Artificial Intelligence Bayesian Networks

Transcription

1 Artificial Intelligence Bayesian Networks Stephan Dreiseitl FH Hagenberg Software Engineering & Interactive Media Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 Overview Representation of uncertain knowledge Constructing Bayesian networks Using Bayesian networks for inference Algorithmic aspects of inference Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

2 A simple Bayesian network example Rain Worms Umbrellas P(Rain, Worms, Umbrellas) = P(Worms Rain)P(Umbrellas Rain)P(Rain) With conditional independence, need only right-hand side to represent joint distribution Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 A simple Bayesian net example (cont.) Rain Worms Umbrellas Intuitively: graphical representation of influence Mathematically: graphical representation of conditional independence assertions Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

3 A more complicated example Burglary Earthquake P(b) P(e) Alarm B E P(a B, E) T T 0.95 T F 0.94 F T 0.24 F F MaryCalls A P(m A) T 0.7 F 0.01 JohnCalls A P(j A) T 0.9 F 0.05 Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 Definition of Bayesian networks A Bayesian network is a directed acyclic graph with random variables as nodes, links that specify directly influences relationships, probability distributions P(X i parents(x i )) for each node X i Graph structure asserts conditional independencies: P(MaryCalls JohnCalls, Alarm, Earthquake, Burglary) = P(MaryCalls Alarm) Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

4 Bayesian networks as joint probabilities P(X 1,..., X n ) = = n P(X i X 1,..., X i 1 ) i=1 n P(X i parents(x i )) i=1 for parents(x i ) {X 1,..., X i 1 } Burglary example: P(b, e, a, m, j) = P(b)P( e)p(a b, e)p( m a)p(j a) = = Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 Conditional independencies in networks Use graphical structure to visualize conditional dependencies and independencies Nodes are dependent if there is information flow between them (along one possible path) Nodes are independent if information flow is blocked (along all possible paths) Distinguish situations with and without evidence (instantiated variables) Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

5 Conditional independencies in networks No evidence: Information flow along a path is blocked iff there is a head-to-head node (blocker) on path No blockers between A and B: A B A B Blocker C between A and B: A C B Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 Conditional independencies in networks Evidence blocks information flow, except at blockers (or their descendents), where it opens information flow Information flow between A and B blocked by evidence: A B A B Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

6 Conditional independencies in networks Information flow between A and B unblocked by evidence: A B A B Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 Conditional independencies in networks A node is conditionally independent of its non-descendents, given its parents P 1 P 2 A X B C C D 1 2 P(X P 1, P 2, A, B, D) = P(X P 1, P 2 ) Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

7 Cond. independencies in networks (cont.) A node is conditionally independent of all other nodes in the network, given its Markov blanket: its parents, children, and children s parents P 1 P 2 A X B C C D 1 2 P(X P 1, P 2, C 1, C 2, A, B, D) = P(X P 1, P 2, C 1, C 2, A, B) Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 Noisy OR For Boolean node X with n Boolean parents, conditional probability table has 2 n entries Noisy OR assumption reduces this number to n: Assume each parent may be inhibited independently Flu Malaria Cold Fever Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

8 Noisy OR (cont.) Need only specify first three entries of table: Flu Malaria Cold P( fever) T F F 0.2 F T F 0.1 F F T 0.6 F F F 1.0 F T T = 0.06 T F T = 0.12 T T F = 0.02 T T T = Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 Building an example network When I go home at night, I want to know if my family is home before I try the doors (perhaps the most convenient door to enter is double locked when nobody is home). Now, often when my wife leaves the house she turns on an outdoor light. However, she sometimes turns on this light if she is expecting a guest. Also, we have a dog. When nobody is home, the dog is put in the back yard. The same is true if the dog has bowel trouble. Finally, if the dog is in the back yard, I will probably hear her barking, but sometimes I can be confused by other dogs barking. F. Jensen, An introduction to Bayesian networks, UCL Press, Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

9 Building an example network (cont.) Relevant entities: Boolean random variables FamilyOut, LightsOn, HearDogBark Causal structure: FamilyOut has direct influence on both LightsOn and HearDogBark, so LightsOn and HearDogBark are conditionally independent given FamilyOut FamilyOut LightsOn HearDogBark Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 Building an example network (cont.) Numbers in conditional probability table derived from previous experience, or subjective belief P(familyout) = 0.2 P(lightson familyout) = 0.99 P(lightson familyout) = 0.1 Run into problem with P(heardogbark familyout): dog may be out because of bowel problems, and barking may be other dogs Network structure needs to be updated to reflect this Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

