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

fuzzy relational equations with generalized connectives and their applications the idea of fuzzy relational equation with generalized connectives is introduced and algorithms of resolution of this class equations are presented in details it is clearly shown that some types of well known fuzzy relational equations sup min inf max can be treated as special cases of a wise class of equations under discussion applicational aspects are carefully investigated presenting the advantages of the proposed approach

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

nested sampling algorithm for subsurface flow model selection uncertainty quantification and nonlinear calibration 1 calibration of subsurface flow models is an essential step for managing ground water aquifers designing of contaminant remediation plans and maximizing recovery from hydrocarbon reservoirs we investigate an efficient sampling algorithm known as nested sampling ns which can simultaneously sample the posterior distribution for uncertainty quantification and estimate the bayesian evidence for model selection model selection statistics such as the bayesian evidence are needed to choose or assign different weights to different models of different levels of complexities in this work we report the first successful application of nested sampling for calibration of several nonlinear subsurface flow problems the estimated bayesian evidence by the ns algorithm is used to weight different parameterizations of the subsurface flow models prior model selection the results of the numerical evaluation implicitly enforced occam s razor where simpler models with fewer number of parameters are favored over complex models the proper level of model complexity was automatically determined based on the information content of the calibration data and the data mismatch of the calibrated model

efficient bayes factor estimation from the reversible jump output summary we propose a class of estimators of the bayes factor which is based on an extension of the bridge sampling identity of meng wong 1996 and makes use of the output of the reversible jump algorithm of green 1995 within this class we give the optimal estimator and also a suboptimal one which may be simply computed on the basis of the acceptance probabilities used within the reversible jump algorithm for jumping between models the proposed estimators are very easily computed and lead to a substantial gain of efficiency in estimating the bayes factor over the standard estimator based on the reversible jump output this is illustrated through a series of monte carlo simulations involving a linear and a logistic regression model

convergence of slice sampler markov chains we analyse theoretical properties of the slice sampler we find that the algorithm has extremely robust geometric ergodicity properties for the case of just one auxiliary variable we demonstrate that the algorithm is stochastically monotone and we deduce analytic bounds on the total variation distance from stationarity of the method by using foster lyapunov drift condition methodology

particle markov chain monte carlo methods summary markov chain monte carlo and sequential monte carlo methods have emerged as the two main tools to sample from high dimensional probability distributions although asymptotic convergence of markov chain monte carlo algorithms is ensured under weak assumptions the performance of these algorithms is unreliable when the proposal distributions that are used to explore the space are poorly chosen and or if highly correlated variables are updated independently we show here how it is possible to build efficient high dimensional proposal distributions by using sequential monte carlo methods this allows us not only to improve over standard markov chain monte carlo schemes but also to make bayesian inference feasible for a large class of statistical models where this was not previously so we demonstrate these algorithms on a non linear state space model and a levy driven stochastic volatility model

contribution to application of energy measure of fuzzy sets abstract the works of de luca termini continued by for example knopfmacher loo and gottwald are the most important on the topic of determination of measures of fuzzy sets the matter is to evaluate how fuzzy a fuzzy set is there are two general concepts of measures of fuzzy set i e entropy and energy measures we show that the special kind of energy measure is better suited than the entropy kind of measure in many practical situations applications of the use of energy measure discussed in detail include decision making fuzzy process control and prediction in fuzzy systems

on monte carlo methods for estimating ratios of normalizing constants recently estimating ratios of normalizing constants has played an important role in bayesian computations applications of estimating ratios of normalizing constants arise in many aspects of bayesian statistical inference in this article we present an overview and discuss the current monte carlo methods for estimating ratios of normalizing constants then we propose a new ratio importance sampling method and establish its theoretical framework we find that the ratio importance sampling method can be better than the current methods for example the bridge sampling method meng and wong and the path sampling method gelman and meng in the sense of minimizing asymptotic relative mean square errors of estimators an example is given for illustrative purposes finally we present two special applications and the general implementation issues for estimating ratios of normalizing constants

