“Objective” low informative priors for Bayesian inference from totally censored Gaussian data

Consider structural elements with random strength that after a suitable transformation has normal distribution with unknown mean μ and known or unknown standard deviation σ. By proof testing of n of these structural elements to a given load level it is observed that none of the elements fail. Given solely this test information the problem is that in order to state anything about either the value of μ when σ is known or about the values of μ and σ when both parameters are unknown, it is necessary to introduce some more information in the form of a suitable prior distribution of the parameters, that is, to use a Bayesian procedure with an informative prior. The paper considers the problem of defining such a prior in an axiomatic (“objective”) way without extending the information represented by the test results by more than very little extra information based on common physical sense. The solution suggested in the paper implies that the posterior distribution of the mean shifts towards larger values when the sample size n increases. However, convergence to a specific value is not obtained as long as no failures are observed among the tests. Moreover it turns out that the posterior distribution of the standard deviation is invariant to the sample size n, that is, no updating of the standard deviation is obtained as long as there are no failures among the tests.