weighted average importance sampling and defensive mixture distributions importance sampling uses observations from one distribution to estimate for another distribution by weighting the observations including the target distribution as one component of a mixture distribution bounds the weights and makes importance sampling more reliable the usual importance sampling estimate is a weighted average with weights that do not sum to 1 we discuss simple normalization and other more efficient normalization methods these innovations make importance sampling useful in a wider variety of problems we demonstrate with a case study of oil inventory reliability at a large utility

kardar parisi zhang equation in the weak noise limit pattern formation and upper critical dimension we extend the previously developed nonperturbative weak noise scheme applied to the noisy burgers equation in one dimension to the kardar parisi zhang equation for a growing interface in arbitrary dimensions by means of the cole hopf transformation we show that the growth morphology can be interpreted in terms of dynamically evolving textures of localized growth modes with superimposed diffusive modes in the cole hopf representation the growth modes are static solutions to the diffusion equation and the nonlinear schrodinger equation subsequently boosted to finite velocity by a galilei transformation we discuss the dynamics of the pattern formation and briefly the superimposed linear modes implementing the stochastic interpretation we discuss kinetic transitions and in particular the preliminary scaling properties pertaining to the pair mode or dipole sector in the dipole sector we obtain the hurst exponent h 3 d 4 d or dynamic exponent zdip 4 d 3 d for the random walk of growth modes below d 3 the dipole growth modes show anomalous diffusion above d 3 the dipole growth modes freeze finally applying derrick s theorem based on constrained minimization we show that the upper critical dimension is d 4 in the sense that growth modes cease to exist above this dimension

bayesian computation via the gibbs sampler and related markov chain monte carlo methods the use of the gibbs sampler for bayesian computation is reviewed and illustrated in the context of some canonical examples other markov chain monte carlo simulation methods are also briefly described and comments are made on the advantages of sample based approaches for bayesian inference summaries

central limit theorem for sequential monte carlo methods and its application to bayesian inference the term sequential monte carlo methods or equivalently particle filters refers to a general class of iterative algorithms that performs monte carlo approximations of a given sequence of distributions of interest t we establish in this paper a central limit theorem for the monte carlo estimates produced by these computational methods this result holds under minimal assumptions on the distributions t and applies in a general framework which encompasses most of the sequential monte carlo methods that have been considered in the literature including the resample move algorithm of gilks and berzuini j r stat soc ser b stat methodol 63 2001 127 146 and the residual resampling scheme the corresponding asymptotic variances provide a convenient measurement of the precision of a given particle filter we study in particular in some typical examples of bayesian applications whether and at which rate these asymptotic variances diverge in time in order to assess the long term reliability of the considered algorithm

on the geometric convergence of the gibbs sampler summary the rate of convergence of the gibbs sampler is discussed the gibbs sampler is a monte carlo simulation method with extensive application to computational issues in the bayesian paradigm conditions for the geometric rate of convergence of the algorithm for discrete and continuous parameter spaces are derived and an illustrative exponential family example is given this paper investigates conditions under which the gibbs sampler gelfand and smith 1990 tanner and wong 1987 geman and geman 1984 converges at a geometric rate the main results appear in sections 2 and 3 where geometric convergence results are established with respect to total variation and supremum norms under fairly natural conditions on the underlying distribution for ease of exposition we shall concentrate on the two most commonly encountered situations where the state space is finite or continuous all our results will establish uniform convergence a strong form of geometric convergence under appropriate regularity conditions uniform convergence is a useful property in its own right but also happens to be a sufficient condition for certain ergodic central limit theorems such results are important for estimation in markov chain simulation but will not be considered in detail here our approach is to apply the theory of markov chains to the specific gibbs sampler case in the finite state space case uniform ergodicity is automatic however the situation is more complicated for continuous state spaces where even well behaved underlying distributions can give rise to markov chains which converge slowly or have unbounded kernels we give two results in this context corollary 2 and corollary 3 which establish uniform convergence under different sets of conditions on the underlying density finally we apply these results to an example of a bayesian hierarchical model where regularity conditions for geometric convergence are naturally satisfied here the hierarchical structure of the model is crucial in permitting application of corollary 3

