCBo+dzyHnsck8zaErKpvikKqu1yp5OS9vNL1e1pnke1VW0+tlhfKauemZfpAS6hBLmMcRCIoZeleg1TEPUHTtc0/LWFfyL12aPIiMRYeNkgG7DF6VZGpyueN9mXsgonxicj8/A4efM3+N+YT0FX9WMhmV/JmIXf5Tj4t5C+u6NlCMrifwvJNw2SLvcGb0ITmReNSVcu41cOp+a9gQXu0n+/KrcVrH16UPdsxncky+41khd4dt9gDWkv159l9Ryq521jHQ3bfPvtQs1OTe3sXxtaEcHhEutLme/F+XKZ+pont3zebDMO87pjPtO0y0XmJLBoX9Yc1sz+Vr/zrZSe/LzSlraET1aVJxlRnzuvDAI+RPoBnNOZ836L7P8VQ35toe4Imyi+uzyj67LKqc7IamQ5dxN4H8CDvkDnfSq+PG5vy1UTmlVXU8hYb8Nzbn2YJbniUzsy/OQ7qK3vnIOs+dmdPtCb9fHsGLn3mu2xALCpnPl59fOcj63czEc8cxcnvOpDLp9rl8IqcWfwOounNyOLzeoz7AFef1ZYa3h1mnc+9xEVX0+8U9Pa9szbEcnfTVSF5O7o3t0pfVX86fP2Kc/LQ3AUPrclFa08qIi07uMWUy83ya8pRbnOXblbO5xpu2LS3lVWDwtkwHc83+tHKyut6dft83OzjYbHe/a3dePFx/8ti8C7ylvlT31Dfw0yP1BJW5C7/a6jf1h3qr/lr7fe3Ptbdrf2vWmzeMzBeq8ln753/gPhah</latexit> X 1 6 ? X 1 X 2 X 3 X 1 X 2 X 3 X 1 X 2 X 3 X 2 X 1 X 3 Brady Neal Markov Equivalence and Main Theorem 12 / 45

  34. Markov Equivalence via Immoral Skeletons Brady Neal Markov Equivalence and Main Theorem 13 / 45

  35. Markov Equivalence via Immoral Skeletons Two important graph qualities that we can use to distinguish graphs: Brady Neal Markov Equivalence and Main Theorem 13 / 45

  36. Markov Equivalence via Immoral Skeletons Two important graph qualities that we can use to distinguish graphs: 1. Immoralities Brady Neal Markov Equivalence and Main Theorem 13 / 45

  37. Markov Equivalence via Immoral Skeletons Two important graph qualities that we can use to distinguish graphs: 1. Immoralities 2. Skeleton Brady Neal Markov Equivalence and Main Theorem 13 / 45

  38. Markov Equivalence via Immoral Skeletons Two important graph qualities that we can use to distinguish graphs: 1. Immoralities 2. Skeleton Theorem: Two graphs are Markov equivalent if and only if they have the same skeleton and same immoralities (Verma & Pearl, 1990; Frydenburg, 1990). Brady Neal Markov Equivalence and Main Theorem 13 / 45

  39. Markov Equivalence via Immoral Skeletons Two important graph qualities that we can use to distinguish graphs: 1. Immoralities 2. Skeleton Theorem: Two graphs are Markov equivalent if and only if they have the same skeleton and same immoralities (Verma & Pearl, 1990; Frydenburg, 1990). Essential graph (aka CPDAG): skeleton + immoralities Brady Neal Markov Equivalence and Main Theorem 13 / 45

  40. <latexit sha1_base64="5wIKTKq+kukbfiQxkoSFOlUFIW0=">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</latexit> Question: What graphs are Markov equivalent to the basic fork graph? X 2 X 1 X 3

  41. <latexit sha1_base64="vcMVYs41M8xH0M+3wQpL73YVA0M=">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</latexit> Question: What graphs are Markov equivalent to the basic immorality? X 1 X 3 X 2

