# Conditional Probability Table (CPT)

### Context

• Bayesian networks are models that consist of two parts:

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

2. 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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