optimal two phase stratified sampling for estimation of the age composition of a fish population the provision of biological advice for the management of a commercial fish stock requires among other things an estimate of the age composition this estimate is usually obtained from length and age information collected by sampling the commercial landings associated with that fish stock the sampling methods used and the types of problems encountered are discussed by quinn et al 1983 and stevenson 1983 the final stage of the sample design is typically a two phase stratified sample of a large catch from a commercial vessel quinn et al 1983 stevenson 1983 southward 1976 at the first phase a large simple random sample srs of size n is selected and each fish is assigned to a stratum consisting of all fish having lengths in a specified range denote by nf the number of sampled fish in length stratum i at the second phase a srs of size ni is chosen from stratum i i 1 i each of these ni fish is aged with nij fish found to be agej j 1 j the problems are i estimation of ipj j 1 j where p j is the population proportion of age j fish ii given n and in i 1 i how to select ini i 1 i and iii for a specified cost function and given budget how to select n smith and sedransk 1982 provide methods for i and ii for ii their approach is flexible in that currently used or proposed allocation schemes are shown to be optimal under appropriate specifications in the current paper we i review the smith sedransk methodology ii show how to determine the optimal value of n and iii exhibit potential gains from using two phase sampling in section 3 we give a numerical illustration of the

order restricted bayesian estimation of the age composition of a population of atlantic cod abstract the management of a commercial fishery often requires estimates of the age composition these estimates are typically based on age and length data obtained from sampling the commercial landings from the fishery and the catches from a research vessel survey we use data from annual research cruises of georges bank conducted by the canadian department of fisheries and oceans to show how inferences for atlantic cod gadus morhua can be improved traditionally two phase stratified sampling is used with fish length or weight as the stratification variable letting pi denote the proportion of fish belonging to length stratum i and ij denote the proportion of fish belonging to age class j in stratum i we use bayesian methods to estimate p j i ij the proportion of fish that are age j specifications of smoothness expressed as unimodal order relations among the ij within and between the length strata are incorporated into the prior distributions uncertainty about both the locations

mixtures of empirical distributions an iterative application of the age length key it is well known that an age length key is capable of providing biased estimates of age distributions when length distributions are applied to age length keys constructed from sampling in a different year from that in which the length data were collected in separate unpublished papers fukuda and chikuni far seas fisheries research laboratory fisheries agency of japan 1974 proposed a method of iterating an age length key that corrects this problem for a length distribution collected in any year their method provides an estimate of the age distribution corresponding to this length distribution which is consistent with an age length key constructed from the estimated age distribution and previously estimated length at age distributions we show that this method is an application of the em algorithm to mixtures of empirical distributions also we give conditions for which these estimates of age distribution are unique maximum likelihood estimates the method is used to estimate age distributions of the catch of greenland turbot reinhardtius hippoglossoides by small japanese freezer trawlers in the eastern bering sea

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

convergence of adaptive mixtures of importance sampling schemes in the design of ecient simulation algorithms one is often beset with a poor choice of proposal distributions although the performances of a given kernel can clarify how adequate it is for the problem at hand a permanent on line modification of kernels causes concerns about the validity of the resulting algorithm while the issue is quite complex and most often intractable for mcmc algorithms the equivalent version for importance sampling algorithms can be validated quite precisely we derive sucient convergence conditions for a wide class of population monte carlo algorithms and show that rao blackwellized versions asymptotically achieve an optimum in terms of a kullback divergence criterion while more rudimentary versions simply do not benefit from repeated updating

geometric convergence rates for stochastically ordered markov chains let n be a markov chain on the state space 0 that is stochastically ordered in its initial state that is a stochastically larger initial state produces a stochastically larger chain at all other times examples of such chains include random walks the number of customers in various queueing systems and a plethora of storage processes a large body of recent literature concentrates on establishing geometric ergodicity of n in total variation that is proving the existence of a limiting probability measure and a number r 1 such that lim n to infty r n sup a in cal b 0 infty vert p x phi n in a pia vert 0 for every deterministic initial state 0 x we seek to identity the largest r that satisfies this relationship a dependent sample path coupling and a foster lyapunov drift inequality are used to derive convergence rate bounds we then show that the bounds obtained are frequently the best possible application of the methods to queues and random walks are included

fixed width output analysis for markov chain monte carlo markov chain monte carlo is a method of producing a correlated sample to estimate features of a target distribution through ergodic averages a fundamental question is when sampling should stop that is at what point the ergodic averages are good estimates of the desired quantities we consider a method that stops the simulation when the width of a confidence interval based on an ergodic average is less than a user specified value hence calculating a monte carlo standard error is a critical step in assessing the simulation output we consider the regenerative simulation and batch means methods of estimating the variance of the asymptotic normal distribution we give sufficient conditions for the strong consistency of both methods and investigate their finite sample properties in various examples

