Artificial Intelligence Assignment 6,Solution

Questions

This will be a non-coding homework. Please upload a digital copy of your solution on CCLE. Submitted les can be in either PDF or plain text. You may also submit a scanned PDF of a handwritten solution, but please ensure that the scanned le is clearly legible.

    1. Consider the following:

An oil well may be drilled on Mr. G’s farm in LA. Based on what has happened to similar farms, we judge the probability of only oil being present to be .3, the probability of only natural gas being present to be .2, and the probability of neither being present to be .5. Oil and gas never occur together. If oil is present, a geological test will give a positive result with probability .8; if natural gas is present, it will give a positive result with probability .2; and if neither are present, the test will be positive with probability .1.

        (a) Model this problem as a Bayesian network over three variables: Oil, Gas, and Test.

        (b) Suppose the test comes back positive. What’s the probability that oil is present?

    2. Consider the Bayesian network in Figure 1:

        (a) Express Pr(A; B; C; D; E; F; G; H) as a multiplication of conditional and marginal probabilities, according to the factorization encoded in the network structure.

        (b) Express Pr(E; F; G; H) in terms of factors instead of (conditional) probabilities.

        (c) Express Pr(a; :b; c; d; :e; f; :g; h) in terms of the parameters in the CPTs (a denotes A = 1 and

:a denotes A = 0). Use placeholder symbols for the parameters that are not shown in the CPTs.

        (d) Compute Pr(:a; b) and Pr(:e j a). Justify your answers. Hint: leaf nodes that are not part of the probability query can be removed from the network without a ecting the computed probability.

        (e) List the Markovian assumptions (also known as topological semantics) encoded in the Bayesian network structure.

        (f) Provide the Markov blanket for variable D.

        (g) Multiply the factors (tables) corresponding to Pr(DjAB) and Pr(EjB).

        (h) Sum out D from the factor (table) computed above.

Submission

Submit all solution  les on Gradescope.

Submit your solution in a  le named hw6.pdf .




1


    • B


C    D    E


F


    • H

A

Pr(A)

B

Pr(B)


B
E
Pr(EjB)








1
1
.1
1

.1


1

.6

1
0
.9
0

.9


0

.4

0
1
.8









0
0
.2








Pr(DjAB)






A
B

D





1
1
1

.7




1
1
0

.3




1
0
1

.6




1
0
0

.4




0
1
1

.2




0
1
0

.8




0
0
1

.8




0
0
0

.2


















Figure 1: A Bayesian network with some of its CPTs.









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