# Conditional Probability Table (CPT)

### Context

Bayesian networks are models that consist of two parts:

A qualitative part to represent the dependencies using a Directed Acyclic Graph (DAG).

A quantitative part, using local probability distributions, for specifying the probabilistic relationships.

A Directed Acyclic Graph (DAG) consists of nodes and directed links:

Nodes represent variables of interest (e.g., the temperature of a device, the gender of a patient, a feature of an object, or the occurrence of an event).

Nodes can correspond to symbolic/categorical variables, numerical variables with discrete values, or discretized continuous variables.

Directed arcs represent statistical (informational) or causal dependencies among the variables. The directions are used to define kinship relations, i.e., parent-child relationships.

For example, in a Bayesian network with an arc from X to Y, X is the parent node of Y, and Y is the child node.

The local probability distributions can be either marginal for nodes without parents (Root Nodes) or conditional for nodes with parents.

In the latter case, the dependencies are quantified by Conditional Probability Tables (CPT) for each node given its parents in the Directed Acyclic Graph (DAG).

Once fully specified, a Bayesian network compactly represents the Joint Probability Distribution (JPD).

Thus, the Bayesian network can be used for computing the posterior probabilities of any subset of nodes given evidence set on any other subset.

### Example

The following illustration shows a simple Bayesian network, which consists of only two nodes and one directed arc.

This Bayesian network represents the Joint Probability Distribution (JPD) of the variables Eye Color and Hair Color in a population of students (Snee, 1974).

Eye Color is a Root Node and, therefore, does not have any Parents. In other words, Eye Color does not depend on any other node.

As a result, the table associated with Eye Color is a Probability Table, i.e., it represents the marginal distribution of Eye Color unconditionally.

On the other hand, the probabilities of Hair Color are only defined conditionally upon the values of its parent node, Eye Color.

Hence, the probabilities of Hair Color are provided in a Conditional Probability Table (CPT).

It is important to point out that this Bayesian network does not imply any causal relationships, even though the arc direction may suggest that to a casual observer.

The arc direction merely defines the parent-child relationship of the nodes for purposes of representing the Joint Probability Distribution (JPD).

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