Lecture 6

# Lecture 6 - Administrivia Probabilistic graphical models...

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Probabilistic graphical models CPSC 532c (Topics in AI) Stat 521a (Topics in multivariate analysis) Lecture 6 Kevin Murphy Wednesday 29 September, 2004 Administrivia Discussion section on Thursday, 3.30-4, in 304 (this week only). Types of probabilistic inference There are several kinds of queries we can make. Suppose the joint is P ( Y,E,W ) = P ( Y,W ) × P ( E | ) . Conditional probability queries (sum-product): P ( Y | E = e ) s w P ( ) × P ( e | ) Most probable explanation (MPE) queries (max-product, MAP): ( y,w ) * = arg max y max w P ( ) × P ( e | ) Maximum A Posteriori (MAP) queries (max-sum-product, marginal MAP) y * = arg max y s w P ( ) × P ( e | ) Inference in Hidden Markov Models (HMM) X3 E1 E2 E3 X1 X2 Conditional probability queries, e.g. estimate current state given past evidence (online Fltering) P ( X t | e 1: t ) = s x 1: t - 1 P ( x 1: t - 1 ,X t | e 1: t ) Most probable explanation (MPE) queries, e.g., most probable sequence of states (Viterbi decoding) x * 1: t = arg max x 1: t P ( x 1: t | e 1: t )

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Word-error rate vs bit error rate Note: Most probable sequence of states not necessarily equal to sequence of most probable states. e.g., X 1 X 2 P ( X 1 ) p 0 . 4 0 . 6 P P ( X 2 | X 1 ) p 0 . 1 0 . 9 0 . 5 0 . 5 P P ( X 1 ,X 2 ) p 0 . 04 0 . 36 0 . 3 0 . 3 P arg max x 1 P ( X 1 ) = 1 , arg max x 1 max x 2 P ( X 1 2 ) = (0 , 1) Viterbi decoding minimizes word error rate x * 1: t = arg max x 1: t P ( x 1: t | e 1: t ) To minimize bit error rate, use most marginally likely state P ( X t | y 1: t ) = s x 1: t - 1 P ( x 1: t - 1 t | e 1: t ) x * t = max x P ( X t = x | e 1: t ) MAP vs Marginal MAP E3 W1 W2 W3 Q1 Q2 Q3 E1 E2 Consider a Dynamic Bayes Net (DBN) for speech recognition, where W = word and Q = phoneme. Most likely sequence of states (Viterbi/ MAP, max-product): arg max q 1: t ,w 1: t P ( q 1: t ,w 1: t | e 1: t ) Most likely sequence of words (Marginal MAP, max-sum-product): arg max w 1: t s q 1: t P ( w 1: t ,q 1: t | e 1: t ) Max-product often used as computationally simpler approximation to max-sum-product (or can use A * decoding).
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