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  • Contigency Table Fit and Deviance Formulas

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Question

What is the math behind Contingency Table Fit (CTF) and Deviance?

Answer

The CTF measures the quality of the representation of the Joint Probability Distribution by your network (with respect to the fully connected network). 

Contingency Table Fit

BayesiaLab's CTF is defined as:


where

 is the entropy of the data with the fully unconnected network U

 is the entropy of the data with the evaluated network B

 is the entropy of the data with the fully connected network F

The fully connected network is a graph in which all nodes have a direct link with all other ones. Therefore, this is the exact representation of the chain rule, without any conditional independence assumptions utilized for representing the joint probability distribution.

  • C is equal to 100 when the joint probability distribution is represented without any approximation, i.e. the same data entropy as the one obtained with the fully connected network
  • C is equal to 0 when the joint probability distribution is represented by considering that all the variables are independent, i.e. data entropy as the one obtained with the fully unconnected network

C can also be negative, if the parameters of the B do not correspond to the dataset.

Deviance

Deviance is defined as:

where N is the size of the dataset.

Example

The Contingency Table Fit (CTF) and Deviance can be computed for the current network by evaluating its overall performance (Analysis | Network Performance | Overall).

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