Bayesian Networks

Bayesian Networks

1. Idea

Directed acyclic graph factorizes joint distribution into local parent-child tables — compact + interpretable:

Absent edge = conditional independence.

2. Example

Rain → WetGrass ← Sprinkler
p(Rain, Sprinkler, Wet) = p(Rain) p(Sprinkler) p(Wet | Rain, Sprinkler)

Knowing Wet couples Rain/Sprinkler (explaining away): if Sprinkler on, Rain less likely.

3. Independence (d-separation, intuition)

• Chain $A → B → C$: $A$ independent of $C$ given $B$.
• Fork $A ← B → C$: $A$ independent of $C$ given $B$.
• Collider $A → B ← C$: $A$ independent of $C$ until $B$ observed, then dependent.

4. Inference + Learning

Exact (variable elimination) for small nets; approximate (sampling, variational) for large. Learn tables by counting (ML/MAP with Dirichlet smoothing); learn graph by scored search (harder).

5. Exam Points

• Write factorization + draw 3-node net — 5-7 marks, very frequent.
• Explaining away + d-separation cases — 5 marks, frequent.
• Parameter counting (tables size) — 3 marks, frequent.