10 Building an example network (cont.) Introduce mediating variable DogOut to model uncertainty with bowel problems and hearing other dogs bark FamilyOut BowelProblems LightsOn DogOut HearDogBark Need: P(DogOut FamilyOut, BowelProblems) P(HearDogBark DogOut) Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 Building an example network (cont.) Obtain the following additional probability tables: FamilyOut P(f) 0.2 BowelProblems P(b) 0.05 LightsOn DogOut F P(l F) T 0.99 F 0.1 F B P(d F, B) T T 0.99 T F 0.88 F T 0.96 F F 0.2 HearDogBark D P(h D) T 0.6 F 0.25 Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

11 Inference in Bayesian networks Given events (instantiated variables) e and no information on hidden variables H, calculate distribution for query variable Q Algorithmically, calculate P(Q e) by marginalizing over H P(Q e) = αp(q, e) = α h P(Q, e, h) with h all possible value combinations of H Distinguish between causal, diagnostic, and intercausal reasoning Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 Types of inference Causal reasoning: query variable is downstream of events P(heardogbark familyout) = 0.56 Diagnostic reasoning: query variable upstream of events P(familyout heardogbark) = Explaining away (intercausal reasoning): knowing effect and possible cause, reduce the probability of other possible causes P(familyout bowelproblems, heardogbark) = P(bowelproblems heardogbark) = P(bowelproblems familyout, heardogbark) = Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

12 Algorithmic aspects of inference Calculating joint distribution computationally expensive Several alternatives for inference in Bayesian networks: Exact inference by enumeration by variable elimination Stochastic inference (Monte Carlo methods) by sampling from the joint distribution by rejection sampling by likelihood weighting by Markov chain Monte Carlo methods Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 Inference by enumeration FamilyOut example (d {d, d}, b {b, b}) P(F l, h) = αp(f, l, h) = α P(F, l, h, d, b ) d b P(f l, h) = α d b P(f )P(b )P(l f )P(d f, b )P(h d ) = α P(f )P(l f ) d P(h d ) b P(b )P(d f, b ) = α ( ) = α Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

13 Inference by enumeration (cont.) Similarily, P( f l, h) = α From P(f l, h) + P( f l, h) = 1, normalization yields P(F l, h) = α (0.111, ) = (0.806, 0.194) Burglary example: P(B j, m) = α P(B) e P(e ) a P(a B, e )P(j a )P(m a ) Last two factors P(j a)p(m a) do not depend on e, but have to be evaluated twice (for e and e) Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 Variable elimination Eliminate repetitive calculations by summing inside out, storing intermediate results (cf. dynamic programming) Burglary example, different query: P(J b) = α P(b) e P(e ) a P(a b, e )P(J a ) m P(m a ) } {{ } = 1 Fact: Any variable that is not an ancestor of the query or evidence variables is irrelevant and can be dropped Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

14 Sampling from the joint distribution Straightforward if there is no evidence in the network: Sample each variable in topological order For nodes without parents, sample from their distribution; for nodes with parents, sample from the conditional distribution With N S (x 1,..., x n ) being the number of times specific realizations (x 1,..., x n ) are generated in N sampling experiments, obtain lim N N S (x 1,..., x n ) N = P(x 1,..., x n ) Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 Example: joint distribution sampling FamilyOut P(f) 0.2 BowelProblems P(b) 0.05 LightsOn DogOut F P(l F) T 0.99 F 0.1 F B P(d F, B) T T 0.99 T F 0.88 F T 0.96 F F 0.2 HearDogBark D P(h D) T 0.6 F 0.25 What is probability of family at home, dog has no bowel problems and isn t out, the light is off, and a dog s barking can be heard? Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

15 Example: joint distribution sampling FamilyOut example: Generate samples from the network by first sampling from FamilyOut and BowelProblems variables, then sampling from all other variables in turn, given sampled parent values Obtain N S ( f, b, l, d, h) = Compare with P( f, b, l, d, h) = = Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 Example: joint distribution sampling Advantage of sampling: easy to generate estimates for other probabilities Standard error of estimates drops as 1/ N, for N = this is N S ( d)/ = ( P( d) = ) N S (f, h) N S ( h) = ( P(f h) = ) Last example: Form of rejection sampling Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

