bayesian analysis of dichotomous quantal response models abstract qualitative response models qrm s are analyzed from the bayesian point of view using diffuse and informative prior distributions exact finite sample bayesian and large sample bayesian and non bayesian estimation results are compared in addition the paper provides 1 plots and discussion of the properties of likelihood functions for qrm s 2 posterior distributions for logit models derivatives and elasticities 3 bayesian prediction procedures for qrm s 4 new estimates for the median and other fractiles of the logistic distribution 5 posterior odds ratios for model selection problems and 6 comparisons of two alternative monte carlo numerical integration procedures it is concluded that asymptotic approximations are not accurate for small to moderate sized samples even when only a single input variable is used and that operational bayesian methods are available for providing both exact small sample and large sample approximate inferences for drm s

discharge rates of medicare stroke patients to skilled nursing facilities bayesian logistic regression with unobserved heterogeneity abstract we determine factors both hospital specific and market area specific associated with hospitals propensities for discharging medicare stroke patients to skilled nursing facilities snf s in california and florida logistic regression is generalized to the case of a betabinomial hierarchical model in which covariate information is included in the hyperparameters of the second stage beta distribution it is found that the posterior mean of the proportion discharged to snf is approximately a weighted average i e shrinkage estimator of the logistic regression estimator and the observed rate we develop fully bayesian inference that takes into account uncertainty about the hyperparameters and we find that this also allows us to test for overdispersion in a natural way the number of observed zeros i e hospitals that sent no stroke patients to a snf is excessive compared to the number expected from a standard logistic regression model and is fit better by the hierarchical betabinomial mod

bayesian gaussian process classification with the em ep algorithm gaussian process classifiers gpcs are bayesian probabilistic kernel classifiers in gpcs the probability of belonging to a certain class at an input location is monotonically related to the value of some latent function at that location starting from a gaussian process prior over this latent function data are used to infer both the posterior over the latent function and the values of hyperparameters to determine various aspects of the function recently the expectation propagation ep approach has been proposed to infer the posterior over the latent function based on this work we present an approximate em algorithm the em ep algorithm to learn both the latent function and the hyperparameters this algorithm is found to converge in practice and provides an efficient bayesian framework for learning hyperparameters of the kernel a multiclass extension of the em ep algorithm for gpcs is also derived in the experimental results the em ep algorithms are as good or better than other methods for gpcs or support vector machines svms with cross validation

expectation consistent approximate inference we propose a novel framework for approximations to intractable probabilistic models which is based on a free energy formulation the approximation can be understood as replacing an average over the original intractable distribution with a tractable one it requires two tractable probability distributions which are made consistent on a set of moments and encode different features of the original intractable distribution in this way we are able to use gaussian approximations for models with discrete or bounded variables which allow us to include non trivial correlations these are neglected in many other methods we test the framework on toy benchmark problems for binary variables on fully connected graphs and 2d grids and compare with other methods such as loopy belief propagation good performance is already achieved by using single nodes as tractable substructures significant improvements are obtained when a spanning tree is used instead

on measuring sensitivity to parametric model misspecification summary in settings where parametric inference is inconsistent under model misspecification the discrepancy between correct and misspecified inferences is compared with the discrepancy between correct and misspecified models to make the comparison tractable large sample and small misspecification approximations are employed the ratio of the approximate discrepancy between inferences to the approximate discrepancy between models is regarded as a relative measure of sensitivity to model misspecification the maximum ratio over a family of correct distributions is determined as a measure of worst case sensitivity as well the distribution producing this maximum can be examined to see how a particular combination of a parametric family and estimand is susceptible to model misspecifications statisticians recognize that in reality no models are really correct in many parametric situations model misspecifications give rise to asymptotically inconsistent estimators of parameters the goal of this work is to quantify how bad an estimator can be relative to the degree of model misspecification the rationale for such an investigation is that misleading inferences arise when a bad answer is produced from an only mildly misspecified model we are less concerned about bad inferences arising from grossly misspecified models as such models are likely to be screened out by diagnostic checks we quantify the sensitivity of a parametric family and estimand to model misspecifications by comparing the kullback leibler kl divergence between correct and misspecified inferences with the kl divergence between correct and misspecified models more specifically the divergence between inferences can be based on either sampling distributions of maximum likelihood estimators or posterior distributions for tractability local approx imations to the kl divergence are employed in a somewhat similar spirit to cook 1986 the maximum ratio of approximate divergences over a class of correct models can be com puted as a worst case measure of sensitivity to misspecifications as well the correct model achieving the maximum can be examined to gain insight into the kind of departures from the model which are damaging this methodology is described fully in the next section then section 3 details an application to some common parametric models for survival data the paper concludes with a discussion in section 4 some technical comments are relegated to appendix a and appendix b

