Bayesian Networks & Joint Factorization

Diseases cause symptoms. Symptoms cause test results. A Bayesian network is just a clean way to draw who causes what — and then ask probabilistic questions on top.

Each node in the picture is a random variable. Each arrow says “this directly influences that”. Attach one small probability table per node — its conditional probability table (CPT) — and you’ve quietly specified the entire joint distribution, in a fraction of the numbers it would take to list the joint outright. This is the model behind real diagnostic systems (medical risk, equipment fault-finding) and the probabilistic graphical models used across ML to reason under uncertainty — the compactness is what makes them tractable.

The DAG and its CPTs

A Bayesian network is a directed acyclic graph (DAG) in which every node X carries a CPT P(X | Parents(X)). No cycles, no self-influence — just parents pointing to children.

The Alarm node needs a CPT with one row per parent combination — four rows here, one for each setting of (B, E):

The joint factorisation

For any Bayes net on n variables:

P(X₁, X₂, …, Xₙ) = ∏ᵢ P(Xᵢ | Parents(Xᵢ))

That’s the headline. Instead of one giant table over all 2ⁿ joint outcomes, you store one small table per node — conditioned only on its parents, not on every earlier variable. That’s where the savings come from.

The corresponding independence statement: each node is independent of its non-descendants given its parents. In the alarm net, once you know B and E, the alarm is independent of any other non-descendant — its parents fully explain it.

Worked example — a 2-node net

A 2-node net Disease → Test with P(D) = 0.3, P(+ | D) = 0.8, P(+ | ¬D) = 0.1. The test reads positive. Compute P(D | +).

The joint factorises as P(D, T) = P(D) · P(T | D). So the joint for the observed outcome D = 1, T = + is:

P(D=1, +) = P(D) · P(+ | D) = 0.30 · 0.80 = 0.24

Posterior by Bayes — the denominator is the total probability of a positive test, which is itself just summing the joint over D:

P(D | +) =        P(+ | D) · P(D)
            ──────────────────────────────────
            P(+ | D)·P(D) + P(+ | ¬D)·P(¬D)

         =       0.80 · 0.30                =  0.24 / 0.31  ≈  0.77
            ──────────────────────────
            0.80·0.30 + 0.10·0.70

So ≈ 0.77 — the same shape as the disease/test problem from the Bayes’ theorem lesson (and GATE DA 2026 Q57). The factorisation just made the joint mechanical to write down before plugging into Bayes.

Drag the prior down and watch the same posterior collapse when the disease is rare — base rates dominate Bayes-net inference too.

How GATE asks this

Usually a NAT: a small 3-or-4-node net with CPTs given, asking for a joint probability of a specific assignment (multiply down the factorisation), or a posterior of one variable given evidence on another (factorise the joint, then Bayes). MSQs ask which independence statements the DAG implies. GATE DA 2026 Q57 was the disease/positive-test posterior above (answer 0.77). GATE DA 2024 ran the same machinery on a slightly larger net.

The recipe never changes: write the joint as a product of CPTs, plug in the observed values, normalise.

Quick check