42 Formal Meaning of a Bayesian Network

Recall : Full joint distribution

$$P(X_1,X_2,\dots ,X_n)=P(X_1)P(X_2\mid X_1)P(X_3\mid X_1,X_2)\cdots P(X_n\mid X_1,X_2,\dots ,X_{n-1})$$

A Bayesian Network, encodes the following limited-dependence assumption :

$$P(X_1,X_2,\dots ,X_n)=\prod _{i=1}^{n}P(X_i\mid \text{Parents}(X_i))$$

Essence of a Bayesian Network :

• A compact representation of a joint probability distribution in terms of conditional distributions.