Bayesian Networks What is the likelihood of X given evidence E? i.e. P(X|E) = ?

Issues Representational Power –allows for unknown, uncertain information Inference –Question: What is Probability of X if E is true. –Processing: in general, exponential Acquisition or Learning –network: human input – probabilities: data+ learning

Bayesian Network Directed Acyclic Graph Nodes are RV’s Edges denote dependencies Root nodes = nodes without predecessors –prior probability table Non-root nodes –conditional probabilites for all predecessors

Pearl Alarm Example B -> A E ->A A->JC A->MC P(B) =.001 P(-B) =.999 P(E) =.002 P(-E) =.998 etc. P(A|B&E) =.95 P(A|B&-E) =.94 P(A|-B&E) =.29 P(A|-B&-E) =.001 P(JC|A) =.90 P(JC|-A) =.05 P(MC|A) =.70 P(MC|-A) =.01

Joint Probability yields all Event = fully specified values for RVs. Prob of event: P(x1,x2,..xn) = P(x1|Parents(X1)*..P(xn|Parents(Xn)) E.g. P(j&m&a&-b&-e) = P(j|a)*P(m|a)*P(a|-b^-e)*P(-b)*P(-e) =.9*.7*.001*.999*..998 = Do this for all events and then sum as needed. Yield exact probability

Summing out P(b) from full joint = sum over all events with b = true (marginalization) –Silly: must be.001. P(MC|JC) = linked by alarm, sum still too much but not so apparent Need to Answer: what RVs are independent of others depending on the evidence. We’re skipping: Market Blankets

Probability Calculation Cost For example, P( 5 boolean variables) requires 2^5 entries. In general 2^n. For Bayes net, only need tables for all conditional probabilities and priors. If max k inputs to a node, and n RVs, then need n*2^k table entries. Computation can be reduced, but difficult.

Approximate Inference Simple Sampling Use BayesNetwork as a generative model Eg. generate examples, via topological order. Generates examples with appropriate distribution. Now use examples to estimate probabilities.

Sampling -> probabilities Generate examples with proper probability density. Use the ordering of the nodes to construct events. Finally use counting to yield an estimate of the exact probability.

Estimate P(JC|MC) 1.Do a large number of times 1.Use prior tables to compute “root events” 2.Say b = f and e = g 3.Use conditional tables to compute internal nodes values (IN ORDER) 4.Say a = false 5.Say JC = false and MC = true 2.Count the number of appropriate events 3.Note many entries irrelevant: In this case only if MC = true is event considered. Markov Chain Monte Carlo will only construct appropriate events.

Confidence of Estimate Given n examples and k are heads. How many examples needed to be 99% certain that k/n is within.01 of the true p. From statistic: Mean = np, Variance = npq For confidence of.99, t = 3.25 (table) 3.25*sqrt(pq/N) N >6,400. Correct probabilities not needed: just correct ordering.

Applications Bayesian Networks extended to decision theory. –decision nodes have actions attached –value nodes indicate expect utility Pathfinder (heckerman): medical diagnosis –adds utility theory (decision theory) –some actions specific tests –60 disease, 130 features Research Arena