Exam Preparation Class

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Presentation transcript:

Exam Preparation Class .

QUESTION 2 on Basic Inference directly from definitions Given is a Bayesian network A  B where A and B are binary variables. Variable A stands for type of shirt with values (a0 = T-shirt; a1 = not T-shirt). Variable B stands for color of shirt with values (b0 = red; b1 = purple). Denote z = p(a0). Let P(b0|a0)=0.5 and let P(b0|a1)=0.25. An observer examines a shirt color. Denote by e the process of examination (which is influenced by internal conceptions and laboratory light conditions). The observer declares: “the likelihood of red is twice the likelihood of purple. Provide a formula for the posterior probability of A after the evidence is given. Namely, a formula for p(a0|e) (as a function of z). .

Answer: Use the network A  BE Set E=e. Set P(e|b0) / P(e/b1) = 2. Write a formula directly from the definition of Bayes network for p(a0,e)= p(a0,b,e)= z p(b0|a0) p(e|b0) + z p(b1|a0) p(e|b1) and for p(a1,e) = p(a1,b,e)= (1-z) p(b0|a1) p(e|b0) + (1-z) p(b1|a1) p(e|b1) Divide the two formulae. It is a function of z. Use the relationship p(a0,e)=1-p(a0,e) to obtain the answer.

QUESTION 3 on d-separation Describe a linear algorithm for the following task. Input: a Bayesian Network D=(V,E), a set of nodes J, and a set of nodes Z. Output: The set of all nodes X that are d-separated from J by Z. (P.S. Due to soundness and completeness theorems, X is the largest set of variables that can be shown to be conditionally independent of J, given Z, based on the graph structure alone.) .

Answer (d-separation: from theorems to algorithms): Use BFS with minor changes and linear preprocessing. Consider a set of legal pairs of edges u – v –w according to d-separation. Namely, a pair is legal if edges meet head-to-head and v is in Z or has a descendant in Z - or - the pair of edges do not meet head-to-head and v is not in Z. Construct a table: Z Set false in all entries. Set true for all v in Z. Iterate to their parents. .

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QUESTION 1: conditional independence properties (I(X,Z,Y) .