Annealed importance sampling

Simulated annealing—moving from a tractable distribution to a distribution of interest via a sequence of intermediate distributions—has traditionally been used as an inexact method of handling isolated modes in Markov chain samplers. Here, it is shown how one can use the Markov chain transitions for such an annealing sequence to define an importance sampler. The Markov chain aspect allows this method to perform acceptably even for high-dimensional problems, where finding good importance sampling distributions would otherwise be very difficult, while the use of importance weights ensures that the estimates found converge to the correct values as the number of annealing runs increases. This annealed importance sampling procedure resembles the second half of the previously-studied tempered transitions, and can be seen as a generalization of a recently-proposed variant of sequential importance sampling. It is also related to thermodynamic integration methods for estimating ratios of normalizing constants. Annealed importance sampling is most attractive when isolated modes are present, or when estimates of normalizing constants are required, but it may also be more generally useful, since its independent sampling allows one to bypass some of the problems of assessing convergence and autocorrelation in Markov chain samplers.     

The Gibbs sampler with particle efficient importance sampling for state-space models*

We consider Particle Gibbs (PG) for Bayesian analysis of non-linear non-Gaussian state-space models. As a Monte Carlo (MC) approximation of the Gibbs procedure, PG uses sequential MC (SMC) importance sampling inside the Gibbs to update the latent states. We propose to combine PG with the Particle Efficient Importance Sampling (PEIS). By using SMC sampling densities which are approximately globally fully adapted to the targeted density of the states, PEIS can substantially improve the simulation efficiency of the PG relative to existing PG implementations. The efficiency gains are illustrated in PG applications to a non-linear local-level model and stochastic volatility models.     

Daisee: Adaptive importance sampling by balancing exploration and exploitation

We study adaptive importance sampling (AIS) as an online learning problem and argue for the importance of the trade-off between exploration and exploitation in this adaptation. Borrowing ideas from the online learning literature, we propose Daisee, a partition-based AIS algorithm. We further introduce a notion of regret for AIS and show that Daisee has $𝒪(\sqrt{T}{(\log T)}^{\frac{3}{4}})$ cumulative pseudo-regret, where $T$ is the number of iterations. We then extend Daisee to adaptively learn a hierarchical partitioning of the sample space for more efficient sampling and confirm the performance of both algorithms empirically.     

Estimating likelihoods for spatio-temporal models using importance sampling

This paper describes how importance sampling can be applied to estimate likelihoods for spatio-temporal stochastic models of epidemics in plant populations, where observations consist of the set of diseased individuals at two or more distinct times. Likelihood computation is problematic because of the inherent lack of independence of the status of individuals in the population whenever disease transmission is distance-dependent. The methods of this paper overcome this by partitioning the population into a number of sectors and then attempting to take account of this dependence within each sector, while neglecting that between-sectors. Application to both simulated and real epidemic data sets show that the techniques perform well in comparison with existing approaches. Moreover, the results confirm the validity of likelihood estimates obtained elsewhere using Markov chain Monte Carlo methods.     

A bayesian treatpent of nonresponse when sampling from a dichotomous population

In the situation where significant nonresponse is present in a sample survey, a prior distribution is developed which can reflect vague prior beliefs about the differences in the attitudes of respondents and nonrespondents. A posterior interval estimate is derived for the population proportion of interest.     

A non-iterative sampling Bayesian method for linear mixed models with normal independent distributions

In this paper, we utilize normal/independent (NI) distributions as a tool for robust modeling of linear mixed models (LMM) under a Bayesian paradigm. The purpose is to develop a non-iterative sampling method to obtain i.i.d. samples approximately from the observed posterior distribution by combining the inverse Bayes formulae, sampling/importance resampling and posterior mode estimates from the expectation maximization algorithm to LMMs with NI distributions, as suggested by Tan et al. [33 Tan, M., Tian, G. and Ng, K. 2003. A noniterative sampling method for computing posteriors in the structure of EM-type algorithms. Statist. Sinica, 13(3): 625–640. [Web of Science ®] , [Google Scholar]]. The proposed algorithm provides a novel alternative to perfect sampling and eliminates the convergence problems of Markov chain Monte Carlo methods. In order to examine the robust aspects of the NI class, against outlying and influential observations, we present a Bayesian case deletion influence diagnostics based on the Kullback–Leibler divergence. Further, some discussions on model selection criteria are given. The new methodologies are exemplified through a real data set, illustrating the usefulness of the proposed methodology.     

A hybrid approach based on saddlepoint and importance sampling methods for bootstrap tail probability estimation

We propose a simple hybrid method which makes use of both saddlepoint and importance sampling techniques to approximate the bootstrap tail probability of an M-estimator. The method does not rely on explicit formula of the Lugannani-Rice type, and is computationally more efficient than both uniform bootstrap sampling and importance resampling suggested in earlier literature. The method is also applied to construct confidence intervals for smooth functions of M-estimands.     

Sequential importance sampling for nonparametric Bayes models: The next generation

There are two generations of Gibbs sampling methods for semiparametric models involving the Dirichlet process. The first generation suffered from a severe drawback: the locations of the clusters, or groups of parameters, could essentially become fixed, moving only rarely. Two strategies that have been proposed to create the second generation of Gibbs samplers are integration and appending a second stage to the Gibbs sampler wherein the cluster locations are moved. We show that these same strategies are easily implemented for the sequential importance sampler, and that the first strategy dramatically improves results. As in the case of Gibbs sampling, these strategies are applicable to a much wider class of models. They are shown to provide more uniform importance sampling weights and lead to additional Rao-Blackwellization of estimators.     

