CSE 473: Artificial Intelligence Spring 2014

Transcription

1 CSE 473: Artificial Intelligence Spring 2014 Hidden Markov Models Hanna Hajishirzi Many slides adapted from Dan Weld, Pieter Abbeel, Dan Klein, Stuart Russell, Andrew Moore & Luke Zettlemoyer 1

2 Outline Probabilistic sequence models (and inference) Probability and Uncertainty Preview Markov Chains Hidden Markov Models Exact Inference Particle Filters

3 Going Hunting 3

4 Hidden Markov Models Markov chains not so useful for most agents Eventually you don t know anything anymore Need observations to update your beliefs X 1 X 2 X 3 X 4 Hidden Markov models (HMMs) Underlying Markov chain over states S You observe outputs (effects) at each time step X 1 X 2 X 3 X 4 X N 5 E 2 E 3 E 4 E N 5

5 Example: Weather HMM An HMM is defined by: Initial distribution: Transitions: Emissions:

6 Ghostbusters HMM P(X 1 ) = uniform P(X X) = usually move clockwise, but sometimes move in a random direction or stay in place P(E X) = same sensor model as before: red means close, green means far away. 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 P(X 1 ) X 1 X 2 X 3 X 4 1/6 1/6 0 1/6 1/2 0 E 3 E P(X X=<1,2>) P(E X) P(red 3) P(orange 3) P(yellow 3) P(green 3) E

7 Hidden Markov Models X 1 X 2 X 3 X 4 X N 5 E 2 E 3 E 4 E N 5 Defines a joint probability distribution: P (X 1,,X 2,E 2,X 3,E 3 )=P (X 1 )P ( X 1 )P (X 2 X 1 )P (E 2 X 2 )P (X 3 X 2 )P (E 3 X 3 )! Ques)ons#to#be#resolved:#! Does#this#indeed#define#a#joint#distribu)on?#! Can#every#joint#distribu)on#be#factored#this#way,#or#are#we#making#some#assump)ons#about#the# joint#distribu)on#by#using#this#factoriza)on?# Y t=2

8 Chain#Rule#and#HMMs# X 1 X 2 X 3 E 2 E 3! From#the#chain#rule,#every#joint#distribu)on#over###########################################can#be#wriden#as:# X 1,,X 2,E 2,X 3,E 3 P (X 1,,X 2,E 2,X 3,E 3 )=P (X 1 )P ( X 1 )P (X 2 X 1, )P (E 2 X 1,,X 2 ) P (X 3 X 1,,X 2,E 2 )P (E 3 X 1,,X 2,E 2,X 3 )! Assuming#that# ## X 2? X 1, E 2? X 1, X 2, X 3? X 1,,E 2 X 2, E 3? X 1,,X 2,E 2 X 3 ##### ##### gives#us#the#expression#posited#on#the#previous#slide:## P (X 1,,X 2,E 2,X 3,E 3 )=P (X 1 )P ( X 1 )P (X 2 X 1 )P (E 2 X 2 )P (X 3 X 2 )P (E 3 X 3 ) 8

9 Chain#Rule#and#HMMs# X 1 X 2 X 3! From#the#chain#rule,#every#joint#distribu)on#over#########################################can#be#wriden#as:# X 1,,...,X T,E T P (X 1,,...,X T,E T )=P (X 1 )P ( X 1 ) TY P (X t X 1,,...,X t 1,E t 1 )P (E t X 1,,...,X t 1,E t 1,X t ) t=2! Assuming#that#for#all#t:##! State#independent#of#all#past#states#and#all#past#evidence#given#the#previous#state,#i.e.:## X t? X 1,,...,X t 2,E t 2,E t 1 X t 1 E 2 E 3! Evidence#is#independent#of#all#past#states#and#all#past#evidence#given#the#current#state,#i.e.:# ##### # E t? X 1,,...,X t 2,E t 2,X t 1,E t 1 X t ######gives#us#the#expression#posited#on#the#earlier#slide:## TY P (X 1,,...,X T,E T )=P(X 1 )P ( X 1 ) P (X t X t 1 )P (E t X t ) t=2 9

10 Implied Conditional Independencies X 1 X 2 X 3 E 2 E 3! Many#implied#condi)onal#independencies,#e.g.,#? X 2,E 2,X 3,E 3 X 1! To#prove#them#! Approach#1:#follow#similar#(algebraic)#approach#to#what#we#did#in#the# Markov#models#lecture#! Approach#2:#directly#from#the#graph#structure#(3#lectures#from#now)#! Intui)on:#If#path#between#U#and#V#goes#through#W,#then# U? V W [Some#fineprint#later]# 10

11 Conditional Independence HMMs have two important independence properties: Markov hidden process, future depends on past via the present Current observation independent of all else given current state X 1 X 2 X 3 X 4 E 2 E 3 E 4 Quiz: Are observations E1, E2 independent? [No, correlated by the hidden state]

12 Real HMM Examples Speech recognition HMMs: Observations are acoustic signals (continuous valued) States are specific positions in specific words (so, tens of thousands) X 1 X 2 X 3 X 4 E 3 E 4

13 Real HMM Examples Machine translation HMMs: Observations are words (tens of thousands) States are translation options X 1 X 2 X 3 X 4 E 3 E 4

14 Real HMM Examples Robot tracking: Observations are range readings (continuous) States are positions on a map (continuous) X 1 X 2 X 3 X 4 E 3 E 4

15 HMM Computations Given joint P(X 1:n,:n ) evidence :n =e 1:n X 1 X 2 X 3 X 4 E 3 E 4 X n E n Inference problems include: Filtering, find P(X t e 1:t ) for current t Smoothing, find P(X t e 1:n ) for past t

16 HMM Computations Given joint P(X 1:n,:n ) evidence :n =e 1:n X 1 X 2 X 3 X 4 E 3 E 4 Inference problems include: Filtering, find P(X t e 1:t ) for current t Smoothing, find P(X t e 1:n ) for past t Most probable explanation, find x* 1:n = argmaxx 1:n P(x 1:n e 1:n )

17 Filtering / Monitoring Filtering, or monitoring, is the task of tracking the distribution B(X)=P(X t e 1:t ) (the belief state) over time We start with B(X) in an initial setting, usually uniform As time passes, or we get observations, we update B(X) The Kalman filter was invented in the 60 s and first implemented as a method of trajectory estimation for the Apollo program

18 Example: Robot Localization Example from Michael Pfeiffer Prob 0 t=0 Sensor model: never more than 1 mistake Motion model: may not execute action with small prob. 1

19 Example: Robot Localization Prob 0 1 t=1

20 Example: Robot Localization Prob 0 1 t=2

21 Example: Robot Localization Prob 0 1 t=3

22 Example: Robot Localization Prob 0 1 t=4

23 Example: Robot Localization Prob 0 1 t=5