CS 188: Artificial Intelligence Spring 2007

CS 188: Artificial IntelligenceSpring 2007 Lecture 14: Bayes Nets III 3/1/2007 Srini Narayanan – ICSI and UC Berkeley

Announcements • Office Hours this week will be on Friday • (11-1). • Assignment 2 grading • Midterm 3/13 • Review 3/8 (next Thursday) • Midterm review materials up over the weekend. • Extended office hours next week • (Thursday 11-1, Friday 2:30-4:30)

Representing Knowledge

Inference • Inference: calculating some statistic from a joint probability distribution • Examples: • Posterior probability: • Most likely explanation: L R B D T T’

Inference in Graphical Models • Queries • Value of information • What evidence should I seek next • Sensitivity Analysis • What probability values are most critical • Explanation • e.g., Why do I need a new starter motor • Prediction • e.g., What would happen if my fuel pump stops working

Inference Techniques • Exact Inference • Inference by enumeration • Variable elimination • Approximate Inference/ Monte Carlo • Prior Sampling • Rejection Sampling • Likelihood weighting • Monte Carlo Markov Chain (MCMC)

Reminder: Alarm Network

Inference by Enumeration • Given unlimited time, inference in BNs is easy • Recipe: • State the marginal probabilities you need • Figure out ALL the atomic probabilities you need • Calculate and combine them • Example:

Example Where did we use the BN structure? We didn’t!

Example • In this simple method, we only need the BN to synthesize the joint entries

Nesting Sums • Atomic inference is extremely slow! • Slightly clever way to save work: • Move the sums as far right as possible • Example:

Evaluation Tree • View the nested sums as a computation tree: • Still repeated work: calculate P(m | a) P(j | a) twice, etc.

Variable Elimination: Idea • Lots of redundant work in the computation tree • We can save time if we carry out the summation right to left and cache all intermediate results into objects called factors • This is the basic idea behind variable elimination

Basic Objects • Track objects called factors • Initial factors are local CPTs • During elimination, create new factors • Anatomy of a factor: Factor argument variables Variables introduced Variables summed out

Basic Operations • First basic operation: join factors • Combining two factors: • Just like a database join • Build a factor over the union of the domains • Example:

Basic Operations • Second basic operation: marginalization • Take a factor and sum out a variable • Shrinks a factor to a smaller one • A projection operation • Example:

Example

General Variable Elimination • Query: • Start with initial factors: • Local CPTs (but instantiated by evidence) • While there are still hidden variables (not Q or evidence): • Pick a hidden variable H • Join all factors mentioning H • Project out H • Join all remaining factors and normalize

Variable Elimination • What you need to know: • VE caches intermediate computations • Polynomial time for tree-structured graphs! • Saves time by marginalizing variables as soon as possible rather than at the end • Approximations • Exact inference is slow, especially when you have a lot of hidden nodes • Approximate methods give you a (close) answer, faster

Sampling • Basic idea: • Draw N samples from a sampling distribution S • Compute an approximate posterior probability • Show this converges to the true probability P • Outline: • Sampling from an empty network • Rejection sampling: reject samples disagreeing with evidence • Likelihood weighting: use evidence to weight samples

Prior Sampling Cloudy Cloudy Sprinkler Sprinkler Rain Rain WetGrass WetGrass

Prior Sampling • This process generates samples with probability …i.e. the BN’s joint probability • Let the number of samples of an event be • Then • I.e., the sampling procedure is consistent

Cloudy C Sprinkler S Rain R WetGrass W Example • We’ll get a bunch of samples from the BN: c, s, r, w c, s, r, w c, s, r, w c, s, r, w c, s, r, w • If we want to know P(W) • We have counts <w:4, w:1> • Normalize to get P(W) = <w:0.8, w:0.2> • This will get closer to the true distribution with more samples • Can estimate anything else, too • What about P(C| r)? P(C| r, w)?

Cloudy C Sprinkler S Rain R WetGrass W Rejection Sampling • Let’s say we want P(C) • No point keeping all samples around • Just tally counts of C outcomes • Let’s say we want P(C| s) • Same thing: tally C outcomes, but ignore (reject) samples which don’t have S=s • This is rejection sampling • It is also consistent (correct in the limit) • c, s, r, w • c, s, r, w • c, s, r, w • c, s, r, w • c, s, r, w

Likelihood Weighting • Problem with rejection sampling: • If evidence is unlikely, you reject a lot of samples • You don’t exploit your evidence as you sample • Consider P(B|a) • Idea: fix evidence variables and sample the rest • Problem: sample distribution not consistent! • Solution: weight by probability of evidence given parents Burglary Alarm Burglary Alarm

Likelihood Sampling Cloudy Cloudy Sprinkler Sprinkler Rain Rain WetGrass WetGrass

Cloudy C S Rain R W = Likelihood Weighting • Sampling distribution if z sampled and e fixed evidence • Now, samples have weights • Together, weighted sampling distribution is consistent

Cloudy C S Rain R W Likelihood Weighting • Note that likelihood weighting doesn’t solve all our problems • Rare evidence is taken into account for downstream variables, but not upstream ones • A better solution is Markov-chain Monte Carlo (MCMC), more advanced • We’ll return to sampling for robot localization and tracking in dynamic BNs

Summary • Exact inference in graphical models is tractable only for polytrees. • Variable elimination caches intermediate results to make the exact inference process more efficient for a given for a given query. • Approximate inference techniques use a variety of sampling methods and can scale to large models. • NEXT: Adding dynamics and change to graphical models

Bayes Net for insurance

CS 188: Artificial Intelligence Spring 2007