arXiv subjects

Dirk P. Kroese

Publications and source records attributed to Dirk P. Kroese.

10 recordsLinked to original sources

arXiv2025

Unbiased and Consistent Nested Sampling via Sequential Monte Carlo

We introduce a new class of sequential Monte Carlo methods which reformulates the essence of the nested sampling method of Skilling (2006) in terms of sequential Monte Carlo techniques. Two new algorithms are proposed, nested sampling via sequential Monte Carlo (NS-SMC) and adaptive nested sampling via sequential Monte Carlo (ANS-SMC). The new framework allows convergence results to be obtained in the setting when Markov chain Monte Carlo (MCMC) is used to produce new samples. An additional benefit is that marginal likelihood (normalising constant) estimates given by NS-SMC are unbiased. In contrast to NS, the analysis of our proposed algorithms does not require the (unrealistic) assumption that the simulated samples be independent. We show that a minor adjustment to our ANS-SMC algorithm recovers the original NS algorithm, which provides insights as to why NS seems to produce accurate estimates despite a typical violation of its assumptions. A numerical study is conducted where the performance of the proposed algorithms and temperature-annealed SMC is compared on challenging problems. Code for the experiments is made available online at https://github.com/LeahPrice/SMC-NS .

stat.CO↗

arXiv2020

Rare Events in Random Geometric Graphs

This work introduces and compares approaches for estimating rare-event probabilities related to the number of edges in the random geometric graph on a Poisson point process. In the one-dimensional setting, we derive closed-form expressions for a variety of conditional probabilities related to the number of edges in the random geometric graph and develop conditional Monte Carlo algorithms for estimating rare-event probabilities on this basis. We prove rigorously a reduction in variance when compared to the crude Monte Carlo estimators and illustrate the magnitude of the improvements in a simulation study. In higher dimensions, we leverage conditional Monte Carlo to remove the fluctuations in the estimator coming from the randomness in the Poisson number of nodes. Finally, building on conceptual insights from large-deviations theory, we illustrate that importance sampling using a Gibbsian point process can further substantially reduce the estimation variance.

math.PR↗

arXiv2019

Unbiased Estimation of the Reciprocal Mean for Non-negative Random Variables

Many simulation problems require the estimation of a ratio of two expectations. In recent years Monte Carlo estimators have been proposed that can estimate such ratios without bias. We investigate the theoretical properties of such estimators for the estimation of $β= 1/\mathbb{E}\, Z$, where $Z \geq 0$. The estimator, $\widehat β(w)$, is of the form $w/f_w(N) \prod_{i=1}^N (1 - w\, Z_i)$, where $w < 2β$ and $N$ is any random variable with probability mass function $f_w$ on the positive integers. For a fixed $w$, the optimal choice for $f_w$ is well understood, but less so the choice of $w$. We study the properties of $\widehat β(w)$ as a function of~$w$ and show that its expected time variance product decreases as $w$ decreases, even though the cost of constructing the estimator increases with $w$. We also show that the estimator is asymptotically equivalent to the maximum likelihood (biased) ratio estimator and establish practical confidence intervals.

math.ST↗

arXiv2019

Perfect Sampling for Gibbs Point Processes Using Partial Rejection Sampling

We present a perfect sampling algorithm for Gibbs point processes, based on the partial rejection sampling of Guo et al. (2017). Our particular focus is on pairwise interaction processes, penetrable spheres mixture models and area-interaction processes, with a finite interaction range. For an interaction range $2r$ of the target process, the proposed algorithm can generate a perfect sample with $O(\log(1/r))$ expected running time complexity, provided that the intensity of the points is not too high.

math.PR↗

arXiv2017

Stratified Splitting for Efficient Monte Carlo Integration

The efficient evaluation of high-dimensional integrals is of importance in both theoretical and practical fields of science, such as data science, statistical physics, and machine learning. However, exact computation methods suffer from the curse of dimensionality. However, due to the curse of dimensionality, deterministic numerical methods are inefficient in high-dimensional settings. Consequentially, for many practical problems, one must resort to Monte Carlo estimation. In this paper, we introduce a novel Sequential Monte Carlo technique called Stratified Splitting. The method provides unbiased estimates and can handle various integrand types including indicator functions, which are used in rare-event probability estimation problems. Moreover, we demonstrate that a variant of the algorithm can achieve polynomial complexity. The results of our numerical experiments suggest that the Stratified Splitting method is capable of delivering accurate results for a variety of integration problems.

math.ST↗

arXiv2015

Fitting Laguerre tessellation approximations to tomographic image data

The analysis of polycrystalline materials benefits greatly from accurate quantitative descriptions of their grain structures. Laguerre tessellations approximate such grain structures very well. However, it is a quite challenging problem to fit a Laguerre tessellation to tomographic data, as a high-dimensional optimization problem with many local minima must be solved. In this paper, we formulate a version of this optimization problem that can be solved quickly using the cross-entropy method, a robust stochastic optimization technique that can avoid becoming trapped in local minima. We demonstrate the effectiveness of our approach by applying it to both artificially generated and experimentally produced tomographic data.

cond-mat.mtrl-sci↗

arXiv2015

Estimating Residual Connectivity for Random Graphs

Computation of the probability that a random graph is connected is a challenging problem, so it is natural to turn to approximations such as Monte Carlo methods. We describe sequential importance resampling and splitting algorithms for the estimation of these probabilities. The importance sampling steps of these algorithms involve identifying vertices that must be present in order for the random graph to be connected, and conditioning on the corresponding events. We provide numerical results demonstrating the effectiveness of the proposed algorithm.

stat.CO↗

arXiv2015

CEoptim: Cross-Entropy R Package for Optimization

The cross-entropy (CE) method is simple and versatile technique for optimization, based on Kullback-Leibler (or cross-entropy) minimization. The method can be applied to a wide range of optimization tasks, including continuous, discrete, mixed and constrained optimization problems. The new package CEoptim provides the R implementation of the CE method for optimization. We describe the general CE methodology for optimization and well as some useful modifications. The usage and efficacy of CEoptim is demonstrated through a variety of optimization examples, including model fitting, combinatorial optimization, and maximum likelihood estimation.

stat.CO↗

arXiv2014

Computing the Drift of Random Walks in Dependent Random Environments

Although the theoretical behavior of one-dimensional random walks in random environments is well understood, the numerical evaluation of various characteristics of such processes has received relatively little attention. This paper develops new theory and methodology for the computation of the drift of the random walk for various dependent random environments, including $k$-dependent and moving average environments.

math.PR↗

arXiv2013

Spatial Process Generation

The generation of random spatial data on a computer is an important tool for understanding the behavior of spatial processes. In this paper we describe how to generate realizations from the main types of spatial processes, including Gaussian and Markov random fields, point processes, spatial Wiener processes, and Levy fields. Concrete MATLAB code is provided.

stat.CO↗