  42. <latexit sha1_base64="9uhgdMlqMmTAPQwzT8Li+1oJ2U=">AV03icrVjJbtGB6lW+JuTgqfemFjB4gBWZXdFi4QGEgsJyiKpkgAO0tN6DIkUSIW0nKqkPoUvTaV+lb9CH6DL29/7/N0OR1ELKQUSIHP7z79vMsBu6Tpy02383r3z7nvf3D9xtqH38yafrN289i4NRZMkTK3CD6EXjKXr+PIkcRJXvgjaXpdVz7vDjs8/xCRrET+MfJZSjPbPvOz3HMhMCvbrZ+PmsK/uOnybO8HXoWMkokpM1Q/uGD8Gtoyn72c+vZ7GiZnIc+Pug23DTIy7WZ720i3HmxN7i1BPNw2TrvSDcbGQdAzHpwbhH64HL0zRXdlLwHN4TmRdJaTHE1JIqc/KNAcMU3Bnod2v2hPaCYDGCIJzorem507zOY6FXNH21rImfTtkitfrW+2W238jPnBrh5sCv17Ety8ac4E7YIhCVGwhNS+CKhsStMEdN1KnZFW4QEOxcpwSIaOZiXYiLWiHZEWJIwTIO6d6nt1MN9emdecagtkiKS/+IKA1xR+PYNO4Bqp4s3yjgLpORgjfreEnPrubpETQRA4LW0WYq9N16fJqrE7Ihm9hrUOWhICwH6ySzT16uvSekIV8vyRMSObqCIaWQRzCaogLCOip/I8+2aASJjAkzRirav0rtKa5wO6hsTLpHEMTVlXQzScfEhWUJXxnER0+UcfyULlX5VvHpTGxzEXVnBuMcEGYrXNCpzrpJZzK7lWInmPFkgi/ETYPiEHRMkQOY75D+HMq5ahFPlmEikn3YEiKDWprz95jPsjmkeO5AnxhRM5BvDuSEurJyvTM9Xp2wTsi+pTmBpDvmiSPhK8I2Rv5sUmsCN6G+PNgo7WDLyFyuR4Nskmzq4m8Q1oxpnyUp5Q/sw0UtqlBDP0tSN+gHxJcW/CK5zXTaINoKHK4hiyfB37A+or7BeP7gxlz2aQJmR40GOAbBkSLJenOLG8L8mKGPXjaq4pPV14LQSXJqSzf8Ye7CJIzqC7DGNbchaoN6XUvsaz0mJansGwfdaV6qgfzj2lWdcVYq0/moVENn5QEktcX2mKOAOtd9Jqlu6HqyTGsm8xo1YX+edfgHC3rx1R5fMqxspGNA1CV+bIXF9u5yJY91EkTsWTf9wnjAFQ9DY9Rm6GWtFaov46ukgDxsPD0dE6oemdp45mZlOCe9pRLeKrOeU3imV+Ip4mcOoP3ivNVHB9q+mw14I6fAlpN9wyVaNaJROZ6rpHYwk/KNGHFL4SvQ9jyxfRvFGZW5f12+PlzHDljE81pVe42cmOx5cdik3rBJmV2Z8caYZE2J2EN6aYm4RX8YdQopPEK7wdDo/qY3BEUEm2mM9zGV1mOvgoPZwDzTmacwVEUlMxhc1dVSOS8X5Y2CV9Pa03wv0yp4NS1DLpCbju4HMXi9qKFLEAem8ApZelxDFWBNUHCl8sV9POx1o50XkRaw+c1lB56jLIq0FX5uFa/BF0w0j5hmp/ewIMX6H8T7ICu6secNrmSN3O6yzfyaU4/vqJnc0rviv7NKV/XUEqsDc4UxjRPa2uKsZ7UYql8qlsTJK0m2f5FVsT9y5kdsW3cmQ683nBVqdTxAD2iu1J/n1x2u5N3aOh6h+3bRhyJItitq5+StcsjOFrBvhj5np8vV6mvRXSr50HcVjUHbMZtQPJdyHZ6SEurE8O+OcnixS7wLFeybL94A4yQMXwO+qFj2mPNYGkr2h/t0v3R6Vb1WufM4a6X5Z5LtHnPdpvTrCunZ1vqE+De2UzkxlGQ9JQvaf6Nlju3qM8AyTleRu5izyjp7iwyK0ntNgeEpU5glzUc8zxezK86vos5qy8lAU4A+TnsbXAunjFX45fnzPIvAnXfKZafnkN69ujuoqPHGvcIWeis0vqMeqEr3ZPnblvFZy/e1QVnLvp7Vsz8sr8Z0+/fZxeR4SR46hTAJqBStyXHaODkGT6N1ijDNnfrJulU7Kit+sbnEhrzN70D3I6lXi7VX65u7s1/f5gfP9lq737TaT/c27+/pL3PXxefitrhLftoX96kyn5BfrcZfjX8a/zb+2zjZSDd+2/hdoV5raJrPROm38cf/PqFV1A=</latexit> Question: What graphs is the following graph Markov equivalent to? A B C D

  43. <latexit sha1_base64="9uhgdMlqMmTAPQwzT8Li+1oJ2U=">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</latexit> Question: What graphs is the following graph Markov equivalent to? A B C D

  44. <latexit sha1_base64="TxnHGw9Y/fS7PZVHqd3DmS5QMSg=">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</latexit> <latexit sha1_base64="9uhgdMlqMmTAPQwzT8Li+1oJ2U=">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</latexit> Question: What graphs is the following graph Markov equivalent to? A A B B C D C D

  45. <latexit 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<latexit 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<latexit 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Question: What graphs is the following graph Markov equivalent to? A A A B B B C D C D C D

  46. <latexit 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<latexit 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<latexit 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<latexit 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Question: What graphs is the following graph Markov equivalent to? A A A A B B B B C D C D C D C D

  47. <latexit sha1_base64="5iw4e3tdeXSoIX/I0DdOufJlW/M=">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</latexit> Question: What graphs is the following graph Markov equivalent to? A B C D

  48. <latexit sha1_base64="3DQ7r19kWAZMzfs23R4DTL741SA=">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</latexit> Question: Give a few graphs that the following graph is Markov F equivalent to: A B G E C D H

  49. Independence-Based Causal Discovery Assumptions Markov Equivalence and Main Theorem The PC Algorithm Can We Do Better? Semi-Parametric Causal Discovery No Identifiability Without Parametric Assumptions Linear Non-Gaussian Setting Nonlinear Additive Noise Setting Brady Neal The PC Algorithm 19 / 45

