Bayesian Networks & Joint Factorization

Last lesson left logic stranded on the shore of certainty, with no word for “usually” and no way to hold a belief at degree 0.9 — exactly what the real world demands. Here is the tool for reasoning in degrees. Diseases cause symptoms; symptoms cause test results. A Bayesian network is a clean way to draw who influences whom, and then ask probabilistic questions on top of the picture.

Each node 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 have quietly specified the entire joint distribution, in a fraction of the numbers it would take to list the joint outright. That compactness is the whole point: it is the model behind real diagnostic systems (medical risk, equipment fault-finding) and the probabilistic graphical models used across ML to reason under uncertainty, and it is what makes such reasoning tractable at all.

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, one for each setting of (B, E):

Alarm’s CPT: one row per parent assignment. The rows need not sum to 1 — each row is its own distribution over A.

The joint factorisation

For any Bayes net on n variables:

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

That is 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 is where the savings come from.

The matching 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 — GATE DA 2026

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

The posterior by Bayes — its denominator the total probability of a positive test, itself just the joint summed 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 value as the disease/test problem from the Bayes’ theorem lesson (and GATE DA 2026, Q57). Drawing it as a network did not change the answer; the factorisation just made the joint mechanical to write down before plugging into Bayes.

Drag the prior down and watch the 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 the 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.

In one breath

A Bayesian network is a DAG whose every node carries a CPT P(X | Parents(X)), and the whole joint factorises as ∏ᵢ P(Xᵢ | Parents(Xᵢ)) — one small table per node instead of a 2ⁿ-entry monster — which encodes the independence that each node is independent of its non-descendants given its parents; you answer a query by multiplying the relevant CPT entries into the joint and applying Bayes, and the standing trap is conditioning a node on all earlier variables instead of only its parents.

Practice

A question to carry forward

A Bayes net stores the whole joint in a handful of small tables — wonderfully compact. But storing the joint and querying it are different acts. Ask the natural question — “given the alarm went off, what is the chance it was a burglary?” — and you must sum the joint over every variable you did not observe (here, the earthquake). For three nodes that is a tiny sum. For a net of thirty variables, the marginal you want hides inside a sum with billions of terms.

Yet stare at that sum and the same sub-expressions repeat over and over. Surely you need not expand the entire joint just to collapse it again. Here is the thread onward: is there a way to compute that marginal exactly — the true probability, no approximation — without ever writing the full joint down, by summing out one variable at a time and reusing the partial results so the work does not explode? That careful, order-sensitive summing-out is the next lesson’s algorithm.