laplace approximation for bessel functions of matrix argument we derive laplace approximations to three functions of matrix argument which arise in statistics and elsewhere matrix bessel av matrix bessel bv and the type ii confluent hypergeometric function of matrix argument we examine the theoretical and numerical properties of the approximations on the theoretical side it is shown that the laplace approximations to av bv and given here together with the laplace approximations to the matrix argument functions 1f1 and 2f1 presented in butler and wood laplace approximations to hyper geometric functions with matrix argument ann statist 2002 satisfy all the important confluence relations and symmetry relations enjoyed by the original functions

wilcoxon type generalized bayesian information criterion we develop a generalized bayesian information criterion for regression model selection the new criterion relaxes the usually strong distributional assumption associated with schwarz s bic by adopting a wilcoxon type dispersion function and appropriately adjusting the penalty term we establish that the wilcoxon type generalized bic preserves the consistency of schwarz s bic without the need to assume a parametric likelihood we also show that it outperforms schwarz s bic with heavier tailed data in the sense that asymptotically it can yield substantially smaller l 2 risk on the other hand when the data are normally distributed both criteria have similar l 2 risk the new criterion function is convex and can be conveniently computed using existing statistical software our proposal provides a flexible yet highly efficient alternative to schwarz s bic at the same time it broadens the scope of wilcoxon inference which has played a fundamental role in classical nonparametric analysis

entropy rigidity of negatively curved manifolds of finite volume we prove the following entropy rigidity result in finite volume if x is a negatively curved manifold with curvature b 2 k x 1 then ent top x n 1 if and only if x is hyperbolic in particular if x has the same length spectrum of a hyperbolic manifold x 0 the it is isometric to x 0 we also give a direct entropy free proof of this fact we compare with the classical theorems holding in the compact case pointing out the main difficulties to extend them to finite volume manifolds

algorithms for hyperbolic quadratic eigenvalue problems we consider the quadratic eigenvalue problem or the qep 2 a b c x 0 where a b and c are hermitian with a positive definite the qep is called hyperbolic if x bx 2 4 x ax x cx for all nonzero x c r we show that a relatively efficient test for hyperbolicity can be obtained by computing the eigenvalues of the qep a hyperbolic qep is overdamped if b is positive definite and c is positive semidefinite we show that a hyperbolic qep whose eigenvalues are necessarily real is overdamped if and only if its largest eigenvalue is nonpositive for overdamped qeps we show that all eigenpairs can be found efficiently by finding two solutions of the corresponding quadratic matrix equation using a method based on cyclic reduction we also present a new measure for the degree of hyperbolicity of a hyperbolic qep

dynamic data driven bayesian gmsfem abstract in this paper we propose a bayesian approach for multiscale problems with the availability of dynamic observational data our method selects important degrees of freedom probabilistically in a generalized multiscale finite element method framework due to scale disparity in many multiscale applications computational models cannot resolve all scales dominant modes in the generalized multiscale finite element method are used as permanent basis functions which we use to compute an inexpensive multiscale solution and the associated uncertainties through our bayesian framework we can model approximate solutions by selecting the unresolved scales probabilistically we consider parabolic equations in heterogeneous media the temporal domain is partitioned into subintervals using residual information and given dynamic data we design appropriate prior distribution for modeling missing subgrid information the likelihood is designed to minimize the residual in the underlying pde problem and the mismatch of observational data using the resultant posterior distribution the sampling process identifies important degrees of freedom beyond permanent basis functions the method adds important degrees of freedom in resolving subgrid information and ensuring the accuracy of the observations

