Bayesian Networks and Probabilistic Inference Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Bayesian Networks and Probabilistic Inference notes for GATE DA: 19 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Bayesian Networks and Probabilistic Inference

Chapter Roadmap

1. Bayesian Network Inference Algorithms and CPT Computation Current Topic. High importance. Master reading Conditional Probability Tables, constructing the joint distribution, and performing inference by enumeration or variable elimination.

2. Bayesian Network Factorization and Conditional Independence Next Topic. Moderate importance. Learn how the graph structure dictates independence via d-separation, reducing computational complexity.

Topic Hero: Inference and CPT Computation

A Bayesian Network is a compact representation of a joint probability distribution. However, to answer real questions (like "What is the probability of a fire given that the alarm is ringing?"), we must perform inference.

This topic bridges the gap between the static graph and dynamic computation. It focuses on two foundational skills:

CPT Computation: Understanding how to read, size, and populate Conditional Probability Tables.
Inference Algorithms: Systematically combining these local tables to compute marginal or conditional probabilities of query variables.

Anatomy of a Conditional Probability Table

Structure of a CPT

For a node $X$ with parents $P a r e n t_{1}, P a r e n t_{2}, \dots, P a r e n t_{k}$ :

The CPT specifies $P (X ∣ P a r e n t_{1}, P a r e n t_{2}, \dots, P a r e n t_{k})$ .
If all variables are boolean, there are $2^{k}$ possible combinations of parent values.
Therefore, the CPT requires $2^{k}$ entries to fully define the conditional distribution.

Independent Parameters

While there are $2^{k}$ entries, the number of independent parameters is also $2^{k}$ for a boolean child. This is because each row represents a distinct conditional distribution $P (X ∣ specific parent assignment)$ , and the complement $P (\neg X ∣ \dots)$ is trivially $1 - P (X ∣ \dots)$ .

General Rule: If a node has $r$ possible values and $k$ parents each with $d$ values, the table has $d^{k}$ rows, and the number of independent parameters is $d^{k} \times (r - 1)$ .

The Chain Rule for Bayesian Networks

Full Joint Distribution

Given a Bayesian Network with variables $X_{1}, X_{2}, \dots, X_{n}$ , the full joint distribution is factorized as:

P (X_{1}, X_{2}, \dots, X_{n}) = i = 1 \prod n P (X_{i} ∣ P a r e n t s (X_{i}))

Why This Matters

Without a network, specifying the joint distribution of $n$ boolean variables requires $2^{n} - 1$ independent parameters.
With a network where each node has at most $k$ parents, we only need $n \times 2^{k}$ parameters.
This exponential reduction is the primary motivation for using Bayesian Networks.

More notes in this unit

Logic, First-Order Representation and Entailment Notes

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Bayesian Networks and Probabilistic Inference Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Bayesian Networks and Probabilistic Inference notes for GATE DA: 19 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice qu

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