## an extension of nested sampling

I was reading [in the Paris métro] Hastings-Metropolis algorithm on Markov chains for small-probability estimation, arXived a few weeks ago by François Bachoc, Lionel Lenôtre, and Achref Bachouch, when I came upon their first algorithm that reminded me much of nested sampling: the following was proposed by Guyader et al. in 2011,

To approximate a tail probability P(H(X)>h),

• start from an iid sample of size N from the reference distribution;
• at each iteration m, select the point x with the smallest H(x)=ξ and replace it with a new point y simulated under the constraint H(y)≥ξ;
• stop when all points in the sample are such that H(X)>h;
• take

$\left(1-\dfrac{1}{N}\right)^{m-1}$

as the unbiased estimator of P(H(X)>h).

Hence, except for the stopping rule, this is the same implementation as nested sampling. Furthermore, Guyader et al. (2011) also take advantage of the bested sampling fact that, if direct simulation under the constraint H(y)≥ξ is infeasible, simulating via one single step of a Metropolis-Hastings algorithm is as valid as direct simulation. (I could not access the paper, but the reference list of Guyader et al. (2011) includes both original papers by John Skilling, so the connection must be made in the paper.) What I find most interesting in this algorithm is that it even achieves unbiasedness (even in the MCMC case!).