Abstract
Multinomial models can be difficult to use when constraints are placed on the probabilities. An exact model checking procedure for such models is developed based on a uniform prior on the full multinomial model. For inference, a nonuniform prior can be used and a consistency theorem is proved concerning a check for prior-data conflict with the chosen prior. Applications are presented and a new elicitation methodology is developed for multinomial models with ordered probabilities.
Highlights
Suppose we have a sample of n from a multinomial(1, θ1, . . . , θk+1) distribution and let T = (T1, . . . , Tk+1) denote the cell counts
Constrained multinomial models arise in a number of interesting contexts and pose some unique challenges
A model checking procedure that allows for evidence in favor of as well as evidence against a constrained multinomial has been presented
Summary
A different approach is taken in Bayarri and Berger (2000) and Castellanos and Bayarri (2007) where the check is based on the conditional distribution of the data given a statistic that is asymptotically minimally sufficient. As noted in Robins et al (2000), there is a problem with posterior predictive checks when based on p-values as these fail to have to have asymptotically uniform distributions when the model is correct and so can be misleading. Posterior distributions are used in inferences about parameters as necessitated by the principle of conditional probability which says that prior beliefs must be updated via conditioning on the observed data This says nothing about how checks on the sampling model or the prior should be conducted. Tail probabilities are used for checking the prior, but these are used as a measure of surprise and not as a measure of evidence as there is no sense in which a prior is true or false, only that it may be inappropriate
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