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 .
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
- 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
Given: Calculate using the entire network
Joint probability using chain rule:
Marginalize over A, B, and D to get P(C):
- Since (sum of probabilities over all possible values of D):
- Since (sum of probabilities over all possible values of B):
Slightly wrong Solution: source