demystify Lindley’s paradox [or not]

Another paper on Lindley’s paradox appeared on arXiv yesterday, by Guosheng Yin and Haolun Shi, interpreting posterior probabilities as p-values. The core of this resolution is to express a two-sided hypothesis as a combination of two one-sided hypotheses along the opposite direction, taking then advantage of the near equivalence of posterior probabilities under some non-informative prior and p-values in the later case. As already noted by George Casella and Roger Berger (1987) and presumably earlier. The point is that one-sided hypotheses are quite friendly to improper priors, since they only require a single prior distribution. Rather than two when point nulls are under consideration. The p-value created by merging both one-sided hypotheses makes little sense to me as it means testing that both θ≥0 and θ≤0, resulting in the proposal of a p-value that is twice the minimum of the one-sided p-values, maybe due to a Bonferroni correction, although the true value should be zero… I thus see little support for this approach to resolving Lindley paradox in that it bypasses the toxic nature of point-null hypotheses that require a change of prior toward a mixture supporting one hypothesis and the other. Here the posterior of the point-null hypothesis is defined in exactly the same way the p-value is defined, hence making the outcome most favourable to the agreement but not truly addressing the issue.

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