bayesian optimization of the estimation of the age composition of a fish population abstract to estimate the age distribution of a fish population a two phase stratified sample design is proposed bayesian methods are employed to determine a optimal estimators of the desired population proportions and b the optimal choice of second phase sample sizes subject to a fixed budget a summary of the results of some numerical examples is presented these examples demonstrate typical expected gains obtained by using the bayesian optima as compared to alternative choices of second phase sample sizes suggested in the substantive literature

combining multiple surrogate models to accelerate failure probability estimation with expensive high fidelity models in failure probability estimation importance sampling constructs a biasing distribution that targets the failure event such that a small number of model evaluations is sufficient to achieve a monte carlo estimate of the failure probability with an acceptable accuracy however the construction of the biasing distribution often requires a large number of model evaluations which can become computationally expensive we present a mixed multifidelity importance sampling mmfis approach that leverages computationally cheap but erroneous surrogate models for the construction of the biasing distribution and that uses the original high fidelity model to guarantee unbiased estimates of the failure probability the key property of our mmfis estimator is that it can leverage multiple surrogate models for the construction of the biasing distribution instead of a single surrogate model alone we show that our mmfis estimator has a mean squared error that is up to a constant lower than the mean squared errors of the corresponding estimators that uses any of the given surrogate models alone even in settings where no information about the approximation qualities of the surrogate models is available in particular our mmfis approach avoids the problem of selecting the surrogate model that leads to the estimator with the lowest mean squared error which is challenging if the approximation quality of the surrogate models is unknown we demonstrate our mmfis approach on numerical examples where we achieve orders of magnitude speedups compared to using the high fidelity model only

limit theorems for weighted samples with applications to sequential monte carlo methods in the last decade sequential monte carlo methods smc emerged as a key tool in computational statistics see e g sequential monte carlo methods in practice 2001 springer new york monte carlo strategies in scientific computing 2001 springer new york complex stochastic systems 2001 109 173 these algorithms approximate a sequence of distributions by a sequence of weighted empirical measures associated to a weighted population of particles which are generated recursively despite many theoretical advances see e g j roy statist soc ser b 63 2001 127 146 ann statist 33 2005 1983 2021 feynman kac formulae genealogical and interacting particle systems with applications 2004 springer ann statist 32 2004 2385 2411 the large sample theory of these approximations remains a question of central interest in this paper we establish a law of large numbers and a central limit theorem as the number of particles gets large we introduce the concepts of weighted sample consistency and asymptotic normality and derive conditions under which the transformations of the weighted sample used in the smc algorithm preserve these properties to illustrate our findings we analyze smc algorithms to approximate the filtering distribution in state space models we show how our techniques allow to relax restrictive technical conditions used in previously reported works and provide grounds to analyze more sophisticated sequential sampling strategies including branching resampling at randomly selected times and so on

geometric convergence and central limit theorems for multidimensional hastings and metropolis algorithms we develop results on geometric ergodicity of markov chains and apply these and other recent results in markov chain theory to multidimensional hastings and metropolis algorithms for those based on random walk candidate distributions we find sufficient conditions for moments and moment generating functions to converge at a geometric rate to a prescribed distribution by phrasing the conditions in terms of the curvature of the densities we show that the results apply to all distributions with positive densities in a large class which encompasses many commonly used statistical forms from these results we develop central limit theorems for the metropolis algorithm converse results showing non geometric convergence rates for chains where the rejection rate is not bounded away from unity are also given these show that the negative definiteness property is not redundant

minorization conditions and convergence rates for markov chain monte carlo abstract general methods are provided for analyzing the convergence of discrete time general state space markov chains such as those used in stochastic simulation algorithms including the gibbs sampler the methods provide rigorous a priori bounds on how long these simulations should be run to give satisfactory results results are applied to two models of the gibbs sampler a bivariate normal model and a hierarchical poisson model with gamma conditionals the methods use the notion of minorization conditions for markov chains

adaptive markov chain monte carlo through regeneration abstract markov chain monte carlo mcmc is used for evaluating expectations of functions of interest under a target distribution this is done by calculating averages over the sample path of a markov chain having as its stationary distribution for computational efficiency the markov chain should be rapidly mixing this sometimes can be achieved only by careful design of the transition kernel of the chain on the basis of a detailed preliminary exploratory analysis of an alternative approach might be to allow the transition kernel to adapt whenever new features of are encountered during the mcmc run however if such adaptation occurs infinitely often then the stationary distribution of the chain may be disturbed we describe a framework based on the concept of markov chain regeneration which allows adaptation to occur infinitely often but does not disturb the stationary distribution of the chain or the consistency of sample path averages

