1CS 188: Artificial IntelligenceFall 2011Lecture 13: Bayes’ Nets10/6/2011Dan Klein – UC BerkeleyProbabilistic Models Models describe how (a portion of) the world works Models are always simplifications May not account for every variable May not account for all interactions between variables “All models are wrong; but some are useful.”– George E. P. Box What do we do with probabilistic models? We (or our agents) need to reason about unknown variables, given evidence Example: explanation (diagnostic reasoning) Example: prediction (causal reasoning) Example: value of information2Model for GhostbustersT B G P(T,B,G)+t +b +g 0.16+t +b ¬g 0.16+t ¬b +g 0.24+t ¬b ¬g 0.04 ¬t +b +g 0.04¬t +b ¬g 0.24¬t ¬b +g 0.06¬t ¬b ¬g 0.06 Reminder: ghost is hidden, sensors are noisy T: Top sensor is redB: Bottom sensor is redG: Ghost is in the top Queries:P( +g) = ??P( +g | +t) = ??P( +g | +t, -b) = ?? Problem: jointdistribution toolarge / complexJoint DistributionIndependence Two variables are independent if: This says that their joint distribution factors into a product two simpler distributions Another form: We write: Independence is a simplifying modeling assumption Empirical joint distributions: at best “close” to independent What could we assume for {Weather, Traffic, Cavity, Toothache}?5Example: Independence N fair, independent coin flips:h 0.5t 0.5h 0.5t 0.5h 0.5t 0.56Example: Independence?T W Pwarm sun 0.4warm rain 0.1cold sun 0.2cold rain 0.3T W Pwarm sun 0.3warm rain 0.2cold sun 0.3cold rain 0.2T Pwarm 0.5cold 0.5W Psun 0.6rain 0.472Conditional Independence P(Toothache, Cavity, Catch) If I have a cavity, the probability that the probe catches in it doesn't depend on whether I have a toothache: P(+catch | +toothache, +cavity) = P(+catch | +cavity) The same independence holds if I don’t have a cavity: P(+catch | +toothache, ¬cavity) = P(+catch| ¬cavity) Catch is conditionally independent of Toothache given Cavity: P(Catch | Toothache, Cavity) = P(Catch | Cavity) Equivalent statements: P(Toothache | Catch , Cavity) = P(Toothache | Cavity) P(Toothache, Catch | Cavity) = P(Toothache | Cavity) P(Catch | Cavity) One can be derived from the other easily8Conditional Independence Unconditional (absolute) independence very rare (why?) Conditional independence is our most basic and robust form of knowledge about uncertain environments: What about this domain: Traffic Umbrella Raining What about fire, smoke, alarm?9The Chain Rule Trivial decomposition: With assumption of conditional independence: Bayes’ nets / graphical models help us express conditional independence assumptions10Ghostbusters Chain RuleT B G P(T,B,G)+t +b +g 0.16+t +b ¬g 0.16+t ¬b +g 0.24+t ¬b ¬g 0.04 ¬t +b +g 0.04¬t +b ¬g 0.24¬t ¬b +g 0.06¬t ¬b ¬g 0.06 Each sensor depends onlyon where the ghost is That means, the two sensors are conditionally independent, given the ghost position T: Top square is redB: Bottom square is redG: Ghost is in the top Givens:P( +g ) = 0.5P( +t | +g ) = 0.8P( +t | ¬g ) = 0.4P( +b | +g ) = 0.4P( +b | ¬g ) = 0.8P(T,B,G) = P(G) P(T|G) P(B|G)Bayes’ Nets: Big Picture Two problems with using full joint distribution tables as our probabilistic models: Unless there are only a few variables, the joint is WAY too big to represent explicitly Hard to learn (estimate) anything empirically about more than a few variables at a time Bayes’ nets: a technique for describing complex joint distributions (models) using simple, local distributions (conditional probabilities) More properly called graphical models We describe how variables locally interact Local interactions chain together to give global, indirect interactions For about 10 min, we’ll be vague about how these interactions are specified12Example Bayes’ Net: Insurance133Example Bayes’ Net: Car14Graphical Model Notation Nodes: variables (with domains) Can be assigned (observed) or unassigned (unobserved) Arcs: interactions Similar to CSP constraints Indicate “direct influence” between variables Formally: encode conditional independence (more later) For now: imagine that arrows mean direct causation (in general, they don’t!)15Example: Coin FlipsX1X2Xn N independent coin flips No interactions between variables: absolute independence16Example: Traffic Variables: R: It rains T: There is traffic Model 1: independence Model 2: rain causes traffic Why is an agent using model 2 better?RT17Example: Traffic II Let’s build a causal graphical model Variables T: Traffic R: It rains L: Low pressure D: Roof drips B: Ballgame C: Cavity18Example: Alarm Network Variables B: Burglary A: Alarm goes off M: Mary calls J: John calls E: Earthquake!194Bayes’ Net Semantics Let’s formalize the semantics of a Bayes’ net A set of nodes, one per variable X A directed, acyclic graph A conditional distribution for each node A collection of distributions over X, one for each combination of parents’ values CPT: conditional probability table Description of a noisy “causal” processA1XAnA Bayes net = Topology (graph) + Local Conditional Probabilities21Probabilities in BNs Bayes’ nets implicitly encode joint distributions As a product of local conditional distributions To see what probability a BN gives to a full assignment, multiply all the relevant conditionals together: Example: This lets us reconstruct any entry of the full joint Not every BN can represent every joint distribution The topology enforces certain conditional independencies22Example: Coin Flipsh 0.5t 0.5h 0.5t 0.5h 0.5t 0.5X1X2XnOnly distributions whose variables are absolutely independent can be represented by a Bayes’ net with no arcs.23Example: TrafficRT+r 1/4¬r3/4+r +t 3/4¬t1/4¬r+t 1/2¬t1/224Example: Alarm NetworkBurglaryEarthqkAlarmJohn callsMary callsB P(B)+b 0.001¬b0.999E P(E)+e 0.002¬e0.998B E A P(A|B,E)+b +e +a 0.95+b +e¬a0.05+b¬e+a 0.94+b¬e ¬a0.06¬b+e +a 0.29¬b+e¬a0.71¬b ¬e+a 0.001¬b ¬e ¬a0.999A J P(J|A)+a +j 0.9+a¬j0.1¬a+j 0.05¬a ¬j0.95A M P(M|A)+a +m 0.7+a¬m0.3¬a+m 0.01¬a ¬m0.99Example: Traffic Causal directionRTr 1/4¬r3/4r t 3/4¬t1/4¬rt 1/2¬t1/2r t 3/16r¬t1/16¬rt 6/16¬r ¬t6/16265Example: Reverse Traffic Reverse
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