Sampling-Based Inference 1 Inference by stochastic simulation - PowerPoint PPT Presentation

Sampling-Based Inference 1

Inference by stochastic simulation Basic idea: 1) Draw N samples from a sampling distribution S 0.5 2) Compute an approximate posterior probability ˆ P 3) Show this converges to the true probability P Coin Outline: – Sampling from an empty network – Rejection sampling: reject samples disagreeing with evidence – Likelihood weighting: use evidence to weight samples – Markov chain Monte Carlo (MCMC): sample from a stochastic process whose stationary distribution is the true posterior 2

Sampling from an empty network function Prior-Sample ( bn ) returns an event sampled from bn inputs : bn , a belief network specifying joint distribution P ( X 1 , . . . , X n ) x ← an event with n elements for i = 1 to n do x i ← a random sample from P ( X i | parents ( X i )) given the values of Parents ( X i ) in x return x 3

Example P(C) .50 Cloudy C P(S|C) C P(R|C) Rain Sprinkler T .10 T .80 F .50 F .20 Wet Grass S R P(W|S,R) T T .99 T F .90 F T .90 F F .01 4

Example P(C) .50 Cloudy C P(S|C) C P(R|C) Rain Sprinkler T .10 T .80 F .50 F .20 Wet Grass S R P(W|S,R) T T .99 T F .90 F T .90 F F .01 5

Example P(C) .50 Cloudy C P(S|C) C P(R|C) Rain Sprinkler T .10 T .80 F .50 F .20 Wet Grass S R P(W|S,R) T T .99 T F .90 F T .90 F F .01 6

Example P(C) .50 Cloudy C P(S|C) C P(R|C) Rain Sprinkler T .10 T .80 F .50 F .20 Wet Grass S R P(W|S,R) T T .99 T F .90 F T .90 F F .01 7

Example P(C) .50 Cloudy C P(S|C) C P(R|C) Rain Sprinkler T .10 T .80 F .50 F .20 Wet Grass S R P(W|S,R) T T .99 T F .90 F T .90 F F .01 8

Example P(C) .50 Cloudy C P(S|C) C P(R|C) Rain Sprinkler T .10 T .80 F .50 F .20 Wet Grass S R P(W|S,R) T T .99 T F .90 F T .90 F F .01 9

Example P(C) .50 Cloudy C P(S|C) C P(R|C) Rain Sprinkler T .10 T .80 F .50 F .20 Wet Grass S R P(W|S,R) T T .99 T F .90 F T .90 F F .01 10

Sampling from an empty network contd. Probability that PriorSample generates a particular event S PS ( x 1 . . . x n ) = Π n i = 1 P ( x i | parents ( X i )) = P ( x 1 . . . x n ) i.e., the true prior probability E.g., S PS ( t, f, t, t ) = 0 . 5 × 0 . 9 × 0 . 8 × 0 . 9 = 0 . 324 = P ( t, f, t, t ) Let N PS ( x 1 . . . x n ) be the number of samples generated for event x 1 , . . . , x n Then we have ˆ lim P ( x 1 , . . . , x n ) = N →∞ N PS ( x 1 , . . . , x n ) /N lim N →∞ = S PS ( x 1 , . . . , x n ) = P ( x 1 . . . x n ) That is, estimates derived from PriorSample are consistent Shorthand: ˆ P ( x 1 , . . . , x n ) ≈ P ( x 1 . . . x n ) 11

Rejection sampling ˆ P ( X | e ) estimated from samples agreeing with e function Rejection-Sampling ( X , e , bn , N ) returns an estimate of P ( X | e ) local variables : N , a vector of counts over X , initially zero for j = 1 to N do x ← Prior-Sample ( bn ) if x is consistent with e then N [ x ] ← N [ x ]+1 where x is the value of X in x return Normalize ( N [ X ]) E.g., estimate P ( Rain | Sprinkler = true ) using 100 samples 27 samples have Sprinkler = true Of these, 8 have Rain = true and 19 have Rain = false . ˆ P ( Rain | Sprinkler = true ) = Normalize ( � 8 , 19 � ) = � 0 . 296 , 0 . 704 � Similar to a basic real-world empirical estimation procedure 12

Analysis of rejection sampling ˆ P ( X | e ) = α N PS ( X, e ) (algorithm defn.) = N PS ( X, e ) /N PS ( e ) (normalized by N PS ( e ) ) ≈ P ( X, e ) /P ( e ) (property of PriorSample ) = P ( X | e ) (defn. of conditional probability) Hence rejection sampling returns consistent posterior estimates Problem: hopelessly expensive if P ( e ) is small P ( e ) drops off exponentially with number of evidence variables! 13

Likelihood weighting Idea: fix evidence variables, sample only nonevidence variables, and weight each sample by the likelihood it accords the evidence function Likelihood-Weighting ( X , e , bn , N ) returns an estimate of P ( X | e ) local variables : W , a vector of weighted counts over X , initially zero for j = 1 to N do x , w ← Weighted-Sample ( bn ) W [ x ] ← W [ x ] + w where x is the value of X in x return Normalize ( W [ X ] ) function Weighted-Sample ( bn , e ) returns an event and a weight x ← an event with n elements; w ← 1 for i = 1 to n do if X i has a value x i in e then w ← w × P ( X i = x i | parents ( X i )) else x i ← a random sample from P ( X i | parents ( X i )) return x , w 14

