Inference in Bayesian networks

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Peter Gibson
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1 Inference in Bayesian networks hapter hapter

2 L L L L omplexity of exact inference Singly connected networks (or polytrees): any two nodes are connected by at most one (undirected) path time and space cost of variable elimination are O(d k n) Multiply connected networks: can reduce 3SA to exact inference NP-hard equivalent to counting 3SA models #P-complete A B D 1. A v B v 2. v D v A B v v D AND hapter

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

4 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 P arents(x i ) in x return x hapter

5 Example P() loudy P(S ).10 P(R ) S R P(W S,R) hapter

6 Example P() loudy P(S ).10 P(R ) S R P(W S,R) hapter

7 Example P() loudy P(S ).10 P(R ) S R P(W S,R) hapter

8 Example P() loudy P(S ).10 P(R ) S R P(W S,R) hapter

9 Example P() loudy P(S ).10 P(R ) S R P(W S,R) hapter

10 Example P() loudy P(S ).10 P(R ) S R P(W S,R) hapter

11 Example P() loudy P(S ).10 P(R ) S R P(W S,R) hapter

12 Sampling from an empty network contd. Probability that PriorSample generates a particular event S P S (x 1... x n ) = Π n i = 1P (x i parents(x i )) = P (x 1... x n ) i.e., the true prior probability E.g., S P S (t, f, t, t) = = = P (t, f, t, t) Let N P S (x 1... x n ) be the number of samples generated for event x 1,..., x n hen we have lim N ˆP (x 1,..., x n ) = lim N P S(x 1,..., x n )/N N = S P S (x 1,..., x n ) = P (x 1... x n ) hat is, estimates derived from PriorSample are consistent Shorthand: ˆP (x1,..., x n ) P (x 1... x n ) hapter

13 Rejection sampling ˆP(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( = true) using 100 samples 27 samples have = true Of these, 8 have = true and 19 have = false. ˆP( = true) = Normalize( 8, 19 ) = 0.296, Similar to a basic real-world empirical estimation procedure hapter

14 Analysis of rejection sampling ˆP(X e) = αn P S (X,e) (algorithm defn.) = N P S (X,e)/N P S (e) (normalized by N P S (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! hapter

15 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 hapter

16 Likelihood weighting example P() loudy P(S ).10 P(R ) S R P(W S,R) w = 1.0 hapter

17 Likelihood weighting example P() loudy P(S ).10 P(R ) S R P(W S,R) w = 1.0 hapter

18 Likelihood weighting example P() loudy P(S ).10 P(R ) S R P(W S,R) w = 1.0 hapter

19 Likelihood weighting example P() loudy P(S ).10 P(R ) S R P(W S,R) w = hapter

20 Likelihood weighting example P() loudy P(S ).10 P(R ) S R P(W S,R) w = hapter

21 Likelihood weighting example P() loudy P(S ).10 P(R ) S R P(W S,R) w = hapter

22 Likelihood weighting example P() loudy P(S ).10 P(R ) S R P(W S,R) w = = hapter

23 Likelihood weighting analysis Sampling probability for WeightedSample is S W S (z,e) = Π l i = 1P (z i parents(z i )) Note: pays attention to evidence in ancestors only somewhere in between prior and posterior distribution loudy Weight for a given sample z,e is w(z,e) = Π m i = 1P (e i parents(e i )) Weighted sampling probability is S W S (z,e)w(z,e) = Π l i = 1P (z i parents(z i )) Π m i = 1P (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 hapter

24 Approximate inference using MM 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 MM-Ask(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 ]) an also choose a variable to sample at random each time hapter

25 he Markov chain With = true, W et = true, there are four states: loudy loudy loudy loudy Wander about for a while, average what you see hapter

26 MM example contd. Estimate P( = true, W et = true) Sample loudy or given its Markov blanket, repeat. ount number of times is true and false in the samples. E.g., visit 100 states 31 have = true, 69 have = false ˆP( = true, W et = true) = Normalize( 31, 69 ) = 0.31, 0.69 heorem: chain approaches stationary distribution: long-run fraction of time spent in each state is exactly proportional to its posterior probability hapter

27 Markov blanket sampling Markov blanket of loudy is and Markov blanket of is loudy,, and W et loudy Probability given the Markov blanket is calculated as follows: P (x i mb(x i )) = P (x i parents(x i ))Π Zj hildren(x i )P (z j parents(z j )) Easily implemented in message-passing parallel systems, brains Main computational problems: 1) Difficult to tell if convergence has been achieved 2) an be wasteful if Markov blanket is large: P (X i mb(x i )) won t change much (law of large numbers) hapter

28 Summary Exact inference by variable elimination: polytime on polytrees, NP-hard on general graphs space = time, very sensitive to topology Approximate inference by LW, MM: LW does poorly when there is lots of (downstream) evidence LW, MM generally insensitive to topology onvergence can be very slow with probabilities close to 1 or 0 an handle arbitrary combinations of discrete and continuous variables hapter

Introduction to Markov Chain Monte Carlo

Introduction to Markov Chain Monte Carlo Monte Carlo: sample from a distribution to estimate the distribution to compute max, mean Markov Chain Monte Carlo: sampling using local information Generic problem