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I'd like to comparare whether partitioning of a dataset is justified. The data is categorical with two levels and the fitted parameter is the prevalence of positives for a certain condition in each partition. I thought doing this by comparing the partioned model against the unpartitioned one.

In one model, one parameter is fitted, simply through the Beta prior conjugate.

In the second, data is split according to some criterion and two parameters are fitted, one for each group.

How do I decide if the partitioning made sense?

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There are variations of the Bayesian Information Criterion that are appropriate for categorical data. For example, Bayesian criterion based model assessment for categorical data. Unfortunately, I dont know of any implementations.

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