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.

Determining Independence given a Bayesian Network (d-Separation)

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.

University of Michigan - Ann Arbor

Multiple methods exist that can be used to determine causal relationships between variables.

Causal Discovery

Pearl, Judea. “The Seven Tools of Causal Inference, with Reflections on Machine Learning.” Communications of the ACM, vol. 62, no. 3, 2019, pp. 54–60., doi:10.1145/3241036.

https://dl.acm.org/doi/pdf/10.1145/3241036

The Seven Tools of Causal Inference, with Reﬂections on Machine Learning

d-separation criterion

Used to infer causal direction by modeling P(y) as a convolution of two non Gaussian distributions, in a linear model X -> Y with non Gaussian noise. P(x) is arguably less Gaussian than P(y), and this 'less' can be measured numerically to give evidence of a direction of causality.

Shimizu et al Method

Based on detecting 'shocks' (local environment changes) and tracing their effects to determine causal direction.

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