  50. The PC Algorithm: Overview Brady Neal The PC Algorithm 20 / 45

  51. <latexit sha1_base64="IynaVgXE4s3SYQJW2SymTdHKk=">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</latexit> True graph The PC Algorithm: Overview A B C D E Brady Neal The PC Algorithm 20 / 45

  52. <latexit sha1_base64="IynaVgXE4s3SYQJW2SymTdHKk=">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</latexit> True graph The PC Algorithm: Overview A B Start with complete undirected graph C D E Brady Neal The PC Algorithm 20 / 45

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  56. <latexit sha1_base64="IynaVgXE4s3SYQJW2SymTdHKk=">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</latexit> <latexit sha1_base64="NmgZoBtMfPRIlhPMF2nLd8BXymQ=">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</latexit> True graph The PC Algorithm: Overview A B Start with complete undirected graph C Three steps: 1. Identify the skeleton D E 2. Identify immoralities and orient them A B C D E Brady Neal The PC Algorithm 20 / 45

  57. <latexit sha1_base64="IynaVgXE4s3SYQJW2SymTdHKk=">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</latexit> <latexit sha1_base64="IynaVgXE4s3SYQJW2SymTdHKk=">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</latexit> True graph The PC Algorithm: Overview A B Start with complete undirected graph C Three steps: 1. Identify the skeleton D E 2. Identify immoralities and orient them 3. Orient qualifying edges that are incident A B on colliders C D E Brady Neal The PC Algorithm 20 / 45

  58. Identifying the Skeleton Brady Neal The PC Algorithm 21 / 45

  59. Identifying the Skeleton Start with complete undirected graph and remove edges X – Y where for some (potentially ⊥ Y | Z <latexit sha1_base64="q0ihpu/VZobt5/GyIm9ZV3HzT4=">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</latexit> X ⊥ empty) conditioning set Z, starting with the empty conditioning set and increasing the size Brady Neal The PC Algorithm 21 / 45

  60. <latexit sha1_base64="IynaVgXE4s3SYQJW2SymTdHKk=">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</latexit> True graph Identifying the Skeleton A B Start with complete undirected graph and remove edges X – Y where for some (potentially ⊥ Y | Z <latexit sha1_base64="q0ihpu/VZobt5/GyIm9ZV3HzT4=">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</latexit> X ⊥ C empty) conditioning set Z, starting with the empty conditioning set and increasing the size D E Brady Neal The PC Algorithm 21 / 45

  61. <latexit sha1_base64="IynaVgXE4s3SYQJW2SymTdHKk=">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</latexit> <latexit 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True graph Identifying the Skeleton A B Start with complete undirected graph and remove edges X – Y where for some (potentially ⊥ Y | Z <latexit 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X ⊥ C empty) conditioning set Z, starting with the empty conditioning set and increasing the size D E A B C D E Brady Neal The PC Algorithm 21 / 45

  62. <latexit 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<latexit 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True graph Identifying the Skeleton A B Start with complete undirected graph and remove edges X – Y where for some (potentially ⊥ Y | Z <latexit 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X ⊥ C empty) conditioning set Z, starting with the empty conditioning set and increasing the size D E <latexit 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A ⊥ ⊥ B A B C D E Brady Neal The PC Algorithm 21 / 45

  63. <latexit 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True graph Identifying the Skeleton A B Start with complete undirected graph and remove edges X – Y where for some (potentially ⊥ Y | Z <latexit 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X ⊥ C empty) conditioning set Z, starting with the empty conditioning set and increasing the size D E ⊥ B | {} <latexit 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A ⊥ A B C D E Brady Neal The PC Algorithm 21 / 45

  64. <latexit 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<latexit 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True graph Identifying the Skeleton A B Start with complete undirected graph and remove edges X – Y where for some (potentially ⊥ Y | Z <latexit 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X ⊥ C empty) conditioning set Z, starting with the empty conditioning set and increasing the size D E ⊥ B | {} <latexit 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A ⊥ A B C D E Brady Neal The PC Algorithm 21 / 45

  65. <latexit 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<latexit 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True graph Identifying the Skeleton A B Start with complete undirected graph and remove edges X – Y where for some (potentially ⊥ Y | Z <latexit 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X ⊥ C empty) conditioning set Z, starting with the empty conditioning set and increasing the size D E ⊥ B | {} <latexit sha1_base64="P6rRl21ORtHYorYMuGk8jrmEFjo=">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</latexit> A ⊥ A B <latexit sha1_base64="wvLSO2j4pP+2woUwkrnLhYy4rHk=">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</latexit> ∀ other pairs ( X, Y ) , ⊥ Y | { C } X ⊥ C D E Brady Neal The PC Algorithm 21 / 45

  66. <latexit 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<latexit 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True graph Identifying the Skeleton A B Start with complete undirected graph and remove edges X – Y where for some (potentially ⊥ Y | Z <latexit 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X ⊥ C empty) conditioning set Z, starting with the empty conditioning set and increasing the size D E ⊥ B | {} <latexit 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A ⊥ A B <latexit 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∀ other pairs ( X, Y ) , ⊥ Y | { C } X ⊥ C D E Brady Neal The PC Algorithm 21 / 45

