On Bayesian analyses and finite mixtures for proportions

When the results of biological experiments are tested for a possible difference between treatment and control groups, the inference is only valid if based upon a model that fits the experimental results satisfactorily. In dominant-lethal testing, foetal death has previously been assumed to follow a variety of models, including a Poisson, Binomial, Beta-binomial and various mixture models. However, discriminating between models has always been a particularly difficult problem. In this paper, we consider the data from 6 separate dominant-lethal assay experiments and discriminate between the competing models which could be used to describe them. We adopt a Bayesian approach and illustrate how a variety of different models may be considered, using Markov chain Monte Carlo (MCMC) simulation techniques and comparing the results with the corresponding maximum likelihood analyses. We present an auxiliary variable method for determining the probability that any particular data cell is assigned to a given component in a mixture and we illustrate the value of this approach. Finally, we show how the Bayesian approach provides a natural and unique perspective on the model selection problem via reversible jump MCMC and illustrate how probabilities associated with each of the different models may be calculated for each data set. In terms of estimation we show how, by averaging over the different models, we obtain reliable and robust inference for any statistic of interest.