1CS 188: Artificial IntelligenceFall 2006Lecture 18: Decision Diagrams10/31/2006Dan Klein – UC BerkeleyAnnouncements Optional midterm On Tuesday 11/21 in class unless conflicts We will count the midterms and final as 1 / 1 / 2, and drop the lowest (or halve the final weight) Will also run grades as if this midterm did not happen You will get the better of the two grades Projects 3.2 up today, grades on previous projects ASAP Total of 3 checkpoints left (including 3.2)2Recap: Sampling Exact inference can be slow Basic idea: sampling Draw N samples from a sampling distribution S Compute an approximate posterior probability Show this converges to the true probability P Benefits: Easy to implement You can get an (approximate) answer fast Last time: Rejection sampling: reject samples disagreeing with evidence Likelihood weighting: use evidence to weight samplesLikelihood 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=true) Idea: fix evidence variables and sample the rest Problem: sample distribution not consistent! Solution: weight by probability of evidence given parentsBurglary AlarmBurglary Alarm3Likelihood SamplingCloudySprinklerRainWetGrassCloudySprinklerRainWetGrassLikelihood Weighting Sampling distribution if z sampled and e fixed evidence Now, samples have weights Together, weighted sampling distribution is consistentCloudyRainCSRW4Likelihood 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 BNsCloudyRainCSRWDecision Networks MEU: choose the action which maximizes the expected utility given the evidence Can directly operationalize this with decision diagrams 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, must be parents, act as observed evidence) Utilities (depend on action and chance nodes)WeatherReportUmbrellaU5Decision Networks Action selection: Instantiate all evidence Calculate posterior over parents of utility node Set action node each possible way Calculate expected utility for each action Choose maximizing actionWeatherReportUmbrellaUExample: Decision NetworksWeatherUmbrellaU0.3rain0.7sunP(W)W0rainleave100sunleavetaketakeA70rain20sunU(A,W)W6Example: Decision NetworksWeatherReportUmbrellaU0rainleave100sunleavetaketakeA70rain20sunU(A,W)W0.3rain0.7sunP(W)W0.8cloud0.2clearP(R|rain)R0.5cloudy0.5clearP(R|sun)RValue of Information Idea: compute value of acquiring each possible piece of evidence Can be done directly from decision network Example: buying oil drilling rights Two blocks A and B, exactly one has oil, worth k Prior probabilities 0.5 each, mutually exclusive Current price of each block is k/2 Probe gives accurate survey of A. Fair price? Solution: compute value of information= expected value of best action given the information minus expected value of best action without information Survey may say “oil in A” or “no oil in A,” prob 0.5 each= [0.5 * value of “buy A” given “oil in A”] +[0.5 * value of “buy B” given “no oil in A”] – 0= [0.5 * k/2] + [0.5 * k/2] - 0 = k/2OilLocDrillLocU7General Formula Current evidence E=e, possible utility inputs s Potential new evidence E’: suppose we knew E’ = e’ BUT E’ is a random variable whose value is currently unknown, so: Must compute expected gain over all possible values (VPI = value of perfect information)VPI Properties Nonnegative in expectation Nonadditive ---consider, e.g., obtaining Ejtwice Order-independent8VPI ExampleWeatherReportUmbrellaUVPI Scenarios Imagine actions 1 and 2, for which U1> U2 How much will information about Ejbe worth?Little – we’re sure action 1 is better.Little – info likely to change our action but not our utilityA lot – either could be much better9Reasoning over Time Often, we want to reason about a sequence of observations Speech recognition Robot localization User attention Need to introduce time into our models Basic approach: hidden Markov models (HMMs) More general: dynamic Bayes’ netsMarkov Models A Markov model is a chain-structured BN Each node is identically distributed (stationarity) Value of X at a given time is called the state As a BN: Parameters: called transition probabilities, specify how the state evolves over time (also, initial probs)X2X1X3X410Conditional Independence Basic conditional independence: Past and future independent of the present Each time step only depends on the previous This is called the (first order) Markov propertyX2X1X3X4Example Weather: States: X = {rain, sun} Transitions: Initial distribution: 1.0 sun What’s the probability distribution after one step?rain sun0.90.90.10.1This is a CPT, not a BN!11Mini-Forward Algorithm Question: probability of being in state x at time t? Slow answer: Enumerate all sequences of length t which end in s Add up their probabilities Better answer: cached incremental belief updateForward simulationExample From initial observation of sun From initial observation of rainP(X1)P(X2)P(X3)P(X∞)P(X1)P(X2)P(X3)P(X∞)12Stationary Distributions If we simulate the chain long enough: What happens? Uncertainty accumulates Eventually, we have no idea what the state is! Stationary distributions: For most chains, the distribution we end up in is independent of the initial distribution Called the stationary distribution of the chain Usually, can only predict a short time outWeb Link Analysis PageRank over a web graph Each web page is a state Initial distribution: uniform over pages Transitions: With prob. c, uniform jump to arandom page (solid lines) With prob. 1-c, follow a randomoutlink (dotted lines) Stationary distribution Will spend more time on highly reachable pages E.g. many ways to get to the Acrobat Reader download page! Somewhat robust to link spam Google 1.0 returned pages
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