What Are BBNs?

• A Bayesian Belief Network is a graphical representation of probabilistic relationships among different random variables.
• These variables can represent anything: diseases, weather conditions, financial markets, or even whether your cat will knock over that glass of water.

Graphical Structure:

• Imagine nodes (circles) connected by arrows (edges).
• Each node represents a random variable (e.g., “Alarm ringing,” “Burglary,” “Fire”).
• The arrows show dependencies between variables.

Conditional Independence:

• BBNs are conditionally independent.
• This means that each node’s probability depends only on its parents (directly connected nodes).

Example: Alarm System

Let’s consider a simple example: an alarm system in a house.

Nodes:

• Alarm (A): Represents whether the alarm rings.
• Burglary (B): Whether a burglary occurred.
• Fire (F): Whether there’s a fire.
• Person 1 (P1): Whether person 1 calls (hearing the alarm).
• Person 2 (P2): Whether person 2 calls (also hearing the alarm).

Probabilities:

 We have observed probabilities for each event:

• P(B=T) = 0.001 (Burglary occurred)
• P(F=T) = 0.002 (Fire occurred)
• P(A|B,T) = 0.95 (Alarm rings given burglary and fire
• P(A|B,T) = 0.95 (Alarm rings given burglary and fire)
• P(A|B,F) = 0.94 (Alarm rings given burglary, no fire)
• P(A|~B,~F) = 0.001 (Alarm rings when no burglary or fire)

Person Nodes:

P1 and P2 call based on the alarm:

• P(P1|A,T) = 0.95 (Person 1 calls if alarm rings)
• P(P2|A,T) = 0.80 (Person 2 calls if alarm rings)

Question:

• What’s the probability that both P1 and P2 call when the alarm rings, but there’s no burglary (B) and no fire (F)?

Solution:

• We compute: P(P1, P2, A, ~B, ~F)
• Using the observed probabilities, we find the joint probability.

Conclusion:

Remember, Baysian belief network's help us reason about complex systems by breaking them down into simpler parts.