large sample joint posterior approximations when full conditionals are approximately normal modern bayesian statistical methods such as gibbs and metropolis hastings sampling were developed to liberate statisticians from the necessity of making large sample assumptions and to facilitate the numerical approximation of problems that had previously been analytically intractable counter to this trend we develop a method for constructing asymptotic joint posterior approximations based on models with k blocks of parameters and where the corresponding properly normalized full conditionals are themselves asymptotically normal we illustrate these techniques by applying them to particular linear and generalized linear mixed models glmms we also consider the relevance of different parameterizations with regard to our asymptotics recent work has indicated that gibbs samplers based on so called centering parameterizations result in better convergence properties for the resulting markov chains our results for the one way random effects model shed some light on this issue for this example we also

nested expectation propagation for gaussian process classification this paper considers probabilistic multinomial probit classification using gaussian process gp priors challenges with multiclass gp classification are the integration over the non gaussian posterior distribution and the increase of the number of unknown latent variables as the number of target classes grows expectation propagation ep has proven to be a very accurate method for approximate inference but the existing ep approaches for the multinomial probit gp classification rely on numerical quadratures or independence assumptions between the latent values associated with different classes to facilitate the computations in this paper we propose a novel nested ep approach which does not require numerical quadratures and approximates accurately all between class posterior dependencies of the latent values but still scales linearly in the number of classes the predictive accuracy of the nested ep approach is compared to laplace variational bayes and markov chain monte carlo mcmc approximations with various benchmark data sets in the experiments nested ep was the most consistent method compared to mcmc sampling but in terms of classification accuracy the differences between all the methods were small from a practical point of view

ergodicity of unipotent flows and kleinian groups let m be a non elementary convex cocompact hyperbolic 3 manifold and be the critical exponent of its fundamental group we prove that a one dimensional unipotent flow for the frame bundle of m is ergodic for the burger roblin measure if and only if 1

high dimensional maximum marginal likelihood item factor analysis by adaptive quadrature although the bock aitkin likelihood based estimation method for factor analysis of dichotomous item response data has important advantages over classical analysis of item tetrachoric correlations a serious limitation of the method is its reliance on fixed point gauss hermite g h quadrature in the solution of the likelihood equations and likelihood ratio tests when the number of latent dimensions is large computational considerations require that the number of quadrature points per dimension be few but with large numbers of items the dispersion of the likelihood given the response pattern becomes so small that the likelihood cannot be accurately evaluated with the sparse fixed points in the latent space in this paper we demonstrate that substantial improvement in accuracy can be obtained by adapting the quadrature points to the location and dispersion of the likelihood surfaces corresponding to each distinct pattern in the data in particular we show that adaptive g h quadrature combined with mean and covariance adjustments at each iteration of an em algorithm produces an accurate fast converging solution with as few as two points per dimension evaluations of this method with simulated data are shown to yield accurate recovery of the generating factor loadings for models of upto eight dimensions unlike an earlier application of adaptive gibbs sampling to this problem by meng and schilling the simulations also confirm the validity of the present method in calculating likelihood ratio chi square statistics for determining the number of factors required in the model finally we apply the method to a sample of real data from a test of teacher qualifications

laplace approximations for posterior expectations when the mode occurs at the boundary of the parameter space abstract this article gives asymptotic expansions for posterior expectations when the mode is on the boundary of the parameter space the idea based on the divergence theorem is to reduce the high dimensional integrals over the parameters space to surface integrals over the boundary of the parameter space and then apply the usual interior mode laplace method to the latter integrals it is shown that these approximations have second order accuracy the method is illustrated with applications to a two sample binomial problem and a random eflects model

computing bayes factors by combining simulation and asymptotic approximations abstract the bayes factor is a ratio of two posterior normalizing constants which may be difficult to compute we compare several methods of estimating bayes factors when it is possible to simulate observations from the posterior distributions via markov chain monte carlo or other techniques the methods that we study are all easily applied without consideration of special features of the problem provided that each posterior distribution is well behaved in the sense of having a single dominant mode we consider a simulated version of laplace s method a simulated version of bartlett correction importance sampling and a reciprocal importance sampling technique we also introduce local volume corrections for each of these in addition we apply the bridge sampling method of meng and wong we find that a simulated version of laplace s method with local volume correction furnishes an accurate approximation that is especially useful when likelihood function evaluations are costly a simple bridge sampli