  67. <latexit sha1_base64="IynaVgXE4s3SYQJW2SymTdHKk=">AWnHicrVjpbtGEN64V+JeSQr4T4uCjR0gBmRVdlukQGAgseSgaJIiARwnqW0EFLmSCPEqubLrCHqsPkyfoK/RmW+XIqmDlINIkLicnW9mdq7lshv7XqparX+vrX308Sefnb9xvrnX3z51dc3b90+TqNR4siXTuRHyeunUrfC+VL5Slfvo4TaQdX7qDts8/+pcJqkXhUfqMpZngd0PvZ7n2IpIb29d+e0K/teOFbe8F3sOWqUyMm6ZT53rT8iV6bT+9OQbk9SZSt5Zt1rb1u2su61Gq1ta7zV3po8WML4aNs6sbvRubR82VOWtR/1rPYZYR4tx3QI05V+dDGL6SzHEz1JF5/oHLMwXLM4VSPxuSgQwYVXHo9ouiG01wNIk0dkZD2bnDirm2tlcp2LucNsYcCpDtxShtzc3W80WPtb8YNcMNoX5PI9u3fhPnApXRMIRIxEIKUKhaOwLW6T0PRG7oiViop2JMdESGnmYl2Ii1gk7Ii5JHDZRh/Tfp7sTQw3pnmWmQDukxadfQkhL3DU8Lo17oOor67cKvMt0jCGbyka9fIDIiqxICodbiMc3Vcl75BzaoVreFXrNajlcSgsB+c0p7dPXpXtEK+f+SOCWNXEIlNHKI5hNVU1hHQlftefbNAJGwSdpxFZX2V1lNc9H9B2SLJvGKSxlWy3x2MQlhGYJW5nHR0yXS/ybVqjtq5LVm67BQ9z1Kpj3iChD8Y5GZclVOovZtZxLGcmTBbqYX4EjJO6UKBEy3yP/ecRzlWHZLIOG5HsYy0xMqhpJP+O+SybY4rnDuxJETUL+eZBT2wqK7c7s9OnaxeyE8KPaW4APeyLBtkjITtB9mZebIA7obsL3Dmw0ZmhN1GZHM8GrYmzq0FyI5rxprK0J7Q/M4u0dWOiWea7I5Cv6S4N+AVzusGYSNYqLM4ha7QxH6f+gr7JaB/prJnM0oDOgLYMUC2DImW69OSWN+PtIoU9eMbqWO6+vBaDCkNaGf/XGAcYE0c0RF0X9DYhR5GW9TrmuK+sWNS0sq+8dCd5rVayD+uXd0Z7UyLkSlWiYr96GpJX42K+YIsN1FrzmG+qenGJ1kxmrurA/7xqco2X7GJXHpxwrF9k4AKosl724eJ2L1rKHOmkgluz7PnHsA9Uz9BS1GRtN64X6a5sqiRAPB9fA5ISud9Z2MTMzJnpgPOUTn65z3pN45i+SaSOnTuG94nyVxEODz3YD7vhjUKtx6isMjah0Xg6U43MJLwjx5xLW8GL6OsZYfpj+rMLOq7A8jL5yTyBmrjKRVpbvIjcUrPxKb1As2KavK/uRIMyXB0lM460p5xbJZd4htIRE4QofT+cntTHoEGViPNbDXFaHuQ0eOp8LzlOTeZpDV5Sa4eCqrtbKebkobzS9GutO872M1fRqLFPOkZue6QcpZL2uwSnEgRFBIUuPalAR9gRN1za/WcG+EHvtyORFYix8VYM0GP0qiJTlc9q7VPogonxCWP+fA8PnqP/TfAEdFU/5lh1JW/muMv38mOv7iZ3NkcEX/5sh3NUiJvcGb0hjzoramOt5LZfOp7o9QdJukj2/6Gpr4L9LXuQ92zWdyTL7DWeF3h30QMaK/Xn+X2HK3m3to5H6L5d9KEmt2K2n5waTlERytsL4U+Z6fL1epr0W41fOmjTgs6o7ZTN0uF5mTwLJ9WXNYc/tb9c63Xnjuyc4raWFH9GBRfpYZ47nzwkjInkB3kHM6a36j7vuMnuomWNtP9ES5S/+PS/vsqlL5ZDcyHbod48k36cdsoOd9OpyY3P+2imd0so6DklD9puY82zOzc+SyuyLiyRdRe9iyTrP3bnTz6wm/Xw7IC595ruskZhXzmJ51fFdLFm/m4lw5shPfh9CcvFUu5q8PGeWv4OoezOy/Lwe07VH/z6qLTW8HWSdj71EUlfT7xT081p7u0I5+8eVQXn7vjO7Rl9Zfmz5+0+zsj4sh59LlJAa1pRYnLTu4xMo8n6Y45eZn+WbpbK7lzdqWFvIqMPL2TQeTZn9af3tzc3f2fd/84HivuftLs/Vib/PhnkXeF18K+6Ie+Sn+IhVeZz8quz9t3awdqTtacb3290Np5sPNOsa9cM5htR+mwc/w9+SH9c</latexit> <latexit sha1_base64="manHgzpfs/psOrYrf/dCaA10eUE=">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</latexit> True graph Identifying the Immoralities A B C D E A B C D E Brady Neal The PC Algorithm 22 / 45

