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What is the math behind CTF and Deviance?
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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:contingencyTableFit.png  where:contingencyTableFit1.png  is the mean of the log-likelihood of the data given the fully unconnected network (UN)contingencyTableFit2.png  is the mean of the log-likelihood of the data given the evaluated network (BBN)contingencyTableFit3.png  is the mean of the log-likelihood of the data given the fully connected network (FCN)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 in the representation of the joint probability distribution.[*:12270or5]C is equal to 100 when the joint probability distribution is represented without any approximation, i.e. the same log likelihood as the one obtained with the fully connected network[/*:m:12270or5][*:12270or5]C is equal to 0 when the joint probability distribution is represented by considering that all the variables are independent, i.e. the same joint probability distribution as the one obtained with the fully unconnected network[/*:m:12270or5]C can also be negative, if the parameters of the BBN do not correspond to the dataset.Deviance:Deviance is defined as:deviance.png where N is the size of the dataset.Fully Unconnected Network:fullyUnconnectedNetwork.png fullyUnconnectedNetwork1.png fullyUnconnectedNetwork2.png Fully Connected Network:fullyConnectedNetwork.png fullyConnectedNetwork1.png BBN Learned:BBNLearned.png BBNLearned1.png 
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