16 Rejection sampling in Bayesian networks Method to approximate conditional probabilities P(X e) of variables X, given evidence e: P(X e) N S(X, e) N S (e) Rejection sampling: Take only those samples that are consistent with the evidence into account Problem with rejection sampling: Number of samples consistent with evidence drops exponentially with number of evidence variables, therefore unusable for real-life networks Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 Likelihood weighting Fix evidence and sample all other variables This overcomes rejection sampling shortcoming by only generating samples consistent with the evidence Problem: Consider situation with P(E = e X = x) = and P(X = x) = 0.9. Then, 90% of samples will have X = x (and fixed E = e), but this combination is very unlikely, since P(E = e X = x) = Solution: Weight each sample by product of conditional probabilities of evidence variables, given its parents Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

17 Example: Likelihood weighting FamilyOut example: Calculate P(F l, d) Iterate the following: sample all non-evidence variables, given the evidence variables, obtaining, e.g., ( f, b, h) calculate weighting factor, e.g. P(l f ) P(d f, b) = = 0.02 Finally, sum and normalize weighting factors for samples (f, l, d) and ( f, l, d) Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 Example: Likelihood weighting (cont). For N = , obtain N S ( f, l, d) = N S (f, l, d) = w( f,l,d) = w(f,l,d) = P(f l, d) /( ) = Correct: P(f l, d) = Disadvantage of likelihood weighting: With many evidence variables, most samples will have very small weights, and few samples with larger weights dominate Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

18 Markov chains A sequence of discrete r.v. X 0, X 1,... is called a Markov chain with state space S iff P(X n = x n X 0 = x 0,..., X n 1 = x n 1 ) = P(X n = x n X n 1 = x n 1 ) for all x 0,..., x n S. Thus, X n is conditionally independent of all variables before it, given X n 1 Specify state transition matrix P with P ij = P(X n = x j X n 1 = x i ) Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 Markov chains Monte Carlo methods Want to obtain samples from given distributions P d (X ) (hard to sample from with other methods) Idea: Construct a Markov chain that, for arbitrary initial state x 0, converges towards a stationary (equilibrium) distribution P d (X ) Then, successive realizations x n, x n+1,... are sampled according to P d (but are not independent!!) Often not clear when convergence of chain has taken place Therefore, discard initial portion of chain (burn-in phase) Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

19 Markov chain example Let S = {1, 2} with state transition matrix P = ( 1 2 Simulate chain for 1000 steps, show N S (1)/N and N S (2)/N for N = 1,..., 1000 with starting state 1 (left) and 2 (right) ) Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 MCMC for Bayesian networks Given evidence e and non-evidence variables X, use Markov chains to sample from the distribution P(X e) Obtain sequence of states x 0, x 1,..., discard initial portion After convergence, samples x k, x k+1,... have desired distribution P(X e) Many variants of Markov chain Monte Carlo algorithms Consider only Gibbs sampling Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

20 Gibbs sampling Fix evidence variables to e, assign arbitrary values to non-evidence variables X Recall: Markov blanket of a variable is parents, children, and children s parents. Iterate the following: pick arbitrary variable X i from X sample from P(X i MarkovBlanket(X i )) new state = old state, with new value of X i Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 Gibbs sampling (cont.) Calculating P(X i MarkovBlanket(X i )): P(x i MarkovBlanket(X i )) = α P(x i parents(x i )) Y i children(x i ) P(y i parents(y i )) With this, calculate P(x i MarkovBlanket(X i )) and P( x i MarkovBlanket(X i )), normalize to obtain P(X i MarkovBlanket(X i )) Sample from this for next value of X i, and thus next state of Markov chain Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

21 Bayesian network MCMC example FamilyOut example: Calculate P(F l, d) Start with arbitrary non-evidence settings (f, b, h) Pick F, sample from P(F l, d, b), obtain f Pick B, sample from P(B f, d), obtain b Pick H, sample from P(H d), obtain h Iterate last three steps times, keep last states Obtain P(f l, d) (correct ) Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43 Comparison of inference algorithms Inference by enumeration computationally prohibitive Variable elimination removes all irrelevant variables Direct sampling from joint distribution: easy when no evidence present Use rejection sampling and likelihood weighting for more efficient calculations Markov chain Monte Carlo methods are most efficient for large networks by calculating new states based on old states, but lose independence of samples Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43

22 Summary Bayesian networks are graphical representations of causal influence among random variables Network structure graphically specifies conditional independence assumptions Need conditional distributions of nodes, given its parents Use noisy OR to reduce number of parameters in tables Reasoning types in Bayesian networks: causal, diagnostic, and explaining away There are exact and approximate inference algorithms Stephan Dreiseitl (Hagenberg/SE/IM) Lecture 11: Bayesian Networks Artificial Intelligence SS / 43