  68. <latexit sha1_base64="IynaVgXE4s3SYQJW2SymTdHKk=">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</latexit> <latexit sha1_base64="manHgzpfs/psOrYrf/dCaA10eUE=">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</latexit> True graph Identifying the Immoralities A B Now for any paths X – Z – Y in our working graph where the following are true: C D E A B C D E Brady Neal The PC Algorithm 22 / 45

  69. <latexit sha1_base64="IynaVgXE4s3SYQJW2SymTdHKk=">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</latexit> <latexit sha1_base64="manHgzpfs/psOrYrf/dCaA10eUE=">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</latexit> True graph Identifying the Immoralities A B Now for any paths X – Z – Y in our working graph where the following are true: C 1. We discovered that there is no edge between X and Y in our previous step. D E A B C D E Brady Neal The PC Algorithm 22 / 45

  70. <latexit sha1_base64="manHgzpfs/psOrYrf/dCaA10eUE=">AXW3icrVhZb9tGEN7YPRIlbZMUrR/6wsYOkACyKrspUiAwkFhyUBRNkQDOVdsIKHElEeJVcmXVEfRDi6LP/Rud+XYpkjpIOYgEicvZmW/uXS47kecmqtn8+8rG5iefvb51Wu16ze+PKrm7duv0rCUdyVL7uhF8ZvOnYiPTeQL5WrPkmiqXtdz5ujNs8fzrcxknbhgcq4tInvl2P3B7btdWRHp3a0OdmTfDSbKHb6P3K4axXJ6sntWs8znrvV76Mhkdn8a0O1Jomwlz6x7rfuWrax7zXrzvjXZae1MH61gfHLfOrE74bm0PNlTlnUQ9qzWGck8WS3TJpmO9MLxvEx7tczhTE/s9gcqkzlcLXM06NlMqEjFsqF4sjp50MR2WoA1yTRORiP5ucOS+Za6Vy7ZO4oN5czZF71Y5tcbZdMkueB452/OB86yo8e6CH2WzGopzpZFYLZhPiV4osHc3t5uNJj7W4mDPDLaF+TwPb137T5wKR4SiK0bCF1IEQtHYE7ZI6Hsi9kRTREQ7ExOixTRyMS/FVNRIdkRckjhsog7pv093J4Ya0D1jJpDukhaPfjFJWuKu4XFo3ANVX1m/leNdpWMCbLbxgq4dg+kTVYkBUavkUs715Tr09Su8VuTDz/DWJU8iUDgO3YLPbp6dK/IQ/6/IE5JI4ekYhp1ieYRVNYR0xXHXmOzQCZsMEnacRWl9ldZjXPh/QdEpZN4wSWsq2WeGryEkCzhK3M4yGnqxH/Ig+1fWVYvZkPLvKuvWDeY6IMxXsaFZHLdOarazWXMsjTJbqYX4EjIO6EKCEq36X4ucRrNUuYbIOG5nsw5cIFdQwyL9iPq3miPK5C3sSZM1CvbnQE5nOyuxO7fTo2gF2TPITmhtAD8eiTvZIYMeo3jSKdXDHdDfGXRc2dufoDXQm57NOPnF1Qk3pBl3hqUjoeOZWqStmxDNMt9d8Rv0S8p7HVHhuq6TbAgLdRUn0BWY3B/QusJx8emfqRzZlFKHDh92DFAtQ6Jl+jQS6/uBvEjQP5BndDVQ9QioNShneMzxtiHT5zREXSPaexAD0tbtNY1xENjx7SglWPjYnVa1Gqh/rh39co4r5XlAnSqZaryAJqa4oHxmDPAduej1jWroV6TE3g3nbOqA/uzVYNrtGgfS2X5KebKQTUOIFXE5Sgu93OZL/vokzpybHvE8cBpHqGnqA3I6Oplu/lumSEPno4uqbmtD9ztrGczMTovsmUh7x6T7nPYln/iRMGzV1iujl58sQj4x8uhvwij8BtVzuFTqrKBvTaDKbKZd3MZKIjx5xjVehFhH8OX72c/KzayL/XHwgVErlhlkNZFd1Abyz0/Ftu0FmxTVRXjyZlmSoynk4jGOzPOHcJl3iG0BEThDp/M5qeVOWgTZWoi1sNc2oeZDS5WPgecp6byNIfuKDXHwV1drpXrclndaHq5rDOr96KspfLMuUctema9SAB1psKOYU8sISfq9LjCqkQe4Kma5vfrmFfgL12ZOoiNha+rpD0scZor0LTlc8q7VNYBWMTE5b54wMieI71b4onoMvGMZNVl4pmJnfxQTHN5MeXjGwm6V8yvpnk+wpJib3BndFY5kVlTzHX80ouXU9Ve4Kk3SR9ftHdVsd/h6LIe7ZjVibL7DdcFXp3PMAaUF9rfV7cd7iT9yr7eITVt4N1KIZmp6R3Xn40tCyDozX8S1Dv2flynf5aJrd+3bSQh2WrYzpTtcuF5iSwal/WHNbC/la+89Vyz3peSXJ7YguLMrOMhM8d4NQvoEuoua01XzC62+z+ipbgrfqQnyj36f1rYZ9dF5ZPdyKzQedx9Qn5IO2QbO+nlcSNz/totnNKOo5IQ/qbmvNsxs3Pksrsi8uQLqN3ObKuc2fh9DOvST/fDohLn/kuKhCzlmOV57f5cj63UyIM0d28vsYyPlT7Xp4Wc2sfgdR9WZk9Xk9omuP/j10W2J426g6D3uJpFVNv1PQz2uthbcjXL/71BVcu5M7t+f0FfHnz9t9nJdHxJHx6HOTgrSm5RFXndwjyCjzfJrglJud5RuFs7nGm7ctydWVb/AOzAomzf5Ue3dze2/+fd/i4NV+Y+nRvPF/vbjfMu8Kr4TtwR9yhOD8Vj6sznFNfuxj+bVzZrm9e/Xdrc6u2dUOzblwxMl+Lwmfrm/8BbRmsIw=</latexit> <latexit