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?

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).

Bayesian Network Example

While causal diagrams are a specific type of Bayesian network where arrows indicate direct causal relations (or the possibility of one), general Bayesian networks do not require causal relationships. The primary difference lies in the meaning of the arrows:

- **Probabilistic Bayesian Networks:** Arrows directed into $$Y$$ mean that the probability of $$Y$$ is governed by conditional probability tables for $$Y$$, given observations of its parent variables.
- **Causal Bayesian Networks:** Conditional probability tables specify the probability of $$Y$$ given interventions on the parent variables.
- **Structural Causal Models:** Lacking conditional probability tables, arrows mean $$Y$$ is a deterministic function of its parents and an exogenous variable $$U_Y$$: $$Y = f_Y(X, A, B, C,…, U_Y)$$.

Both probabilistic and causal Bayesian networks specify probabilities for $$Y$$, not specific values.

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