Example 1

Decompose step by step according to the chain rule into as many terms as possible and simplify, if and are independent of and .

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Given that and are independent of and , we can decompose using the chain rule and then simplify using this independence property.

Chain rule decomposition

Since and are independent of and :

Substitute:

Since and are independent:

Finally

Example 2

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  • Specify the probability by using the single terms of the Bayesian belief network (hint: First reorder the terms!)
  • Now let be an unknown and the probability for be inquired.Write down the probability by using the single terms of the Bayesian belief network.

Example 3

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Given: Calculate using the entire network

Joint probability using chain rule:

Marginalize over A, B, and D to get P(C):

  1. Since (sum of probabilities over all possible values of D):

  1. Since (sum of probabilities over all possible values of B):


Slightly wrong Solution: source