Bayes net criteria

The formal criteria for a Probabilistic Graphical Model (PGM) to be a Bayesian network (mostly useful for proofs):

• The state space must be finite

• Each node is a variable

  • If the node $X_i$ doesn’t have a parent then it has a probability function (called a potential)
  $$p_{X_i}$$

  Note this is a marginal distribution for the variable

• If the node $X_i$ has parents ${\Pi }_i$, then it has a potential

$$p_{X_i|{\Pi }_i}$$

• The factorization represented by the BN is minimal
  • The nodes are ordered so that a node’s parents are a subset of the nodes before it in the ordering
  • for a node $X_j$, its set of parents ${\Pi }_j$ is the smallest set so that $X_i$ is conditionally independent of its non-parents.

$$X_j\perp {\Pi }_j^c|{\Pi }_j$$

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References

@koski2011