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SP11 cs188 lecture 18 -- HMMs I 6PP

SP11 cs188 lecture 18 -- HMMs I 6PP - Announcements CS 188...

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1 CS 188: Artificial Intelligence Spring 2011 Lecture 18: HMMs and Particle Filtering 4/4/2011 Pieter Abbeel --- UC Berkeley Many slides over this course adapted from Dan Klein, Stuart Russell, Andrew Moore Announcements §੿ W4 out, due next week Monday §੿ P4 out, due next week 2 Announcements §੿ Course contest §੿ Fun! (And extra credit.) §੿ Regular tournaments §੿ Instructions posted soon! 3 Outline §੿ Markov Models ( = a particular Bayes net) §੿ Hidden Markov Models (HMMs) §੿ Representation ( = another particular Bayes net) §੿ Inference §੿ Forward algorithm ( = variable elimination) §੿ Particle filtering ( = likelihood weighting with some tweaks) §੿ Why do we study them? §੿ Widespread use for reasoning over time or space §੿ Concept: Stationary distribution 5 Reasoning over Time §੿ Often, we want to reason about a sequence of observations §੿ Speech recognition §੿ Robot localization §੿ User attention §੿ Medical monitoring §੿ Need to introduce time into our models §੿ Basic approach: hidden Markov models (HMMs) §੿ More general: dynamic Bayes nets 6 Markov Models §੿ A Markov model is a chain-structured BN §੿ Each node is identically distributed (stationarity) §੿ Value of X at a given time is called the state §੿ As a BN: §੿ Parameters: called transition probabilities or dynamics, specify how the state evolves over time (also, initial probs) X 2 X 1 X 3 X 4
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2 Conditional Independence §੿ Basic conditional independence: §੿ Past and future independent of the present §੿ Each time step only depends on the previous §੿ This is called the (first order) Markov property §੿ Note that the chain is just a (growing) BN §੿ We can always use generic BN reasoning on it if we truncate the chain at a fixed length X 2 X 1 X 3 X 4 8 Example: Markov Chain §੿ Weather: §੿ States: X = {rain, sun} §੿ Transitions: §੿ Initial distribution: 1.0 sun §੿ What s the probability distribution after one step? rain sun 0.9 0.9 0.1 0.1 This are two new representations of a CPT, not BNs! 9 sun rain sun rain 0.1 0.9 0.9 0.1 Query: P(X t ) §੿ Question: probability of being in state x at time t? §੿ Slow answer: §੿ Enumerate all sequences of length t which end in s §੿ Add up their probabilities §੿ = join on X 1 through X t-1 followed by sum over X 1 through X t-1 10 Mini-Forward Algorithm §੿ Question: What s P(X) on some day t? §੿ An instance of variable elimination! (In order X 1 , X 2 , ) sun rain sun rain sun rain sun rain Forward simulation 11 Example §੿ From initial observation of sun §੿ From initial observation of rain P( X 1 ) P( X 2 ) P( X 3 ) P( X ) P( X 1 ) P( X 2 ) P( X 3 ) P( X ) 12 Stationary Distributions §੿ If we simulate the chain long enough: §੿ What happens?
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