sha1_base64="IynaVgXE4s3SYQJW2SymTdHKk=">AWnHicrVjpbtGEN64V+JeSQr4T4uCjR0gBmRVdlukQGAgseSgaJIiARwnqW0EFLmSCPEqubLrCHqsPkyfoK/RmW+XIqmDlINIkLicnW9mdq7lshv7XqparX+vrX308Sefnb9xvrnX3z51dc3b90+TqNR4siXTuRHyeunUrfC+VL5Slfvo4TaQdX7qDts8/+pcJqkXhUfqMpZngd0PvZ7n2IpIb29d+e0K/teOFbe8F3sOWqUyMm6ZT53rT8iV6bT+9OQbk9SZSt5Zt1rb1u2su61Gq1ta7zV3po8WML4aNs6sbvRubR82VOWtR/1rPYZYR4tx3QI05V+dDGL6SzHEz1JF5/oHLMwXLM4VSPxuSgQwYVXHo9ouiG01wNIk0dkZD2bnDirm2tlcp2LucNsYcCpDtxShtzc3W80WPtb8YNcMNoX5PI9u3fhPnApXRMIRIxEIKUKhaOwLW6T0PRG7oiViop2JMdESGnmYl2Ii1gk7Ii5JHDZRh/Tfp7sTQw3pnmWmQDukxadfQkhL3DU8Lo17oOor67cKvMt0jCGbyka9fIDIiqxICodbiMc3Vcl75BzaoVreFXrNajlcSgsB+c0p7dPXpXtEK+f+SOCWNXEIlNHKI5hNVU1hHQlftefbNAJGwSdpxFZX2V1lNc9H9B2SLJvGKSxlWy3x2MQlhGYJW5nHR0yXS/ybVqjtq5LVm67BQ9z1Kpj3iChD8Y5GZclVOovZtZxLGcmTBbqYX4EjJO6UKBEy3yP/ecRzlWHZLIOG5HsYy0xMqhpJP+O+SybY4rnDuxJETUL+eZBT2wqK7c7s9OnaxeyE8KPaW4APeyLBtkjITtB9mZebIA7obsL3Dmw0ZmhN1GZHM8GrYmzq0FyI5rxprK0J7Q/M4u0dWOiWea7I5Cv6S4N+AVzusGYSNYqLM4ha7QxH6f+gr7JaB/prJnM0oDOgLYMUC2DImW69OSWN+PtIoU9eMbqWO6+vBaDCkNaGf/XGAcYE0c0RF0X9DYhR5GW9TrmuK+sWNS0sq+8dCd5rVayD+uXd0Z7UyLkSlWiYr96GpJX42K+YIsN1FrzmG+qenGJ1kxmrurA/7xqco2X7GJXHpxwrF9k4AKosl724eJ2L1rKHOmkgluz7PnHsA9Uz9BS1GRtN64X6a5sqiRAPB9fA5ISud9Z2MTMzJnpgPOUTn65z3pN45i+SaSOnTuG94nyVxEODz3YD7vhjUKtx6isMjah0Xg6U43MJLwjx5xLW8GL6OsZYfpj+rMLOq7A8jL5yTyBmrjKRVpbvIjcUrPxKb1As2KavK/uRIMyXB0lM460p5xbJZd4htIRE4QofT+cntTHoEGViPNbDXFaHuQ0eOp8LzlOTeZpDV5Sa4eCqrtbKebkobzS9GutO872M1fRqLFPOkZue6QcpZL2uwSnEgRFBIUuPalAR9gRN1za/WcG+EHvtyORFYix8VYM0GP0qiJTlc9q7VPogonxCWP+fA8PnqP/TfAEdFU/5lh1JW/muMv38mOv7iZ3NkcEX/5sh3NUiJvcGb0hjzoramOt5LZfOp7o9QdJukj2/6Gpr4L9LXuQ92zWdyTL7DWeF3h30QMaK/Xn+X2HK3m3to5H6L5d9KEmt2K2n5waTlERytsL4U+Z6fL1epr0W41fOmjTgs6o7ZTN0uF5mTwLJ9WXNYc/tb9c63Xnjuyc4raWFH9GBRfpYZ47nzwkjInkB3kHM6a36j7vuMnuomWNtP9ES5S/+PS/vsqlL5ZDcyHbod48k36cdsoOd9OpyY3P+2imd0so6DklD9puY82zOzc+SyuyLiyRdRe9iyTrP3bnTz6wm/Xw7IC595ruskZhXzmJ51fFdLFm/m4lw5shPfh9CcvFUu5q8PGeWv4OoezOy/Lwe07VH/z6qLTW8HWSdj71EUlfT7xT081p7u0I5+8eVQXn7vjO7Rl9Zfmz5+0+zsj4sh59LlJAa1pRYnLTu4xMo8n6Y45eZn+WbpbK7lzdqWFvIqMPL2TQeTZn9af3tzc3f2fd/84HivuftLs/Vib/PhnkXeF18K+6Ie+Sn+IhVeZz8quz9t3awdqTtacb3290Np5sPNOsa9cM5htR+mwc/w9+SH9c</latexit> True graph Identifying the Immoralities A B Now for any paths X – Z – Y in our working graph where the following are true: C 1. We discovered that there is no edge between X and Y in our previous step. D E 2. Z was not in the conditioning set that makes X and Y conditionally independent. A B C D E Brady Neal The PC Algorithm 22 / 45

