Intro to AI: Lecture 8. Volker Sorge. Introduction. A Bayesian Network. Inference in. Bayesian Networks. Bayesian Networks.

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2 Specifying Probability Distributions Specifying a probability for every atomic event is impractical We have already seen it can be easier to specify probability distributions by using (conditional) independence Bayesian (Belief) s allow us to specify any distribution, to specify such distributions concisely if there is (conditional) independence, in a natural way

3 Idea of a Bayesian Fix set of random variables {X 1,..., X n }. If every variable takes k values, we have to compute k n conditional probabilities to get the complete set of probability distributions. tries to avoid this by representing direct influences between random variables and restricting the necessary probability distributions that need to be computed to those direct influences.

4 Setting Up a Bayesian Every random variable {X 1,..., X n } is a node in the network. Influences are given by directed edges between nodes. Each node holds the joint probability distribution of with its parents nodes. If we do this naively, we can still end up computing close to k n probabilities. If we exploit conditional independence, we can reduce complexity to kn. Every node of a Bayesian is conditionallly independent of its non-descendants given its parents.

5 Example: Bayesian Net P(A).6 A Oversleeps C B Pershore closed D P(B).2 Volker late Mark late A B P(C) T T.9 T F.7 F T.8 F F.2 E Committee cancelled C D P(E) T T.9 T F.4 F T.5 F F.3 B P(D) T.3 F.4

6 Probabilistic Inference: Goal Compute the probability distribution for some event given some evidence. More formally: Let Q be a set of query variables, let E be a set of evidence variables, compute P(Q E). Here evidence means that we know the exact event for the variables in E. E.g., we know that Volker has overslept, how likely is it that the committee will be cancelled?

7 Types of Inference Diagnostic Inferences From effects to causes. How likely is a cause for some observed event? Causal Inferences From causes to effects. How likely will some observed events cause some other event? Intercausal Inferences Between causes of a common effect. How likely is that if we know one cause for an event, that some other cause is also happening? This is also sometimes called explaining away. Mixed Inferences Combining two or more of the above.

8 Inferences in Bayesian Nets A schematic overview for queries Q and observed evidence E: Diagnostic Causal Intercausal Mixed Q E E Q E Q E Q E

9 Inference Examples Diagnostic P(B E) = [<.21,.79 >] Causal P(E A) = [<.533,.467 >] Intercausal P(C D) = [<.557,.443 >] Mixed P(C B, E) = [<.904,.096 >] Computed with

10 Notation P stands for simple probability. P stands for a probability distribution (i.e., a set of probabilities). P(A B) denotes the probability of A under the condition B. P(A B) denotes the probability distribution for A under the condition B. P(A, B) is the not yet normalised distribution for A under the condition B. That is, αp(a, B) = P(A B). Finally small letters stand for probability variables that have to be summed out (sometimes called nuisance variables).

11 Example Computation P(B E = T ) = αp(b, E = T ) We compute P(B, E) by summing out the remaining variables A, C, D. We will write a, c, d for the respective events. This means we have to compute a c d P(B, E, a, c, d), which is the (not normalised) distribution of B under the assumption that E = T, while summing out a, c, d. A simple example how to sum out is: a P(B, a) = P(B A) + P(B A).

12 The great advantage of a Bayesian network is that we effectively can use all conditional probabilities given to express the term P(B, E, a, c, d) as follows: P(B, E) = a P(B, E, a, c, d) = c a d We observe that all the probability distributions on the right hand side are indeed fully given in the network. The summing out works as follows: P(B, E) = P(B, E, a, c, d) = a c d a c d = P(B)P(a)P(d B)P(c B, a)p(e c, d) a c d = P(B)P(A)P(d B)P(c B, A)P(E c, d) c d + P(B)P( A)P(d B)P(c B, A)P(E c, d) c d = P(B)P(A)P(d B)P(C B, A)P(E C, d) P(B)P(a)P(d B)P(c B, a c d P(B)P(a)P(d B)P(c B, a

13 Questions In the above Bayesian, give examples for the following concepts independent events, conditionally independent events, and dependent events. Consider again the example network. Compute the mixed inference with respect to observed evidence that no one overslept and the committee was cancelled (i.e., A = T and E = T ). How likely is it that Volker was late?