Bayesian Belief Network Traffic Predictor

Minneapolis, US | Rainy 15°C

Interactive Directed Acyclic Graph (DAG)

Click state names to set/toggle evidence

Select a Node

Click node header to inspect CPT

Click on any node's title in the network diagram above to view its Conditional Probability Table (CPT) and modify its properties.

Prediction Summary

Traffic Level: Low

Accident Risk: 3.19%

Capacity Constraint: Full Capacity

Evidence Prob P(E): 1.0000

Playground Sandbox

In a Bayesian network, observing evidence propagates probabilities through the graph, updating all nodes. A key feature of Bayesian networks is "explaining away" (also known as diagnostic inference in v-structures).

Tip: Observe how the probability of an Accident drops in Step 2 because the Road Construction explains why there is congestion, reducing the mathematical likelihood of an accident!

Reasoning Log

BBN Playground initialized. Try setting evidence above or running the steps.

BBN Mathematical Foundations

Bayesian Belief Networks encode the Joint Probability Distribution of all variables. Because of the network's conditional independencies, the full joint probability factors into the product of local conditional distributions:

\[P(D, T, W, C, A, BD, RC, TC) = P(D) \cdot P(T) \cdot P(W) \cdot P(C|D) \cdot P(A|T,W) \cdot P(BD|D,T) \cdot P(RC|C,A) \cdot P(TC|BD,RC,W)\]

When you set evidence, the engine filters this joint table, normalizes the remaining cells, and marginalizes to find the posteriors:

\[P(\text{Query} \mid \text{Evidence}) = \frac{P(\text{Query}, \text{Evidence})}{P(\text{Evidence})} = \frac{\sum_{\text{Others}} P(\text{All Variables} \cap \text{Evidence})}{\sum_{\text{All}} P(\text{All Variables} \cap \text{Evidence})}\]

Live Calculus

01

What is a BBN?

A Bayesian Belief Network is a Directed Acyclic Graph (DAG) whose nodes represent random variables and whose edges represent conditional dependencies. A BBN allows us to combine historical data with expert knowledge to make decisions under uncertainty.

02

CPT (Conditional Probability Tables)

Every node has a Conditional Probability Table (CPT) that defines its probability distribution given its parents. If a node has no parents (like Weather), its CPT contains simple prior probabilities. If it has parents, the table defines the distribution for every state combination of its parents.

03

Belief Propagation

When evidence is set, probabilities are updated using Bayes' Theorem. Crucially, evidence propagates both forwards (predictive reasoning, from cause to effect) and backwards (diagnostic reasoning, from effect to cause). Try clicking Traffic Congestion = High to see the Accident probability rise!

04

D-Separation & Independence

BBNs utilize graphical structure to determine whether variables are conditionally independent. For example, once we know the Road Capacity and Base Demand, Traffic Congestion becomes conditionally independent of whether there is an Accident or Road Construction.