Two events $$A$$ and $$B$$ are marginally independent if their joint probability factors into their individual probabilities: $$P(A \cap B)=P(A)P(B).$$ Thus, their independence holds without conditioning on another event or variable.

University of Michigan - Ann Arbor

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    1. Marginal Independence
    2.Conditional Independence

Notions of Independence in Bayesian Networks

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.

Conditional Independence

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.

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