Consider the Bayesian Network on below. Suppose each variable has two possible o

Question

Consider the Bayesian Network on below. Suppose each variable has two possible outcomes:

a) Without conditional independence assumptions as reflected in this network, how many parameters need to be estimated for the full joint distribution?

b) Given the conditional independence assumptions as indicated in the Beyesian network, one needs to specify conditional probability tables for each variable. How many parameters need to be estimated in total for these variables?

c) What is the formula to calculate the full joint probability based on this network?

Explanation / Answer

A)

when specifying an arbitrary multinomial distribution over a k dimensional space, we have k - 1 independent parameters: the last probability is fully determined by the first k - 1. In the case where we have an arbitrary joint distribution over n binary random variables, the number of independent parameters is 2n - 1.

B)

Based on these independence assumptions. we can show that the model factorizes as: n P(C,X 1 , ... ,X n ).
Thus, in this model, we can represent the joint distribution using a small set of factors: a prior distribution P( C),
one for each of the n finding variables. These factors can be encoded using a very small number of parameters.Thus, the number of parameters is linear in the number of variables, as opposed to exponential for the explicit representation of the joint.

c) Calculate the probability for certain events to be in a certain state knowing all conditional probabilities.

For example, for variable Ai I do not know P{Ai=ai} but I know P{Ai=ai|B1=b1…Bn=bn}. Where B1,B2…Bn are parents of Ai.

I want to calculate P{Ai=ai}. Consider that No restrictions are considered so this variable Ai can have parents and children. As well as B1…Bn.

In the sum, the first part: P{Ai=ai|B1=b1…Bn=bn} can be calculated using potentials.

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