Robust importance sampling for some typical types of utility-based shortfall risk measures using exponential twisting and kernel density techniques

A robust algorithm for utility-based shortfall risk (UBSR) measures is developed by combining the kernel density estimation with importance sampling (IS) using exponential twisting techniques. The optimal bandwidth of the kernel density is obtained by minimizing the mean square error of the estimators. Variance is reduced by IS where exponential twisting is applied to determine the optimal IS distribution. Conditions for the best distribution parameters are derived based on the piecewise polynomial loss function and the exponential loss function. The proposed method not only solves the problem of sampling from the kernel density but also reduces the variance of the UBSR estimator.     

Optimal estimators for the importance sampling method

The Monte Carlo method gives some estimators to evaluate the expectation [ILM0001] based on samples from either the true density f or from some instrumental density. In this paper, we show that the Riemann estimators introduced by Philippe (1997) can be improved by using the importance sampling method. This approach produces a class of Monte Carlo estimators such that the variance is of order O(n ^?2). The choice of an optimal estimator among this class is discussed. Some simulations illustrate the improvement brought by this method. Moreover, we give a criterion to assess the convergence of our optimal estimator to the integral of interest.     

Efficient variance reduction approach based on the variation of the input importance

ABSTRACT

To reduce the output variance, the variance-based importance analysis can provide an efficient way by reducing the variance of the ‘important’ inputs. But with the reduction of the variance of those ‘important’ inputs, the input importance will change and it is no longer the most efficient way to reduce the variance of those ‘important’ inputs alone. Thus, analyst needs to consider reducing the variance of other inputs to obtain a more efficient way. This work provides a graphical solution for analyst to decide how to reduce the input variance to achieve the targeted reduction of the output variance efficiently. Furthermore, by the importance sampling-based approach, the graphical solution can be obtained with only a single group of samples, which can decrease the computational cost greatly.     

Saddlepoint approximations to sensitivities of tail probabilities of random sums and comparisons with Monte Carlo estimators

This article proposes computing sensitivities of upper tail probabilities of random sums by the saddlepoint approximation. The considered sensitivity is the derivative of the upper tail probability with respect to the parameter of the summation index distribution. Random sums with Poisson or Geometric distributed summation indices and Gamma or Weibull distributed summands are considered. The score method with importance sampling is considered as an alternative approximation. Numerical studies show that the saddlepoint approximation and the method of score with importance sampling are very accurate. But the saddlepoint approximation is substantially faster than the score method with importance sampling. Thus, the suggested saddlepoint approximation can be conveniently used in various scientific problems.     

Hit and run ARMS: adaptive rejection Metropolis sampling with hit and run random direction

An algorithm for sampling from non-log-concave multivariate distributions is proposed, which improves the adaptive rejection Metropolis sampling (ARMS) algorithm by incorporating the hit and run sampling. It is not rare that the ARMS is trapped away from some subspace with significant probability in the support of the multivariate distribution. While the ARMS updates samples only in the directions that are parallel to dimensions, our proposed method, the hit and run ARMS (HARARMS), updates samples in arbitrary directions determined by the hit and run algorithm, which makes it almost not possible to be trapped in any isolated subspaces. The HARARMS performs the same as ARMS in a single dimension while more reliable in multidimensional spaces. Its performance is illustrated by a Bayesian free-knot spline regression example. We showed that it overcomes the well-known ‘lethargy’ property and decisively find the global optimal number and locations of the knots of the spline function.     

Adaptive rejection sampling with fixed number of nodes

The adaptive rejection sampling (ARS) algorithm is a universal random generator for drawing samples efficiently from a univariate log-concave target probability density function (pdf). ARS generates independent samples from the target via rejection sampling with high acceptance rates. Indeed, ARS yields a sequence of proposal functions that converge toward the target pdf, so that the probability of accepting a sample approaches one. However, sampling from the proposal pdf becomes more computational demanding each time it is updated. In this work, we propose a novel ARS scheme, called Cheap Adaptive Rejection Sampling (CARS), where the computational effort for drawing from the proposal remains constant, decided in advance by the user. For generating a large number of desired samples, CARS is faster than ARS.     

Markov chain importance sampling with applications to rare event probability estimation

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.     

Importance sampling with the generalized exponential power density

In this paper, the generalized exponential power (GEP) density is proposed as an importance function in Monte Carlo simulations in the context of estimation of posterior moments of a location parameter. This density is divided in five classes according to its tail behaviour which may be exponential, polynomial or logarithmic. The notion of p-credence is also defined to characterize and to order the tails of a large class of symmetric densities by comparing their tails to those of the GEP density.The choice of the GEP density as an importance function allows us to obtain reliable and effective results when p-credences of the prior and the likelihood are defined, even if there are conflicting sources of information. Characterization of the posterior tails using p-credence can be done. Hence, it is possible to choose parameters of the GEP density in order to have an importance function with slightly heavier tails than the posterior. Simulation of observations from the GEP density is also addressed.     

Stochastic comparisons of series and parallel systems with independent heterogeneous lower-truncated Weibull components

Abstract

In this paper, we discuss stochastic comparisons of series and parallel systems with independent heterogeneous lower-truncated Weibull components. When a system with possibly different shape and scale parameters and its matrix of parameters changes to another matrix in a certain mathematical sense, we study the hazard rate order of lifetimes of series systems and the usual stochastic order of lifetimes of parallel systems.     

Bounds and approximations for pairwise comparisons of independent multinormal means

Properties and relationships of some commonly used probability bounds, along with other recently developed bounds and approximations, are evaluated for their performance with pairwise comparisons. The comparisons are of independent sample means obtained from normal random variables with a common variance. Computational methods are presented and numerical results are used to further evaluate the performance of the bounds.