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We discovered that there is no edge between X and Y in our previous step. D E 2. Z was not in the conditioning set that makes X and Y conditionally independent. A B Then, we know X – Z – Y forms an immortality. ⊥ B | {} <latexit sha1_base64="P6rRl21ORtHYorYMuGk8jrmEFjo=">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</latexit> A ⊥ C D E Brady Neal The PC Algorithm 22 / 45

  73. <latexit sha1_base64="IynaVgXE4s3SYQJW2SymTdHKk=">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</latexit> <latexit sha1_base64="qfjNYC8ZtInOScZrliQHryStpSY=">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</latexit> True graph Identifying the Immoralities A B Now for any paths X – Z – Y in our working graph where the following are true: C 1. We discovered that there is no edge between X and Y in our previous step. D E 2. Z was not in the conditioning set that makes X and Y conditionally independent. A B Then, we know X – Z – Y forms an immortality. ⊥ B | {} <latexit sha1_base64="P6rRl21ORtHYorYMuGk8jrmEFjo=">AUjnicrVhZb9tGEN6kV+JeTuqn9oWNUqAPtCq5aZ0XoWntBEHRA7gq40CQyJXFiFeJSmrDiGgv6aPbf9O/01nvl2KpA5SCixB5Gp25pvZuZbLfug6cdJq/Xfr9jvf+B3fubn340cefLp97/5pHIwjS5YgRtE5/1eLF3HlyeJk7jyPIxkz+u78qw/OuD5sysZxU7gHyfXoXzt9S59Z+BYvYRIF9uf/2h0Hd+WoaSLnxg/GV3PsY1u2p1ebDdazRY+xuKgrQcNoT9Hwb27f4usEUgLDEWnpDCFwmNXdETMX1fibZoiZBor0VKtIhGDualmIotkh0TlySOHlFHdL2kf6801af/jBlD2iItLv0ikjTEV5rHpvEAVHVn/UaBd5WOFNhs4zXd+xrTI2oihkStk8s415fr09erWXVCa3iM1Tq0khAU9oNVWvOA7i79T2iFfL0mTkjm6QiGlEc4mqKwjorvyPtmiEj0wCdpxFZX2V1lNc8H9B0RVo/GMSxlWw3xTMfFh2YJW5nHRUxXI/5BK1T2VWENZmtwEHe1CuY9JspIvKFRGblKZzG7VnMlGnm6RBfzJ+DwiTsmSoDMd8h/DnGUc9UiTNbRQyQvsZYQGdTUyD9jPsvmkOK5C3tiRM1AvjnQE+rKyu3O7HTp3gd2RPIpzQ2h31hkj0S2BGyN/OiCe6I/k3wz4KN1hy9icrkeJq0Js4uk3ADmnFmWMoTyp+ZRcq6lGiG/u6KX6BfUtxNeIXz2iTZABaqLI6hy9ex71BfYb94dGUqezajmNDhwY4hsmVEtFyfQmJ939AqYtSPq1FTurvwWgUE9rZPxOMPayJIzqG7gmNbehaYN6XVPsazumJa3sGwfdaVGrgfzj2lWdcV4ry/moVENnZQeaWuKRXjFHgO0ues3S3VD15Birm85Z1Yf9edfgHC3bx1J5fMqxspGNQ0iVcdmLy9e5bC17qBMTsWTfXxJHB1IDTY9Rm6HWtFWovwNdJQHiYeHu6ZxQ9c7aJnMzKdE97SmX+FSd857EM78TZg851YX3ivNViE+1fLYbcMdPQa2WO0VlWUjGqWzmWp5ByMJ/6gR1zhfB1iLV8OfsZhZl1sW8Gz19A5IxNK6DZyY/nKj0WDekGDsqrsT40UyI8nYQ0fjfEi4zDuCFp8oXOHpbH5aG4NDoky1xwaYy+owt8FB57PB2dWZpzhURSVzHFzV1Vo5L5fljaJXy9qzfC/LKnq1LFOukJuO7gcxsM5r5BLEgSW8QpYe10gF2BMUXdn86xr2+dhrxzovIm3hWY2khx6jVhXoqnxRa1+CLhpn7DMb2/hwSv0vymegDb1Yy6bOTNXO76rXyay0829Gwu6W3o31zyTY2kxN7gzGgs87K2pjrqJZL5VPdniBpN8meX1S1mbj2yYu8Z9u6Mxl6v+GsULtjBz3AXKs/L+47XMnt2joeo/v20YciaLYraufkxtDyCI7XWF+MfM/Pl+vU1zK59fPmAHFY1h2zGfUEkj+FZKeHuLA/OcDPTxYpngIneifLngd3kQEqhs+pF76gZ6wpNH1Lz3dtuj4r7XrovI5a6z7ZRF3j5D3ab86xL62OW6oT0O7pTNTWcdT0pD9pvp0mXO7+gywCmkTvcuRVdbZC2eReU3qaXNIXOoEdl2DmOfxcrzq+C5HVm9KApwA8nPYTSAXz5jr4eU5s/qNQN17itWn5DuA7q6Oix5j1E1rno7J6jDrhq6eng4V3FZy/e1QVnLvpg/tz+sr486fS5xex8SR86hTAJpRSsirjpHh5BJ9NijDNnfrJulk7KCm/etriQV57G6+h+JPVusXWx3WjPv31bHJzuNdvfNVsvHzWePNZv5u6IL8QD8TX5aV8oco8Ir9a4k/xl/hH/LuzvfP9TmfnB8V6+5aW+UyUPjvP/weblf0J</latexit> A ⊥ C D E Brady Neal The PC Algorithm 22 / 45