on robust analysis of a normal location parameter pericchi and smith considered a normal location parameter problem with double exponential and student t prior distributions these two prior distributions both belong to the class of scale mixtures of normal distributions and are useful in providing a robust analysis of the normal location parameter problem in this paper we extend the analysis to other scale mixtures of normal distributions such as the exponential power and the symmetric stable distributions

estimation for the half logistic distribution based on multiply type ii hybrid censoring abstract in this paper some estimators of the scale parameter of the half logistic distribution are derived under the multiply type ii hybrid censoring scheme we propose three methods first we obtain the maximum likelihood estimator of the scale parameter of the half logistic distribution second we propose some approximate maximum likelihood estimators of the scale parameter by employing three different types of taylor series expansions lastly we obtain the bayes estimators by employing some prior distributions and loss functions the interval estimations such as the asymptotic confidence interval and the credible interval and the highest posterior density interval are obtained to assess the performance of the proposed estimators we obtain the mean squared error average length of 95 interval and coverage probability through monte carlo simulation and applies to the real dataset

an assessment of estimation methods for generalized linear mixed models with binary outcomes two main classes of methodology have been developed for addressing the analytical intractability of generalized linear mixed models likelihood based methods and bayesian methods likelihood based methods such as the penalized quasi likelihood approach have been shown to produce biased estimates especially for binary clustered data with small clusters sizes more recent methods using adaptive gaussian quadrature perform well but can be overwhelmed by problems with large numbers of random effects and efficient algorithms to better handle these situations have not yet been integrated in standard statistical packages bayesian methods although they have good frequentist properties when the model is correct are known to be computationally intensive and also require specialized code limiting their use in practice in this article we introduce a modification of the hybrid approach of capanu and begg 2011 biometrics 67 371 380 as a bridge between the likelihood based and bayesian approaches by employing bayesian estimation for the variance components followed by laplacian estimation for the regression coefficients we investigate its performance as well as that of several likelihood based methods in the setting of generalized linear mixed models with binary outcomes we apply the methods to three datasets and conduct simulations to illustrate their properties simulation results indicate that for moderate to large numbers of observations per random effect adaptive gaussian quadrature and the laplacian approximation are very accurate with adaptive gaussian quadrature preferable as the number of observations per random effect increases the hybrid approach is overall similar to the laplace method and it can be superior for data with very sparse random effects copyright 2013 john wiley sons ltd

inverse groundwater modeling for hydraulic conductivity estimation using bayesian model averaging and variance window 1 this study proposes a bayesian model averaging bma method to address parameter estimation uncertainty arising from nonuniqueness in parameterization methods bma is able to incorporate multiple parameterization methods for prediction through the law of total probability and to obtain an ensemble average of hydraulic conductivity estimates two major issues in applying bma to hydraulic conductivity estimation are discussed the first problem is using occam s window in usual bma applications to measure approximated posterior model probabilities occam s window only accepts models in a very narrow range tending to single out the best method and discard other good methods we propose a variance window to replace occam s window to cope with this problem the second problem is the kashyap information criterion kic in the approximated posterior model probabilities which tends to prefer highly uncertain parameterization methods by considering the fisher information matrix with sufficient amounts of observation data the bayesian information criterion bic is a good approximation and is able to avoid controversial results from using kic this study adopts multiple generalized parameterization gp methods such as the bma models to estimate spatially correlated hydraulic conductivity numerical examples illustrate the issues of using kic and occam s window and show the advantages of using bic and the variance window in bma application finally we apply bma to the hydraulic conductivity estimation of the 1500 foot sand in east baton rouge parish louisiana

dynamics and zeta functions on conformally compact manifolds in this note we study the dynamics and associated zeta functions of conformally compact manifolds with variable negative sectional curvatures we begin with a discussion of a larger class of manifolds known as convex co compact manifolds with variable negative curvature applying results from dynamics on these spaces we obtain optimal meromorphic extensions of weighted dynamical zeta functions and asymptotic counting estimates for the number of weighted closed geodesics a meromorphic extension of the standard dynamical zeta function and the prime orbit theorem follow as corollaries finally we investigate interactions between the dynamics and spectral theory of these spaces

