**Sankhya: The Indian Journal of Statistics**

1996, Volume 58, Series A, Pt. 2, pp. 292--310

**A NEW LOOK AT BAYESIAN POINT NULL HYPOTHESIS TESTING**

By

SANJIB BASU, *Northern Illinois University*

SUMMARY. Bayesian point null hypothesis testing for $H_0 : \theta = \theta_0 $ typically uses a prior with a point mass at $\theta_0$. The alternative method of this article does not require such spiked priors and uses a posterior probability of the HPD set, which contains $\theta_0$ and has the smallest volume among such HPD sets, as a measure of evidence against $H_0$. For parametric prior classes, lower bounds are derived on the evidence of $H_0$. When the prior class is nonparametric, the lower bounds are connected to the smallest volume sets of given posterior probability contents. For arbitrary contamination classes and density ratio classes, these smallest volume sets are explicitly derived. When only symmetric unimodal contaminations are allowed, the problem of finding these sets are reduced to univariate minimization problems, which then can be attacked numerically. Several examples illustrate applications of these methods.

*AMS (1991) subject classification.* 62F15, 62F03, 62G35, 62F25.

*Key words and phrases.* Bayes factor, contamination, density ratio class, HPD set, Lebesgue measure, lower bound on probabilities, point null hypothesis, posterior, $p$-value, robust Bayes, symmetry, unimodality.