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Temporal Inference 16-385 Computer Vision (Kris Kitani) Carnegie Mellon University Basic Inference Tasks Filtering Prediction P ( X t | e 1: t ) P ( X t + k | e 1: t ) Posterior probability over the current Posterior probability over a future

  1. Temporal Inference 16-385 Computer Vision (Kris Kitani) Carnegie Mellon University

  2. Basic Inference Tasks Filtering Prediction P ( X t | e 1: t ) P ( X t + k | e 1: t ) Posterior probability over the current Posterior probability over a future state, given all evidence up to state, given all evidence up to present present Smoothing Best Sequence arg max P ( X 1: t | e 1: t ) P ( X k | e 1: t ) X 1: t Posterior probability over a past state, Best state sequence given all evidence given all evidence up to present up to present

  3. Filtering P ( X t | e 1: t ) Posterior probability over the current state, given all evidence up to present Where am I now?

  4. Filtering Can be computed with recursion (Dynamic Programming) X P ( X t +1 | e 1: t +1 ) ∝ P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) observation model motion model prior X t posterior

  5. Filtering Can be computed with recursion (Dynamic Programming) X P ( X t +1 | e 1: t +1 ) ∝ P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) observation model motion model X t What is this?

  6. Filtering Can be computed with recursion (Dynamic Programming) X P ( X t +1 | e 1: t +1 ) ∝ P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) X t same type of ‘message’

  7. Filtering Can be computed with recursion (Dynamic Programming) X P ( X t +1 | e 1: t +1 ) ∝ P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) X t same type of ‘message’ called a belief distribution Bel ( x t ) sometimes people use this annoying notation instead: a belief is a reflection of the systems (robot, tracker) knowledge about the state X

  8. Filtering Can be computed with recursion (Dynamic Programming) X P ( X t +1 | e 1: t +1 ) ∝ P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) X t Where does this equation come from? (scary math to follow…)

  9. Filtering Can be computed with recursion (Dynamic Programming) X P ( X t +1 | e 1: t +1 ) ∝ P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) X t just splitting up the notation here P ( X t +1 | e 1: t +1 ) = P ( X t +1 | e t +1 , e 1: t )

  10. Filtering Can be computed with recursion (Dynamic Programming) X P ( X t +1 | e 1: t +1 ) ∝ P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) X t P ( X t +1 | e 1: t +1 ) = P ( X t +1 | e t +1 , e 1: t ) Apply Bayes' rule (with evidence)

  11. Filtering Can be computed with recursion (Dynamic Programming) X P ( X t +1 | e 1: t +1 ) ∝ P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) X t P ( X t +1 | e 1: t +1 ) = P ( X t +1 | e t +1 , e 1: t ) = P ( e t +1 | X t +1 , e 1: t ) P ( X t +1 | e 1: t ) Apply Markov assumption on P ( e t +1 | e 1: t ) observation model

  12. Filtering Can be computed with recursion (Dynamic Programming) X P ( X t +1 | e 1: t +1 ) ∝ P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) X t P ( X t +1 | e 1: t +1 ) = P ( X t +1 | e t +1 , e 1: t ) = P ( e t +1 | X t +1 , e 1: t ) P ( X t +1 | e 1: t ) P ( e t +1 | e 1: t ) Condition on the = α P ( e t +1 | X t +1 ) P ( X t +1 | e 1: t ) previous state X t X

  13. Filtering Can be computed with recursion (Dynamic Programming) X P ( X t +1 | e 1: t +1 ) ∝ P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) X t P ( X t +1 | e 1: t +1 ) = P ( X t +1 | e t +1 , e 1: t ) = P ( e t +1 | X t +1 , e 1: t ) P ( X t +1 | e 1: t ) P ( e t +1 | e 1: t ) = α P ( e t +1 | X t +1 ) P ( X t +1 | e 1: t ) X = α P ( e t +1 | X t +1 ) P ( X t +1 | X t , e 1: t ) P ( X t | e 1: t ) X t Apply Markov assumption on motion model X

