Contingency Table Fit

Definition

Contingency Table Fit (CTF) measures the quality of the representation of the Joint Probability Distribution by a Bayesian network $B$ compared to a complete (i.e., fully-connected) network $C$.

BayesiaLab's CTF is defined as:

$${C_B} = 100 \times \frac{{{H_U}({\cal D}) - {H_B}({\cal D})}}{{{H_U}({\cal D}) - {H_C}({\cal D})}}$$

where

• ${{H_U}({\cal D})}$ is the entropy of the data with the unconnected network $U$.

• ${{H_B}({\cal D})}$ is the entropy of the data with the evaluated network $B$.

• ${{H_C}({\cal D})}$ is the entropy of the data with the complete (i.e., fully connected) network $C$. In the complete network, all nodes are directly connected to all other nodes. Therefore, the complete network $C$ is an exact representation of the chain rule. As such, it does not utilize any conditional independence assumptions for representing the Joint Probability Distribution.

Interpretation

• ${C_B}$ is equal to 100 if the Joint Probability Distribution is represented without any approximation, i.e., the entropy of the evaluated network $B$ is the same as that obtained with the complete network $C$.

• ${C_B}$ is equal to 0 if the Joint Probability Distribution is represented by considering that all the variables are independent, i.e., the entropy of the evaluated network B is the same as the one obtained with the unconnected network $U$.

• ${C_B}$ can also be negative if the parameters of network $B$ do not correspond to the dataset.