Point process-based Monte Carlo estimation

Abstract

This paper addresses the issue of estimating the expectation of a real-valued random variable of the form X = g(U) where g is a deterministic function and U can be a random finite-or infinite-dimensional vector. Using recent results on rare event simulation, we propose a unified framework for dealing with both probability and mean estimation for such random variables, i.e. linking algorithms such as Tootsie Pop Algorithm (TPA) or Last Particle Algorithm with nested sampling. Especially, it extends nested sampling as follows: first the random variable X does not need to be bounded any more: it gives the principle of an ideal estimator with an infinite number of terms that is unbiased and always better than a classical Monte Carlo estimator -in particular it has a finite variance as soon as there exists k ∈ R > 1 such that E X k < ∞. Moreover we address the issue of nested sampling termination and show that a random truncation of the sum can preserve unbiasedness while increasing the variance only by a factor up to 2 compared to the ideal case. We also build an unbiased estimator with fixed computational budget which supports a Central Limit Theorem and discuss parallel implementation of nested sampling, which can dramatically reduce its computational cost. Finally we extensively study the case where X is heavy-tailed.

Rare event simulation and splitting for discontinuous random variables

“…In this section, we recall common results from [17,19,28] in the framework of [30]. Let us consider X = S(U) ∈ R a real-valued random variable with distribution µ X where S is a deterministic function (for instance the output of a computer code) and U a random finite-or infinite-dimensional vector with known distribution µ U .…”

“…It has the smallest variance amongst all AMS [7]; especially Simonnet [28] and Guyader et al [17] showed that the random number of iterations of the algorithm follows a Poisson law when X is continuous. Moreover Walter [30], following Huber and Schott [19], brought an original insight in terms of a random walk of the real-valued random variable X = S(U) linked with a Poisson Process with parameter 1. Indeed the estimator is found to be the Minimal Variance Unbiased Estimator (MVUE) of the exponential of a parameter of a Poisson law with N iid.…”

Dynamic nested sampling: an improved algorithm for parameter estimation and evidence calculation

“…and rare event simulation (Walter, 2017). The original development of the nested sampling algorithm was motivated by evidence calculation, but the MultiNest (Feroz and Hobson, 2008;Feroz et al, 2008Feroz et al, , 2013 and PolyChord (Handley et al, 2015a,b) software packages are now extensively used for parameter estimation from posterior samples (such as in DES Collaboration, 2018).…”

Advances in Monte Carlo methodology

“…Further to this, [52,Remark 1] shows that this estimator results in superior estimates over exp(−t/N) in terms of variance so long as p t > exp(−1). In light of this, Walter suggests using Riemann sum quadrature using these alternative values for p t as it will result in a superior NS estimator.…”

“…However, this is somewhat out of place with the rest of the quadrature. Using point process theory and techniques from the literature on unbiased estimation, Walter [52] proposes an unbiased version of NS. However, this unbiasedness relies on the assumption of independent sampling, and comes with a cost of additional variance.…”