Likelihood weighting example P(C) .50 Cloudy C P(S|C) C P(R|C) Rain T .10 Sprinkler T .80 F .50 F .20 Wet Grass S R P(W|S,R) T T .99 T F .90 F T .90 F F .01 w = 1 . 0 15

Likelihood weighting example P(C) .50 Cloudy C P(S|C) C P(R|C) Rain T .10 Sprinkler T .80 F .50 F .20 Wet Grass S R P(W|S,R) T T .99 T F .90 F T .90 F F .01 w = 1 . 0 16

Likelihood weighting example P(C) .50 Cloudy C P(S|C) C P(R|C) Rain T .10 Sprinkler T .80 F .50 F .20 Wet Grass S R P(W|S,R) T T .99 T F .90 F T .90 F F .01 w = 1 . 0 17

Likelihood weighting example P(C) .50 Cloudy C P(S|C) C P(R|C) Rain T .10 Sprinkler T .80 F .50 F .20 Wet Grass S R P(W|S,R) T T .99 T F .90 F T .90 F F .01 w = 1 . 0 × 0 . 1 18

Likelihood weighting example P(C) .50 Cloudy C P(S|C) C P(R|C) Rain T .10 Sprinkler T .80 F .50 F .20 Wet Grass S R P(W|S,R) T T .99 T F .90 F T .90 F F .01 w = 1 . 0 × 0 . 1 19

Likelihood weighting example P(C) .50 Cloudy C P(S|C) C P(R|C) Rain T .10 Sprinkler T .80 F .50 F .20 Wet Grass S R P(W|S,R) T T .99 T F .90 F T .90 F F .01 w = 1 . 0 × 0 . 1 20

Likelihood weighting example P(C) .50 Cloudy C P(S|C) C P(R|C) Rain T .10 Sprinkler T .80 F .50 F .20 Wet Grass S R P(W|S,R) T T .99 T F .90 F T .90 F F .01 w = 1 . 0 × 0 . 1 × 0 . 99 = 0 . 099 21

Likelihood weighting analysis Sampling probability for WeightedSample is S WS ( z , e ) = Π l i = 1 P ( z i | parents ( Z i )) Note: pays attention to evidence in ancestors only Cloudy ⇒ somewhere “in between” prior and posterior distribution Rain Sprinkler Wet Weight for a given sample z , e is Grass w ( z , e ) = Π m i = 1 P ( e i | parents ( E i )) Weighted sampling probability is S WS ( z , e ) w ( z , e ) = Π l i = 1 P ( z i | parents ( Z i )) Π m i = 1 P ( e i | parents ( E i )) = P ( z , e ) (by standard global semantics of network) Hence likelihood weighting returns consistent estimates but performance still degrades with many evidence variables because a few samples have nearly all the total weight 22

Approximate inference using MCMC “State” of network = current assignment to all variables. Generate next state by sampling one variable given Markov blanket Sample each variable in turn, keeping evidence fixed function Gibbs-Sampling ( X , e , bn , N ) returns an estimate of P ( X | e ) local variables : N [ X ] , a vector of counts over X , initially zero Z , the nonevidence variables in bn x , the current state of the network, initially copied from e initialize x with random values for the variables in Y for j = 1 to N do for each Z i in Z do sample the value of Z i in x from P ( Z i | mb ( Z i )) given the values of MB ( Z i ) in x N [ x ] ← N [ x ] + 1 where x is the value of X in x return Normalize ( N [ X ] ) Can also choose a variable to sample at random each time 23

The Markov chain With Sprinkler = true, WetGrass = true , there are four states: Cloudy Cloudy Rain Rain Sprinkler Sprinkler Wet Wet Grass Grass Cloudy Cloudy Rain Rain Sprinkler Sprinkler Wet Wet Grass Grass Wander about for a while, average what you see 24

MCMC example contd. Estimate P ( Rain | Sprinkler = true, WetGrass = true ) Sample Cloudy or Rain given its Markov blanket, repeat. Count number of times Rain is true and false in the samples. E.g., visit 100 states 31 have Rain = true , 69 have Rain = false ˆ P ( Rain | Sprinkler = true, WetGrass = true ) = Normalize ( � 31 , 69 � ) = � 0 . 31 , 0 . 69 � Theorem: chain approaches stationary distribution: long-run fraction of time spent in each state is exactly proportional to its posterior probability 25

Markov blanket sampling Markov blanket of Cloudy is Cloudy Sprinkler and Rain Markov blanket of Rain is Rain Sprinkler Cloudy , Sprinkler , and WetGrass Wet Grass Probability given the Markov blanket is calculated as follows: i | parents ( X i )) Π Z j ∈ Children ( X i ) P ( z j | parents ( Z j )) P ( x ′ i | mb ( X i )) = P ( x ′ Easily implemented in message-passing parallel systems, brains Main computational problems: 1) Difficult to tell if convergence has been achieved 2) Can be wasteful if Markov blanket is large: P ( X i | mb ( X i )) won’t change much (law of large numbers) 26

MCMC analysis: Outline Transition probability q ( x → x ′ ) Occupancy probability π t ( x ) at time t Equilibrium condition on π t defines stationary distribution π ( x ) Note: stationary distribution depends on choice of q ( x → x ′ ) Pairwise detailed balance on states guarantees equilibrium Gibbs sampling transition probability: sample each variable given current values of all others ⇒ detailed balance with the true posterior For Bayesian networks, Gibbs sampling reduces to sampling conditioned on each variable’s Markov blanket 27