Baysian estimation of P(X>x) from a small sample of Gaussian data

The classical statistical uncertainty problem of estimation of upper tail probabilities on the basis of a small sample of observations of a Gaussian random variable is considered. Predictive posterior estimation is discussed, adopting the standard statistical model with diffuse priors of the two normal distribution parameters. Rarely the uncertainty of the predictive estimate itself is quantified in practice. By considering the exceedance probability as a random variable over the posterior probability distribution of the parameters, an explicit expression for the distribution of this random variable is obtained. It is shown that the usual elementary estimate based on the normal distribution is very close to the median of this distribution. For increasing exceedance level the distribution skewness increases so that the predictive estimate, which is equal to the mean of the distribution, comes further and further out in the upper tail of the distribution. The dual frequentist's confidence interval approach is shown to have difficulties not present for the Bayesian approach.