https://towardsdatascience.com/introduction-to-bayesian-networks-81031eeed94e

University of Michigan - Ann Arbor

A Bayesian network's structure is made up of nodes (which can be multi-variate), edges between paired nodes, and a conditional probability distribution in each nodes. The nodes in a Bayesian network represent random variables and edges represent the conditional dependence of these nodes. We try to use this structure to model the conditional probability distribution of outcome variables (which are often causal) after observing new evidence.

How do Bayesian Networks work?

Reference for "How Bayesian Networks work"

As defined in "The Book of Why", causal diagrams are Bayesian networks where every arrow indicates either a direct causal relation or the possibility of a direct causal relation in the direction that the arrow is facing. Unlike causal diagrams, Bayesian networks do not have to be based on causal relations.

The meaning of the arrows is a major difference between Bayesian networks and causal diagrams. 

In a probabilistic Bayesian network, the arrows into $Y$ mean that the probability of Y is governed by the conditional probability tables for $Y$, given observations of its parent variables. The same is true for causal Bayesian networks, except that the conditional probability tables specify the probability of $Y$ given interventions on the parent variables. Both models specify probabilities for $Y$, not a specific value of $Y$. 

In a structural causal model, there are no conditional probability tables. The arrows simply mean $Y$ is a function of its parents, as well as the exogenous variable $U_Y$:
$Y = f_Y(X, A, B, C,…, U_Y)$


Relation between Bayesian Networks and Causal Diagrams

This Bayesian network example shows the causal relationships between different seasons (X1), "whether it is raining (X2), whether the sprinkler is on (X3), whether pavement is wet (X4), and whether pavement is slippery (X5)". Since there is no direct link between X1 and X5, we can assume that season does not have a direct causal influence on how slippery the pavement is. Instead, it is influenced by how wet the pavement is (X4).

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