intermittency of height fluctuations in stationary state of the kardar parisi zhang equation with infinitesimal surface tension in 1 1 dimensions the kardar parisi zhang kpz equation with infinitesimal surface tension dynamically develops sharply connected valley structures within which the height derivative is not continuous we discuss the intermittency issue in the problem of stationary state forced kpz equation in 1 1 dimensions it is proved that the moments of height increments c a behave as x 1 x 2 xi a with xi a a for length scales x 1 x 2 sigma the length scale sigma is the characteristic length of the forcing term we have checked the analytical results by direct numerical simulation

sequential monte carlo samplers summary we propose a methodology to sample sequentially from a sequence of probability distributions that are defined on a common space each distribution being known up to a normalizing constant these probability distributions are approximated by a cloud of weighted random samples which are propagated over time by using sequential monte carlo methods this methodology allows us to derive simple algorithms to make parallel markov chain monte carlo algorithms interact to perform global optimization and sequential bayesian estimation and to compute ratios of normalizing constants we illustrate these algorithms for various integration tasks arising in the context of bayesian inference

damped finite time singularity driven by noise we consider the combined influence of linear damping and noise on a dynamical finite time singularity model for a single degree of freedom we find that the noise effectively resolves the finite time singularity and replaces it by a first passage time distribution or absorbing state distribution with a peak at the singularity and a long time tail the damping introduces a characteristic cross over time in the early time regime the probability distribution and first passage time distribution show a power law behavior with scaling exponent depending on the ratio of the nonlinear coupling strength to the noise strength in the late time regime the behavior is controlled by the damping the study might be of relevance in the context of hydrodynamics on a nanometer scale in material physics and in biophysics

rates of convergence of the hastings and metropolis algorithms we apply recent results in markov chain theory to hastings and metropolis algorithms with either independent or symmetric candidate distributions and provide necessary and sufficient conditions for the algorithms to converge at a geometric rate to a prescribed distribution pi in the independence case in mathbb r k these indicate that geometric convergence essentially occurs if and only if the candidate density is bounded below by a multiple of pi in the symmetric case in mathbb r only we show geometric convergence essentially occurs if and only if pi has geometric tails we also evaluate recently developed computable bounds on the rates of convergence in this context examples show that these theoretical bounds can be inherently extremely conservative although when the chain is stochastically monotone the bounds may well be effective

recursive monte carlo filters algorithms and theoretical analysis recursive monte carlo filters also called particle filters are a powerful tool to perform the computations in general state space models we discuss and compare the accept reject version with the more common sampling importance resampling version of the algorithm in particular we show how auxiliary variable methods and stratification can be used in the accept reject version and we compare dierent resampling techniques in a second part we show laws of large numbers and a central limit theorem for these monte carlo filters by simple induction arguments that need only weak conditions we also show that under stronger conditions the required sample size is independent of the length of the observed series ams 2000 subject classifications primary 62m09 secondary 60g35 60j22 65c05

asymptotic behavior of the gibbs sampler mild sufficient conditions for the ergodicity of the gibbs sampler are obtained the geometric ergodicity of the gibbs sampler is then discussed where the main tool is the drift criterion this criterion requires the construction of a generalized energy function so that the chain is dissipating energy at a suitable rate whenever it lies outside the center of the state space for the two variable case a useful recipe is provided for constructing appropriate energy functions these results are illustrated with four examples

bayesian estimation of finite population parameters in categorical data models incorporating order restrictions this note describes a bayesian method for estimation of finite population parameters in general population surveys where acceptable regression type models are typically unavailable a categorical data model is adopted as in ericson 1969 section 4 however specifications of smoothness are incorporated into the prior distribution these smoothness conditions are expressed as unimodal or possibly multi modal order relations among the category probabilities emphasis is placed on posterior inference about the finite population mean of independent interest is the methodology for evaluating the posterior moments and probabilities using monte carlo integration with importance sampling

canonical phase space approach to the noisy burgers equation presenting a general phase space approach to stochastic processes we analyze in particular the fokker planck equation for the noisy burgers equation and discuss the time dependent and stationary probability distributions in one dimension we derive the long time skew distribution approaching the symmetric stationary gaussian distribution in the short time regime we discuss heuristically the nonlinear soliton contributions and derive an expression for the distribution in accordance with the directed polymer replica model and asymmetric exclusion model results

