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. That means one probability for every possible combination of values all the variables could take at once.

And you have specified it in a fraction of the numbers it would take to list that 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. 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

In words: to get the probability of one complete configuration, walk every node. For each node, look up that node’s own probability given whatever its parents are doing in this configuration, and multiply those numbers together. Written out, 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 — a non-descendant being anything you cannot reach by following arrows forward out of that node. In the alarm net, once you know B and E, the alarm is independent of any other non-descendant; its parents fully explain it.

Read that sentence carefully, because two things trip people up and the alarm net shows both. First, no arrow between two nodes does not mean they are independent — they can still move together through a shared ancestor upstream. Second, and this is the surprising direction: conditioning can create dependence that was not there before. Burglary and Earthquake have no edge and no common ancestor, so on their own they really are independent. But learn that the alarm rang, and they stop being — news of an earthquake now explains the noise, which makes a burglary less likely. Independence in a Bayes net is always a claim about a particular conditioning set, never a property of the pair on its own.

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 has a denominator: the total probability of a positive test. That denominator is 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. It asks for one of:

• the joint probability of a specific assignment (multiply down the factorisation)
• 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))
• has a whole joint that factorises as ∏ᵢ P(Xᵢ | Parents(Xᵢ)) — one small table per node instead of a 2ⁿ-entry monster
• encodes the independence that each node is independent of its non-descendants given its parents
• lets you answer a query by multiplying the relevant CPT entries into the joint and applying Bayes

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? Could that be done 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.