  74. <latexit 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<latexit 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Orienting Qualifying Edges Incident on Colliders True graph A B C A A B B D E C C D E D E Brady Neal The PC Algorithm 23 / 45

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Orienting Qualifying Edges Incident on Colliders True graph Idea: use fact that we discovered all immoralities to orient more edges A B C A A B B D E C C D E D E Brady Neal The PC Algorithm 23 / 45

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<latexit 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Orienting Qualifying Edges Incident on Colliders True graph Idea: use fact that we discovered all immoralities to orient more edges A B Any edge part of a partially directed path of the <latexit 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Z − Y form , where there is no edge connecting X <latexit 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X → Z − Y C and Y can be oriented as <latexit 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Z → Y A A B B D E C C D E D E Brady Neal The PC Algorithm 23 / 45

  77. <latexit 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<latexit 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<latexit 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Orienting Qualifying Edges Incident on Colliders True graph Idea: use fact that we discovered all immoralities to orient more edges A B Any edge part of a partially directed path of the <latexit 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Z − Y form , where there is no edge connecting X <latexit 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X → Z − Y C and Y can be oriented as <latexit 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Z → Y A A B B D E C C D E D E Brady Neal The PC Algorithm 23 / 45

  78. Removing Assumptions Brady Neal The PC Algorithm 24 / 45

  79. Removing Assumptions No assumed causal sufficiency: FCI algorithm (Spirtes et al., 2001) Brady Neal The PC Algorithm 24 / 45

  80. Removing Assumptions No assumed causal sufficiency: FCI algorithm (Spirtes et al., 2001) No assumed acyclicity: CCD algorithm (Richardson, 1996) Brady Neal The PC Algorithm 24 / 45

  81. Removing Assumptions No assumed causal sufficiency: FCI algorithm (Spirtes et al., 2001) No assumed acyclicity: CCD algorithm (Richardson, 1996) Neither causal sufficiency nor acyclicity: SAT-based causal discovery (Hyttinen et al., 2013; 2014) Brady Neal The PC Algorithm 24 / 45

  82. Hardness of Conditional Independence Testing Brady Neal The PC Algorithm 25 / 45

  83. Hardness of Conditional Independence Testing Independence-based causal discovery algorithms rely on accurate conditional independence testing. Brady Neal The PC Algorithm 25 / 45

  84. Hardness of Conditional Independence Testing Independence-based causal discovery algorithms rely on accurate conditional independence testing. Conditional independence testing is simple if we have infinite data. Brady Neal The PC Algorithm 25 / 45

  85. Hardness of Conditional Independence Testing Independence-based causal discovery algorithms rely on accurate conditional independence testing. Conditional independence testing is simple if we have infinite data. However, it is a quite hard problem with finite data, and it can sometimes require a lot of data to get accurate test results (Shah & Peters, 2020). Brady Neal The PC Algorithm 25 / 45

  86. <latexit 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<latexit 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Questions: 1. What are the essential graphs of the following graphs? A A B X 2 X 1 X 3 C B X 1 X 3 X 2 C D D

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Questions: 1. What are the essential graphs of the following graphs? A A B X 2 X 1 X 3 C B X 1 X 3 X 2 C D D

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Questions: 1. What are the essential graphs of the following graphs? A A B X 2 X 1 X 3 C B X 1 X 3 X 2 C D D

  89. <latexit 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Questions: 1. What are the essential graphs of the following graphs? A A B X 2 X 1 X 3 C B X 1 X 3 X 2 C D D

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Questions: 1. What are the essential graphs of the following graphs? A A B X 2 X 1 X 3 C B X 1 X 3 X 2 C D D

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