on identification in fuzzy systems and its applications in control problems abstract the problem of identification in fuzzy systems described by the use of fuzzy equation is considered the identification method and its performance index is also presented the formal procedure of the identification algorithm is illustrated by means of a numerical example the possibility of using the proposed algorithm for the solution of a control problem is given as well

nested sampling statistical physics and the potts model abstract we present a systematic study of the nested sampling algorithm based on the example of the potts model this model which exhibits a first order phase transition for q 4 exemplifies a generic numerical challenge in statistical physics the evaluation of the partition function and thermodynamic observables which involve high dimensional sums of sharply structured multi modal density functions it poses a major challenge to most standard numerical techniques such as markov chain monte carlo in this paper we will demonstrate that nested sampling is particularly suited for such problems and it has a couple of advantages for calculating the partition function of the potts model with n sites a one run stops after o n moves so it takes o n 2 operations for the run b only a single run is required to compute the partition function along with the assignment of confidence intervals c the confidence intervals of the logarithmic partition function decrease with 1 n and d a single run allows to compute quantities for all temperatures while the autocorrelation time is very small irrespective of temperature thermodynamic expectation values of observables which are completely determined by the bond configuration in the representation of fortuin and kasteleyn like the helmholtz free energy the internal energy as well as the entropy and heat capacity can be calculated in the same single run needed for the partition function along with their confidence intervals in contrast thermodynamic expectation values of magnetic properties like the magnetization and the magnetic susceptibility require sampling the additional spin degree of freedom results and performance are studied in detail and compared with those obtained with multi canonical sampling eventually the implications of the findings on a parallel implementation of nested sampling are outlined

markov chain monte carlo methods for computing bayes factors a comparative review the problem of calculating posterior probabilities for a collection of competing models and associated bayes factors continues to be a formidable challenge for applied bayesian statisticians current approaches that take advantage of modern markov chain monte carlo computing methods include those that attempt to sample over some form of the joint space created by the model indicators and the parameters for each model others that sample over the model space alone and still others that attempt to estimate the marginal likelihood of each model directly because the collection of these is equivalent to the collection of model probabilities themselves we review several methods and compare them in the context of three examples a simple regression example a more challenging hierarchical longitudinal model and a binary data latent variable model we find that the joint model parameter space search methods perform adequately but can be difficult to program and tune whereas the marginal likelihood methods o

an approach to the analysis of fuzzy systems this paper deals with a formal description of ill defined processes fuzzy systems by the use of fuzzy relational equations it is pointed out that fuzzy relational equations form a generalized version of the difference equations widely considered in control theory some equivalence between these two kinds of description is presented basic problems of fuzzy systems e g identification prediction sensitivity and stability are shown and numerical algorithms are given indices of each method are introduced especially the degree of fuzziness the sensitivity index which makes it possible to express the quality of each of them

point collocation scheme in silencers with temperature gradient and mean flow this work presents a mathematical approach based on the point collocation technique to compute the transmission loss of perforated dissipative silencers with transversal temperature gradients and mean flow three dimensional wave propagation is considered in silencer geometries with arbitrary but axially uniform cross section to reduce the computational requirements of a full multidimensional finite element calculation a method is developed combining axial and transversal solutions of the wave equation first the finite element method is employed in a two dimensional problem to extract the eigenvalues and associated eigenvectors for the silencer cross section mean flow as well as transversal temperature gradients and the corresponding thermal induced material heterogeneities are included in the model in addition an axially uniform temperature field is taken into account its value being the inlet outlet average a point collocation technique is then used to match the acoustic fields pressure and axial acoustic velocity at the geometric discontinuities between the silencer chamber and the inlet and outlet pipes transmission loss predictions are compared favorably with a general three dimensional finite element approach offering a reduction in the computational effort sound attenuation in perforated dissipative silencers including temperature gradients and mean flow is computed a two dimensional finite element eigenvalue problem is solved for a silencer cross section with transversal thermal variations a point collocation scheme is presented to match the acoustic pressure and axial velocity at the silencer geometrical discontinuities a significant reduction in the computational requirements is obtained compared to a full three dimensional finite element approach

