---
title: "Causal net, belief network, Bayes"
subject: "Artificial Intelligence"
chapter: 4
key: "semester_6/artificial_intelligence:causal-net-belief-network"
source: https://www.bctengineer.com/q/semester_6/artificial_intelligence/causal-net-belief-network
site: BCT Engineering
---

# Causal net, belief network, Bayes

Artificial Intelligence · Chapter 4: Knowledge Representation, Inference & Reasoning

- Asked in: 2071 Magh, 2+5 marks; 2073 Bhadra, 7 marks (asked 2 times)
- Video: [1.  Bayesian Belief Network | BBN | Solved Numerical Example | Burglar Alarm System by Mahesh Huddar](https://www.youtube.com/watch?v=hEZjPZ-Ze0A) — Mahesh Huddar
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/artificial_intelligence/causal-net-belief-network
- Whole chapter: https://www.bctengineer.com/md/semester_6/artificial_intelligence/chapter-4

## Question

What is a causal net (network)? Explain reasoning in a belief network (how does Bayes theorem calculate the probability in a causal net) with suitable example calculation.

## Short answer

A causal net is a DAG with conditional probability tables. Reasoning uses Bayes’ theorem: for wet grass evidence, P(Rain|Wet) = P(Rain,Wet)/P(Wet) = 0.1818/0.2538 ≈ 0.716, illustrating diagnostic reasoning.

## Answer

#### Causal network (belief network)
A **causal network**, also called a **Bayesian belief network**, is a directed acyclic graph (DAG) in which each node is a random variable (event or condition) and a directed arc from A to B means "A directly influences / causes B". Every node stores a **conditional probability table (CPT)**, P(node | parents). A node with no parents has a prior probability. The joint probability is
**P(X1, ..., Xn) = Π P(Xi | Parents(Xi))**.

Compared to a full joint table, it needs far fewer probabilities because of the conditional independence between nodes.

#### Reasoning with Bayes' theorem
**Bayes' theorem:** P(H | E) = P(E | H) · P(H) / P(E), where P(E) is obtained by summing the joint probability over all possible values of the hidden variables. In a belief network, we compute the required joint probabilities from the CPTs, add them (marginalise), and divide. Reasoning may be causal (cause → effect), diagnostic (effect → cause) or intercausal.

#### Example
Rain (R) and Sprinkler (S) can cause Wet Grass (W).

```text
   Rain         Sprinkler
  P(R)=0.2       P(S)=0.1
      \           /
       v         v
        Wet Grass
```
CPT P(W = true | R, S): R,S = (T,T): 0.99; (T,F): 0.90; (F,T): 0.90; (F,F): 0.00.

**Query:** the grass is wet; what is the probability that it rained, P(R | W)?

Joint probabilities P(R,S,W) = P(R)·P(S)·P(W | R,S):
- R, S: 0.2 × 0.1 × 0.99 = 0.0198
- R, ¬S: 0.2 × 0.9 × 0.90 = 0.1620
- ¬R, S: 0.8 × 0.1 × 0.90 = 0.0720
- ¬R, ¬S: 0.8 × 0.9 × 0.00 = 0

P(R, W) = 0.0198 + 0.1620 = 0.1818

P(W) = 0.1818 + 0.0720 + 0 = 0.2538

**P(R | W) = P(R, W) / P(W) = 0.1818 / 0.2538 ≈ 0.716**

The prior probability of rain was 0.2, but seeing the wet grass raises it to about 0.72 (diagnostic reasoning). If we also learn that the sprinkler was on, P(R | W, S) = 0.0198 / (0.0198 + 0.0720) ≈ 0.216, which shows that the sprinkler "explains away" the wet grass (intercausal reasoning).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is a causal (belief) network?
   A. An undirected graph of rules
   B. A directed acyclic graph of random variables where each node has a conditional probability table
   C. A list of facts without probabilities
   D. A decision tree with utilities
   Answer: B. Arcs show direct influence, and each node stores P(node | parents). (Key point: Structure)

2. How is the joint probability computed in a belief network?
   A. As the sum of all CPT entries
   B. As the product of P(Xi | Parents(Xi)) over all nodes
   C. As the maximum of the priors
   D. As the average of the CPTs
   Answer: B. P(X1,...,Xn) = Π P(Xi | Parents(Xi)). (Key point: Joint probability)

3. In the example, how is P(R | W) obtained?
   A. P(R, W) / P(W), by summing joint probabilities over the hidden variable
   B. P(W | R) only
   C. P(R) times P(S)
   D. 1 - P(R)
   Answer: A. P(R,W) = 0.1818 and P(W) = 0.2538, giving about 0.716. (Key point: Computing P(R|W))

4. Why does P(R | W, S) fall to about 0.216 compared with 0.716?
   A. Learning the sprinkler was on explains the wet grass, reducing the need for rain
   B. The sprinkler prevents rain
   C. Wet grass becomes impossible
   D. The prior of rain changes to 0
   Answer: A. This intercausal reasoning means another cause (sprinkler) explains the evidence. (Key point: Explaining away)

## Related in Artificial Intelligence

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- [Bayes: flu, measles and rash](https://www.bctengineer.com/question.md?key=semester_6%2Fartificial_intelligence%3Abayes-flu-measles-rash) — Artificial Intelligence, chapter 4

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