Given a diagram of a Bayesian Network, we can use it to determine the marginal and conditional independence of events in the network (assuming of course, that it is a complete network).
Let us assume that we are trying to determine the marginal independence of events A and B and their conditional independence w.r.t event C.
Algorithm : 
1. Retain only that part of the graph that contains A, B, and nodes that point to them.
2. Connect all pairs of unconnected nodes, where both nodes point to a common node. Make the graph undirected. 
3. If there is no path between A and B, they are marginally independent. If there is no path between A and B after removing C, then A and B are conditionally independent given C.

University of Michigan - Ann Arbor

    1. Marginal Independence
    2.Conditional Independence

Notions of Independence in Bayesian Networks

Provides connection between causes and probabilities. Helps enumerate testable implications of a causal model i.e can allow it to be represented in a compact set which allows for queries to be made into that specific set and thus more realistic estimates.

d-separation criterion

This implies that given any two events A and B, 
$$P(A, B) = P(A) . P(B)$$
That is, the probability that both events occur is the product of the probabilities that each of the events occur.

Marginal Independence

This implies that given a system where only three events A, B and C, occur, 

$$P(A, B| C) = P(A|C) . P(B|C)$$

That is, knowing the value of C does not help us better ascertain the values of A and B.

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