marginal likelihood from the gibbs output abstract in the context of bayes estimation via gibbs sampling with or without data augmentation a simple approach is developed for computing the marginal density of the sample data marginal likelihood given parameter draws from the posterior distribution consequently bayes factors for model comparisons can be routinely computed as a by product of the simulation hitherto this calculation has proved extremely challenging our approach exploits the fact that the marginal density can be expressed as the prior times the likelihood function over the posterior density this simple identity holds for any parameter value an estimate of the posterior density is shown to be available if all complete conditional densities used in the gibbs sampler have closed form expressions to improve accuracy the posterior density is estimated at a high density point and the numerical standard error of resulting estimate is derived the ideas are applied to probit regression and finite mixture models

computational and inferential difficulties with mixture posterior distributions abstract this article deals with both exploration and interpretation problems related to posterior distributions for mixture models the specification of mixture posterior distributions means that the presence of k modes is known immediately standard markov chain monte carlo mcmc techniques usually have difficulties with well separated modes such as occur here the mcmc sampler stays within a neighborhood of a local mode and fails to visit other equally important modes we show that exploration of these modes can be imposed using tempered transitions however if the prior distribution does not distinguish between the different components then the posterior mixture distribution is symmetric and standard estimators such as posterior means cannot be used we propose alternatives for bayesian inference for permutation invariant posteriors including a clustering device and alternative appropriate loss functions

hybrid nested sampling algorithm for bayesian model selection applied to inverse subsurface flow problems a hybrid nested sampling hns algorithm is proposed for efficient bayesian model calibration and prior model selection the proposed algorithm combines nested sampling ns algorithm hybrid monte carlo hmc sampling and gradient estimation using stochastic ensemble method sem ns is an efficient sampling algorithm that can be used for bayesian calibration and estimating the bayesian evidence for prior model selection nested sampling has the advantage of computational feasibility within the nested sampling algorithm a constrained sampling step is performed for this step we utilize hmc to reduce the correlation between successive sampled states hmc relies on the gradient of the logarithm of the posterior distribution which we estimate using a stochastic ensemble method based on an ensemble of directional derivatives sem only requires forward model runs and the simulator is then used as a black box and no adjoint code is needed the developed hns algorithm is successfully applied for bayesian calibration and prior model selection of several nonlinear subsurface flow problems

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

on bayesian analysis of mixtures with an unknown number of components with discussion new methodology for fully bayesian mixture analysis is developed making use of reversible jump markov chain monte carlo methods that are capable of jumping between the parameter subspaces corresponding to different numbers of components in the mixture a sample from the full joint distribution of all unknown variables is thereby generated and this can be used as a basis for a thorough presentation of many aspects of the posterior distribution the methodology is applied here to the analysis of univariate normal mixtures using a hierarchical prior model that offers an approach to dealing with weak prior information while avoiding the mathematical pitfalls of using improper priors in the mixture context

bayesian model choice asymptotics and exact calculations abstract model determination is a fundamental data analytic task here we consider the problem of choosing amongst a finite with loss of generality we assume two set of models after briefly reviewing classical and bayesian model choice strategies we present a general predictive density which includes all proposed bayesian approaches we are aware of using laplace approximations we can conveniently assess and compare asymptotic behavior of these approaches concern regarding the accuracy of these approximation for small to moderate sample sizes encourages the use of monte carlo techniques to carry out exact calculations a data set fit with nested non linear models enables comparison between proposals and between exact and asymptotic values

finding the observed information matrix when using the em algorithm a procedure is derived for extracting the observed information matrix when the em algorithm is used to find maximum likelihood estimates in incomplete data problems the technique requires computation of a complete data gradient vector or second derivative matrix but not those associated with the incomplete data likelihood in addition a method useful in speeding up the convergence of the em algorithm is developed two examples are presented

fuzzy relation equations on a finite set abstract the family of all the solutions of a fuzzy relation equation on a finite set is considered it is characterized by the set of all the lower solutions which can be obtained by a combinatorial algorithm