adaptive approximate bayesian computation sequential techniques can be adapted to the abc algorithm to enhance its efficiency for instance when sisson et al 2007 introduced the abc prc algorithm the goal was to improve upon existing abc mcmc algorithms marjoram et al 2003 while the abc prc method is based upon the theoretical developments of del moral et al 2006 the application to the setting of approximate bayesian computation induces a bias in the approximation to the posterior distribution of interest as we demonstrate in this paper via both theoretical reasoning and experimental results it is however possible to devise an alternative version based on genuine importance sampling arguments that we call abc pmc in connection with the population monte carlo method introduced in cappe et al 2004 furthermore the abc pmc algorithm is simpler than the abc prc algorithm in that it does not require any backward transition kernel and proposes an automatic scaling of the forward kernel in this paper we demonstrate the applicability of abc pmc and compare its performances with abc prc

multifidelity importance sampling estimating statistics of model outputs with the monte carlo method often requires a large number of model evaluations this leads to long runtimes if the model is expensive to evaluate importance sampling is one approach that can lead to a reduction in the number of model evaluations importance sampling uses a biasing distribution to sample the model more efficiently but generating such a biasing distribution can be difficult and usually also requires model evaluations a different strategy to speed up monte carlo sampling is to replace the computationally expensive high fidelity model with a computationally cheap surrogate model however because the surrogate model outputs are only approximations of the high fidelity model outputs the estimate obtained using a surrogate model is in general biased with respect to the estimate obtained using the high fidelity model we introduce a multifidelity importance sampling mfis method which combines evaluations of both the high fidelity and a surrogate model it uses a surrogate model to facilitate the construction of the biasing distribution but relies on a small number of evaluations of the high fidelity model to derive an unbiased estimate of the statistics of interest we prove that the mfis estimate is unbiased even in the absence of accuracy guarantees on the surrogate model itself the mfis method can be used with any type of surrogate model such as projection based reduced order models and data fit models furthermore the mfis method is applicable to black box models i e where only inputs and the corresponding outputs of the high fidelity and the surrogate model are available but not the details of the models themselves we demonstrate on nonlinear and time dependent problems that our mfis method achieves speedups of up to several orders of magnitude compared to monte carlo with importance sampling that uses the high fidelity model only

convergence properties of the gibbs sampler for perturbations of gaussians 1993 and smith and roberts 1993 a host of other applications of these methods in the context of hierarchical bayesian models is described in all these examples the question arises as to the rate of convergence of the markov chain there is no unified theory to answer this question especially in the case of continuous densities on r n and currently many different approaches are being attempted for the different types of simulation methods one direction of research uses doeblin type bounds on bounded sets together with geometric drift conditions or some form of lyapounov functions to obtain rather general results on exponential convergence for continuous n densities on r see mengersen and tweedie 1996 for the metropolis algorithm and rosenthal 1995 for the gibbs sampler these results are then applied to certain special cases and yield bounds on the rate of convergence in variational norm distance another approach analyzes a discretized version of the problem using general discrete state space markov chain techniques see frieze kannan and polson 1994 finally there are attempts to use spectral analysis to obtain rates of convergence in the l sense where the markov chain operator is considered 2 as a bounded operator in the hilbert space of square integrable functions

the calculation of posterior distributions by data augmentation comment a noniterative sampling importance resampling alternative to the data augmentation algorithm for creating a few imputations when fractions of missing information are modest the sir algorithm

bayesian analysis of constrained parameter and truncated data problems using gibbs sampling abstract constrained parameter problems arise in a wide variety of applications including bioassay actuarial graduation ordinal categorical data response surfaces reliability development testing and variance component models truncated data problems arise naturally in survival and failure time studies ordinal data models and categorical data studies aimed at uncovering underlying continuous distributions in many applications both parameter constraints and data truncation are present the statistical literature on such problems is very extensive reflecting both the problems widespread occurrence in applications and the methodological challenges that they pose however it is striking that so little of this applied and theoretical literature involves a parametric bayesian perspective from a technical viewpoint this perhaps is not difficult to understand the fundamental tool for bayesian calculations in typical realistic models is multidimensional numerical integration which often is problem

three dimensional forced burgers turbulence supplemented with a continuity equation we investigate turbulent limit of the forced burgers equation supplemented with a continuity equation in three dimensions the scaling exponent of the conditional two point correlation function of density i e r x1 r x2 udu ux12x2u 2a 3 is calculated self consistently in the nonuniversal region from which we obtain a 353 also we derive an equation governing the evolution of the probability density function pdf of longitudinal velocity increments in length scale from which a possible mechanism for the dependence of the inertial pdf to one point urms is developed