**A** new arXival on nested sampling: “A statistical test for nested sampling algorithms” by Johannes Buchner. The point of the test is to check if versions of the nested sampling algorithm that fail to guarantee increased likelihood (or nesting) at each step are not missing parts of the posterior mass. and hence producing biased evidence approximations. This applies to MultiNest for instance. This version of nest sampling evaluates the above-threshold region by drawing hyper-balls around the remaining points. A solution which is known to fail in one specific but meaningful case. Buchner’s arXived paper proposes an hyper-pyramid distribution for which the volume of any likelihood constrained set is known. Hence allowing for a distribution test like Kolmogorov-Smirnov. Confirming the findings of Beaujean and Caldwell (2013). The author then proposes an alternative to MultiNest that is more robust but also much more costly as it computes distances between all pairs of bootstrapped samples. This solution passes the so-called “shrinkage test”, but it is orders of magnitude less efficient than MultiNest. And also simply shows that its coverage is fine for a specific target rather than all possible targets. I wonder if a solution to the problem is at all possible given that evaluating a support or a convex hull is a complex problem which complexity explodes with the dimension.

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