Precise Testing for Hardy-Weinberg Equilibrium in a Biological Population: An Objective Bayesian Analysis

Many of the problems which traditionally have been formulated in terms of hypothesis testing are really complex decision problems on model choice, whose appropriate solution naturally depends on the structure of the problem. In this paper a probability model for the formation of genotypes from two alleles is given and expressed in terms of two parameters, $\alpha$ and $\beta$ ; $\alpha=0$ corresponding to Hardy-Weinberg equilibrium (Lindley, 1988). A particular scientific hypothesis of genetical equilibrium is discussed, special attention is paid to considering that in some genetical applications the proportion of \textbf{A} alleles is known fairly precisely before sampling, the posterior distribution of $\alpha$ considering $\beta$ known is found providing estimation of $\alpha$. The corresponding precise hypothesis testing problem is considered from a decision-theoretical viewpoint, where the null hypothesis is rejected if the null model is expected to be too far from the true model in the logarithmic divergence (Kullback-Leibler) sense. The results are illustrated using examples with data previously analyzed in the literature