1CS 188: Artificial IntelligenceFall 2008Lecture 17: Bayes Nets IVLecture 17: Bayes Nets IV10/28/2008Dan Klein – UC Berkeley12Inference Inference: calculating some statistic from a joint probability distribution Examples:Posterior probability:R BLPosterior probability: Most likely explanation:TDT’33Mini-Examples: VE Mini-Alarm Network B: there is a burglary A: the alarm soundsC: neighbor callsABB Pb 0.1¬b0.9C: neighbor calls4CB A Pb a 0.8b¬a0.2¬ba 0.1¬b ¬a0.9A C Pa c 0.7a¬c0.3¬ac 0.5¬a ¬c0.54Basic Operation: Join First basic operation: join factors Combining two factors: Just like a database joinBuild a factor over the union of the variables involvedBuild a factor over the union of the variables involved Example: Computation for each entry: pointwise products55Basic Operation: Eliminate Second basic operation: marginalization Take a factor and sum out a variable Shrinks a factor to a smaller oneA projectionoperationA projectionoperation Example: Definition:66Example: P(A)B Pb0.1B AA, BStartJoin on B Sum out B7A B Pa b 0.08a¬b0.09¬ab 0.02¬a ¬b0.81B A Pb a 0.8b¬a0.2¬ba 0.1¬b ¬a0.9b0.1¬b0.9AA Pa 0.17¬a0.837Example: Multiple JoinsB A Pb a 0.8b¬a0.2B Pb 0.1¬b0.9BJoin on BB, AB A Pb a 0.08b¬a0.028¬ba 0.1¬b ¬a0.9A¬cA C Pa c 0.7a¬c0.3¬ac 0.5¬a ¬c0.5C¬ba 0.09¬b ¬a0.81A C Pa c 0.7a¬c0.3¬ac 0.5¬a ¬c0.58Example: Multiple JoinsJoin on AB, ACB A Pb a 0.08b¬a0.02B, A, CBACP9b¬a0.02¬ba 0.09¬b ¬a0.81A C Pa c 0.7a¬c0.3¬ac 0.5¬a ¬c0.5BACPb a c 0.056b a¬c0.024b¬ac 0.010b¬a ¬c0.010¬ba c 0.063¬ba¬c0.027¬b ¬ac 0.405¬b ¬a ¬c0.4059Query: P(C)Sum out BB, A, CB A C Pbac0.056A, CA C Pa c 0.119a¬c0.05110bac0.056b a¬c0.024b¬ac 0.010b¬a ¬c0.010¬ba c 0.063¬ba¬c0.027¬b ¬ac 0.405¬b ¬a ¬c0.405Sum out A¬ac 0.415¬a ¬c0.415CC Pc 0.534¬c0.46610Variable Elimination Why is inference by enumeration so slow? You join up the whole joint distribution before you sum out the hidden variables You do lots of computations irrelevant to the evidenceYou end up repeating a lot of workYou end up repeating a lot of work Idea: interleave joining and marginalizing! Marginalize variables as soon as possible Called “Variable Elimination” Still NP-hard, but usually much faster than inference by enumeration1111P(C) : Marginalizing EarlySum out BB, ACB A Pb a 0.08b¬a0.02ACA Pa 0.17¬a0.8312b¬a0.02¬ba 0.09¬b ¬a0.81A C Pa c 0.7a¬c0.3¬ac 0.5¬a ¬c0.5¬a0.83A C Pa c 0.7a¬c0.3¬ac 0.5¬a ¬c0.512Marginalizing EarlyACA Pa 0.17¬a0.83Join on AA, CA C Pa c 0.119a¬c0.05113¬a0.83A C Pa c 0.7a¬c0.3¬ac 0.5¬a ¬c0.5Sum out Aa¬c0.051¬ac 0.415¬a ¬c0.415CC Pc 0.534¬c0.46613General 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 Sum out H Join all remaining factors and normalize1414Example: P(B|a)B Pb0.1B a, BStart / SelectJoin on B Normalize15A B Pa b 0.08a¬b0.09B A Pb a 0.8b¬a0.2¬ba 0.1¬b ¬a0.9b0.1¬b0.9aA B Pa b 8/17a¬b9/1715Example: P(B|¬c)B A Pb a 0.8b¬a0.2B Pb 0.1¬b0.9BJoin on AB Pb 0.1¬b0.9A,¬cB16¬ba 0.1¬b ¬a0.9A¬cA C Pa c 0.7a¬c0.3¬ac 0.5¬a ¬c0.5A,¬cB A C Pb a¬c0.24b¬a ¬c0.10¬ba¬c0.03¬b ¬a ¬c0.4516Example: P(B|¬c)B Pb 0.1¬b0.9A, ¬cBSum out AB Pb 0.1¬cB17B A C Pb a¬c0.24b¬a ¬c0.10¬ba¬c0.03¬b ¬a ¬c0.45¬b0.9¬cB C Pb¬c0.34¬b ¬c0.4817Example: P(B|¬c)Join on BB Pb0.1BB, ¬cB C Pb¬c0.03418b0.1¬b0.9¬cB C Pb¬c0.34¬b ¬c0.48b¬c0.034¬b ¬c0.423NormalizeB C Pb¬c0.073¬b ¬c0.92718Variable Elimination What you need to know: Should be able to run it on small examples, understand the factor creation / reduction flow Better than enumeration: VE caches intermediate computations Saves time by marginalizing variables as soon as possible rather than at the endthan at the end Polynomial time for tree-structured graphs – sound familiar? We will see special cases of VE later You’ll have to implement the special cases Approximations Exact inference is slow, especially with a lot of hidden nodes Approximate methods give you a (close, wrong?) answer, faster1919Sampling 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 samples2020Prior SamplingCloudySprinklerRainCloudySprinklerRainSprinklerRainWetGrassSprinklerRainWetGrass2121Prior 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 consistent2222Example We’ll get a bunch of samples from the BN:c, ¬s, r, wc, s, r, w¬c, s, r, ¬wc, ¬s, r, wCloudySprinklerRainCSRc, ¬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)?WetGrassW2323Rejection Sampling Let’s say we want P(C) No point keeping all samples around Just tally counts of C outcomesLet’s say we want P(C| s)CloudySprinklerRainWetGrassCSRWLet’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 for conditional probabilities (i.e., correct in the limit)c, ¬s, r, wc, s, r, w¬c, s, r, ¬wc, ¬s, r, w¬c, s, ¬r, w2424Likelihood 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)BurglaryAlarm Idea: fix evidence variables and sample the rest Problem: sample distribution not consistent! Solution: weight by probability of evidence given parentsBurglary AlarmBurglaryAlarm2525Likelihood SamplingCloudySprinklerRainCloudySprinklerRainSprinklerRainWetGrassSprinklerRainWetGrass2626Likelihood Weighting Sampling distribution if z sampled and e fixed evidence Now, samples have weightsCloudyRainCSR Together, weighted sampling distribution is
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