  14. Filtering Can be computed with recursion (Dynamic Programming) X P ( X t +1 | e 1: t +1 ) ∝ P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) X t P ( X t +1 | e 1: t +1 ) = P ( X t +1 | e t +1 , e 1: t ) = P ( e t +1 | X t +1 , e 1: t ) P ( X t +1 | e 1: t ) P ( e t +1 | e 1: t ) = α P ( e t +1 | X t +1 ) P ( X t +1 | e 1: t ) X = α P ( e t +1 | X t +1 ) P ( X t +1 | X t , e 1: t ) P ( X t | e 1: t ) X t X = α P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) X t

  15. Hidden Markov Model example ‘In the trunk of a car of a sleepy driver’ model left right binary random variable (left lane or right lane) x 2 x 4 x 1 x 0 x 3 x = { x left , x right }

  16. From a hole in the car you can see the ground x 2 x 4 x 1 x 0 x 3 e 1 e 2 e 3 e 4 binary random variable (center lane is yellow or road is gray) e = { e gray , e yellow }

  17. P ( x t | x t − 1 ) x right x left x right x left 0.7 0.3 x left What needs P ( x 0 ) 0.5 0.5 to sum to one? 0.3 0.7 x right x 2 x 4 x 1 x 0 x 3 e 1 e 2 e 3 e 4 P ( e t | x t ) x right x left 0.9 0.2 e yellow This is filtering! 0.1 0.8 e gray What’s the probability of being in the left lane at t=4?

  18. x left x right x right x left x right P ( x 0 ) P ( x t | x t − 1 ) P ( e t | x t ) x left x left e yellow 0.5 0.5 0.7 0.3 0.9 0.2 x right e gray 0.3 0.7 0.1 0.8 X P ( X t +1 | e 1: t +1 ) ∝ P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) Filtering: X t p ( x 1 | e 1 = e yellow ) =? What is the belief distribution if I see yellow at t=1 X p ( x 1 ) = x 0 p ( x 1 | x 0 ) p ( x 0 ) Prediction step: p ( x 1 | e 1 ) = α p ( e 1 | x 1 ) p ( x 1 ) Update step:

  19. x left x right x right x left x right P ( x 0 ) P ( x t | x t − 1 ) P ( e t | x t ) x left x left e yellow 0.5 0.5 0.7 0.3 0.9 0.2 x right e gray 0.3 0.7 0.1 0.8 X P ( X t +1 | e 1: t +1 ) ∝ P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) Filtering: X t What is the belief distribution if I see yellow at t=1 p ( x 1 | e 1 = e yellow ) =? X p ( x 1 ) = x 0 p ( x 1 | x 0 ) p ( x 0 ) Prediction step: = [0 . 7 0 . 3](0 . 5) + [0 . 3 0 . 7](0 . 5)  0 . 7 �  0 . 5  0 . 5 � � 0 . 3 = = 0 . 3 0 . 7 0 . 5 0 . 5

  20. x left x right x right x left x right P ( x 0 ) P ( x t | x t − 1 ) P ( e t | x t ) x left x left e yellow 0.5 0.5 0.7 0.3 0.9 0.2 x right e gray 0.3 0.7 0.1 0.8 X P ( X t +1 | e 1: t +1 ) ∝ P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) Filtering: X t What is the belief distribution if I see yellow at t=1 p ( x 1 | e 1 = e yellow ) =? p ( x 1 | e 1 ) = α p ( e 1 | x 1 ) p ( x 1 ) Update step:

