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Berkeley COMPSCI 188 - Lecture 16: Bayes Nets IV

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1Announcements Midterm Next TUESDAY, 10/25, 5-8pm Prep page is on the web (practice exams, etc) Topical review sessions: see prep page Overall review: in class Thursday If you have a conflict, we should already know about it! Written 3 Due this Friday but fixes not due until NEXT Friday P1, P2, W1 in glookupCS 188: Artificial IntelligenceFall 2011Lecture 16: Bayes Nets IV10/18/2011Dan Klein – UC Berkeley2Approximate Inference3Approximate Inference Simulation has a name: sampling Sampling is a hot topic in machine learning,and it’s really simple Basic idea: Draw N samples from a sampling distribution S Compute an approximate posterior probability Show this converges to the true probability P Why sample? Learning: get samples from a distribution you don’t know Inference: getting a sample is faster than computing the right answer (e.g. with variable elimination)4SAF3Prior SamplingCloudySprinklerRainWetGrassCloudySprinklerRainWetGrass5+c 0.5-c 0.5+c+s 0.1-s 0.9-c+s 0.5-s 0.5+c+r 0.8-r 0.2-c+r 0.2-r 0.8+s+r+w 0.99-w 0.01-r+w 0.90-w 0.10-s+r+w 0.90-w 0.10-r+w 0.01-w 0.99Samples:+c, -s, +r, +w-c, +s, -r, +w…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 consistent64Example First: 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 Example: we want to know P(W) We have counts <+w:4, -w:1> Normalize to get approximate 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| +w)? P(C| +r, +w)? P(C| -r, -w)? Fast: can use fewer samples if less time (what’s the drawback?)CloudySprinklerRainWetGrassCSRW7Rejection Sampling Let’s say we want P(C) No point keeping all samples around Just tally counts of C as we go 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 called rejection sampling It is also consistent for conditional probabilities (i.e., correct in the limit)+c, -s, +r, +w+c, +s, +r, +w-c, +s, +r, -w+c, -s, +r, +w-c, -s, -r, +wCloudySprinklerRainWetGrassCSRW85Sampling Example There are 2 cups.  The first contains 1 penny and 1 quarter The second contains 2 quarters Say I pick a cup uniformly at random, then pick a coin randomly from that cup. It's a quarter (yes!). What is the probability that the other coin in that cup is also a quarter?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 parentsBurglary AlarmBurglary Alarm10-b, -a-b, -a-b, -a-b, -a+b, +a-b +a-b, +a-b, +a-b, +a+b, +a6Likelihood Weighting11+c 0.5-c 0.5+c+s 0.1-s 0.9-c+s 0.5-s 0.5+c+r 0.8-r 0.2-c+r 0.2-r 0.8+s+r+w 0.99-w 0.01-r+w 0.90-w 0.10-s+r+w 0.90-w 0.10-r+w 0.01-w 0.99Samples:+c, +s, +r, +w…CloudySprinklerRainWetGrassCloudySprinklerRainWetGrassLikelihood Weighting Sampling distribution if z sampled and e fixed evidence Now, samples have weights Together, weighted sampling distribution is consistentCloudyRCSW127Likelihood Weighting Likelihood weighting is good We have taken evidence into account as we generate the sample E.g. here, W’s value will get picked based on the evidence values of S, R More of our samples will reflect the state of the world suggested by the evidence Likelihood weighting doesn’t solve all our problems Evidence influences the choice of downstream variables, but not upstream ones (C isn’t more likely to get a value matching the evidence) We would like to consider evidence when we sample every variable13CloudyRainCSRWMarkov Chain Monte Carlo* Idea: instead of sampling from scratch, create samples that are each like the last one. Procedure: resample one variable at a time, conditioned on all the rest, but keep evidence fixed. E.g., for P(B|+c): Properties: Now samples are not independent (in fact they’re nearly identical), but sample averages are still consistent estimators! What’s the point: both upstream and downstream variables condition on evidence.14+a +c+b+a +c-b-a +c-b8Decision Networks MEU: choose the action which maximizes the expected utility given the evidence Can directly operationalize this with decision networks Bayes nets with nodes for utility and actions Lets us calculate the expected utility for each action New node types: Chance nodes (just like BNs) Actions (rectangles, cannot have parents, act as observed evidence) Utility node (diamond, depends on action and chance nodes)WeatherForecastUmbrellaU15[DEMO: Ghostbusters]Decision Networks Action selection: Instantiate all evidence Set action node(s) each possible way Calculate posterior for all parents of utility node, given the evidence Calculate expected utility for each action Choose maximizing actionWeatherForecastUmbrellaU169Example: Decision NetworksWeatherUmbrellaUW P(W)sun 0.7rain 0.3A W U(A,W)leave sun 100leave rain 0take sun 20take rain 70Umbrella = leaveUmbrella = takeOptimal decision = leaveDecisions as Outcome Trees Almost exactly like expectimax / MDPs What’s changed?18U(t,s)Weather Weather{}U(t,r) U(l,s) U(l,r)10Evidence in Decision Networks Find P(W|F=bad) Select for evidence First we join P(W) and P(bad|W) Then we normalizeWeatherForecastW P(W)sun 0.7rain 0.3F P(F|rain)good 0.1bad 0.9F P(F|sun)good 0.8bad 0.2W P(W)sun 0.7rain 0.3W P(F=bad|W)sun 0.2rain 0.9W P(W,F=bad)sun 0.14rain 0.27W P(W | F=bad)sun 0.34rain 0.66UmbrellaUExample: Decision NetworksWeatherForecast=badUmbrellaUA W U(A,W)leave sun 100leave rain 0take sun 20take rain 70W P(W|F=bad)sun 0.34rain 0.66Umbrella = leaveUmbrella = takeOptimal decision = take2011Decisions as Outcome Trees21U(t,s)W | {b} W | {b}U(t,r) U(l,s) U(l,r){b}Value of Information Idea: compute value of acquiring evidence Can be done directly from decision network Example: buying oil drilling rights Two blocks A and B, exactly one has oil, worth k You can drill in one location Prior probabilities


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Berkeley COMPSCI 188 - Lecture 16: Bayes Nets IV

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