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## Sequential Monte Carlo for counting vertex covers in general graphs

### Statistics and Computing (2016-05-01) 26: 591-607 , May 01, 2016

In this paper we describe a sequential importance sampling (SIS) procedure for counting the number of vertex covers in general graphs. The optimal SIS proposal distribution is the uniform over a suitably restricted set, but is not implementable. We will consider two proposal distributions as approximations to the optimal. Both proposals are based on randomization techniques. The first randomization is the classic probability model of random graphs, and in fact, the resulting SIS algorithm shows polynomial complexity for random graphs. The second randomization introduces a probabilistic relaxation technique that uses Dynamic Programming. The numerical experiments show that the resulting SIS algorithm enjoys excellent practical performance in comparison with existing methods. In particular the method is compared with *cachet*—an exact model counter, and the state of the art *SampleSearch*, which is based on Belief Networks and importance sampling.

## Non-asymptotic Bandwidth Selection for Density Estimation of Discrete Data

### Methodology and Computing in Applied Probability (2008-04-26) 10: 435-451 , April 26, 2008

We propose a new method for density estimation of categorical data. The method implements a non-asymptotic data-driven bandwidth selection rule and provides model sparsity not present in the standard kernel density estimation method. Numerical experiments with a well-known ten-dimensional binary medical data set illustrate the effectiveness of the proposed approach for density estimation, discriminant analysis and classification.

## Efficient Monte Carlo simulation via the generalized splitting method

### Statistics and Computing (2012-01-01) 22: 1-16 , January 01, 2012

We describe a new Monte Carlo algorithm for the consistent and unbiased estimation of multidimensional integrals and the efficient sampling from multidimensional densities. The algorithm is inspired by the classical splitting method and can be applied to general static simulation models. We provide examples from rare-event probability estimation, counting, and sampling, demonstrating that the proposed method can outperform existing Markov chain sampling methods in terms of convergence speed and accuracy.

## The Generalized Cross Entropy Method, with Applications to Probability Density Estimation

### Methodology and Computing in Applied Probability (2011-03-01) 13: 1-27 , March 01, 2011

Nonparametric density estimation aims to determine the sparsest model that explains a given set of empirical data and which uses as few assumptions as possible. Many of the currently existing methods do not provide a sparse solution to the problem and rely on asymptotic approximations. In this paper we describe a framework for density estimation which uses information-theoretic measures of model complexity with the aim of constructing a sparse density estimator that does not rely on large sample approximations. The effectiveness of the approach is demonstrated through an application to some well-known density estimation test cases.

## Markov chain importance sampling with applications to rare event probability estimation

### Statistics and Computing (2013-03-01) 23: 271-285 , March 01, 2013

We present a versatile Monte Carlo method for estimating multidimensional integrals, with applications to rare-event probability estimation. The method fuses two distinct and popular Monte Carlo simulation methods—Markov chain Monte Carlo and importance sampling—into a single algorithm. We show that for some applied numerical examples the proposed Markov Chain importance sampling algorithm performs better than methods based solely on importance sampling or MCMC.

## An Efficient Algorithm for Rare-event Probability Estimation, Combinatorial Optimization, and Counting

### Methodology and Computing in Applied Probability (2008-12-01) 10: 471-505 , December 01, 2008

Although importance sampling is an established and effective sampling and estimation technique, it becomes unstable and unreliable for high-dimensional problems. The main reason is that the likelihood ratio in the importance sampling estimator degenerates when the dimension of the problem becomes large. Various remedies to this problem have been suggested, including heuristics such as resampling. Even so, the consensus is that for large-dimensional problems, likelihood ratios (and hence importance sampling) should be avoided. In this paper we introduce a new adaptive simulation approach that does away with likelihood ratios, while retaining the multi-level approach of the cross-entropy method. Like the latter, the method can be used for rare-event probability estimation, optimization, and counting. Moreover, the method allows one to sample exactly from the target distribution rather than asymptotically as in Markov chain Monte Carlo. Numerical examples demonstrate the effectiveness of the method for a variety of applications.