  21. x left x right x right x left x right P ( x 0 ) P ( x t | x t − 1 ) P ( e t | x t ) x left x left e yellow 0.5 0.5 0.7 0.3 0.9 0.2 x right e gray 0.3 0.7 0.1 0.8 X P ( X t +1 | e 1: t +1 ) ∝ P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) Filtering: X t What is the belief distribution if I see yellow at t=1 p ( x 1 | e 1 = e yellow ) =? p ( x 1 | e 1 ) = α p ( e 1 | x 1 ) p ( x 1 ) Update step: = α (0 . 9 0 . 2) . ∗ (0 . 5 0 . 5) observed yellow  0 . 9 �  0 . 5  0 . 45 � � 0 . 0 = α = 0 . 0 0 . 2 0 . 5 0 . 1  0 . 818 � more likely to be in which lane? ≈ 0 . 182

  22. x left x right x right x left x right P ( x 0 ) P ( x t | x t − 1 ) P ( e t | x t ) x left x left e yellow 0.5 0.5 0.7 0.3 0.9 0.2 x right e gray 0.3 0.7 0.1 0.8 X P ( X t +1 | e 1: t +1 ) ∝ P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) Filtering: X t What is the belief distribution if I see yellow at t=1 p ( x 1 | e 1 = e yellow ) =? Summary X p ( x 1 ) = x 0 p ( x 1 | x 0 ) p ( x 0 ) Prediction step: �  � 0 . 5 = 0 . 5  p ( x 1 | e 1 ) = α p ( e 1 | x 1 ) p ( x 1 ) Update step:  0 . 818 � ≈ 0 . 182

  23. x left x right x right x left x right P ( x 0 ) P ( x t | x t − 1 ) P ( e t | x t ) x left x left e yellow 0.5 0.5 0.7 0.3 0.9 0.2 x right e gray 0.3 0.7 0.1 0.8 X P ( X t +1 | e 1: t +1 ) ∝ P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) Filtering: X t p ( x 2 | e 1 , e 2 ) =? What if you see yellow again at t=2

  24. x left x right x right x left x right P ( x 0 ) P ( x t | x t − 1 ) P ( e t | x t ) x left x left e yellow 0.5 0.5 0.7 0.3 0.9 0.2 x right e gray 0.3 0.7 0.1 0.8 X P ( X t +1 | e 1: t +1 ) ∝ P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) Filtering: X t p ( x 2 | e 1 , e 2 ) =? What if you see yellow again at t=2 X p ( x 2 | e 1 ) = x 1 p ( x 2 | x 1 ) p ( x 1 | e 1 ) Prediction step:  �  �  � p ( x 1 | e 1 , e 2 ) = α p ( e 1 | x 1 ) p ( x 1 ) Update step:  �  �

  25. x left x right x right x left x right P ( x 0 ) P ( x t | x t − 1 ) P ( e t | x t ) x left x left e yellow 0.5 0.5 0.7 0.3 0.9 0.2 x right e gray 0.3 0.7 0.1 0.8 X P ( X t +1 | e 1: t +1 ) ∝ P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) Filtering: X t p ( x 2 | e 1 , e 2 ) =? What if you see yellow again at t=2 X p ( x 2 | e 1 ) = x 1 p ( x 2 | x 1 ) p ( x 1 | e 1 ) Prediction step:  0 . 7 �  0 . 818  0 . 627 � � 0 . 3 = = 0 . 3 0 . 7 0 . 182 0 . 373 Why does the probability of being in the left lane go down?

  26. x left x right x right x left x right P ( x 0 ) P ( x t | x t − 1 ) P ( e t | x t ) x left x left e yellow 0.5 0.5 0.7 0.3 0.9 0.2 x right e gray 0.3 0.7 0.1 0.8 X P ( X t +1 | e 1: t +1 ) ∝ P ( e t +1 | X t +1 ) P ( X t +1 | X t ) P ( X t | e 1: t ) Filtering: X t p ( x 2 | e 1 , e 2 ) =? What if you see yellow again at t=2 p ( x 2 | e 1 , e 2 ) = α p ( e 2 | x 2 ) p ( x 2 | e 1 ) Update step:  0 . 9 �  0 . 627 � 0 . 0 = α 0 . 0 0 . 2 0 . 373  0 . 883 � ≈ 0 . 117 Why does the probability of being in the left lane go up?

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