on secant approximations to cumulative distribution functions summary we investigate the properties of an approximation called the secant approximation to the cumulative distribution function where the density is of a broad parametric class formulae for higher order terms are derived that give the approximation explicitly in terms of functions that define the density in all cases examined the first order secant approximation is monotonic and has the correct limits at both ends of the support resulting in the important property that the approximation itself is a cumulative distribution function we give conditions under which the approximation has bounded relative errors throughout the support the superiority of the secant approximation is illustrated in the particular case of the skew t family of distributions also known as pearson s type iv family a collection of formulae useful for the systematic evaluation of these approximations is given in appendix 2

a simple predictive density function abstract an easily computable predictive density is considered although involving a plain maximum likelihood technique it coincides with the posterior predictive density with a noninformative prior for standard situations achievements are comparable with classical inference

specification tests based on mcmc output abstract two test statistics are proposed to determine model specification after a model is estimated by an mcmc method the first test is the mcmc version of ios a test and its asymptotic null distribution is normal the second test is motivated from the power enhancement technique of fan et al 2015 it combines a component j 1 that tests a null point hypothesis in an expanded model and a power enhancement component j 0 obtained from the first test it is shown that j 0 converges to zero when the null model is correctly specified and diverges when the null model is misspecified also shown is that j 1 is asymptotically 2 distributed suggesting that the second test is asymptotically pivotal when the null model is correctly specified the main feature of the first test is that no alternative model is needed the second test has several properties first its size distortion is small and hence bootstrap methods can be avoided second it is easy to compute from mcmc output and hence is applicable to a wide range of models including latent variable models for which frequentist methods are difficult to use third when the test statistic rejects the null model and j 1 takes a large value the test suggests the source of misspecification the finite sample performance is investigated using simulated data the method is illustrated in a linear regression model a linear state space model and a stochastic volatility model using real data

a variational maximization maximization algorithm for generalized linear mixed models with crossed random effects we present a variational maximization maximization algorithm for approximate maximum likelihood estimation of generalized linear mixed models with crossed random effects e g item response models with random items random raters or random occasion specific effects the method is based on a factorized variational approximation of the latent variable distribution given observed variables which creates a lower bound of the log marginal likelihood the lower bound is maximized with respect to the factorized distributions as well as model parameters with the proposed algorithm a high dimensional intractable integration is translated into a two dimensional integration problem we incorporate an adaptive gauss hermite quadrature method in conjunction with the variational method in order to increase computational efficiency numerical studies show that under the small sample size conditions that are considered the proposed algorithm outperforms the laplace approximation

importance weighted marginal bayesian posterior density estimation abstract markov chain sampling schemes generate dependent observations i 0 i n from a full joint posterior distribution data frequently only certain marginals of this full posterior density are of interest thus an interesting problem is how to estimate the marginal posterior densities based on the dependent observations i 0 i n from data we propose a new importance weighted marginal density estimation iwmde method an iwmde is obtained by averaging many dependent observations of the ratio of the full joint posterior densities multiplied by a weighting conditional density w the asymptotic properties for the iwmde and the guidelines for choosing a weighting conditional density w are also considered a bivariate normal model and a constrained linear multiple regression model are used to illustrate how to derive the iwmde s for the marginal posterior densities

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

estimating bayes factors via posterior simulation with the laplace metropolis estimator abstract the key quantity needed for bayesian hypothesis testing and model selection is the integrated or marginal likelihood of a model we describe a way to use posterior simulation output to estimate integrated likelihoods we describe the basic laplace metropolis estimator for models without random effects for models with random effects we introduce the compound laplace metropolis estimator we apply this estimator to data from the world fertility survey and show it to give accurate results batching of simulation output is used to assess the uncertainty involved in using the compound laplace metropolis estimator the method allows us to test for the effects of independent variables in a random effects model and also to test for the presence of the random effects

limit sets of discrete groups of isometries of exotic hyperbolic spaces let be a geometrically finite discrete group of isometries of hyperbolic space hf where f r c h or o in which case n 2 we prove that the critical exponent of equals the hausdorff dimension of the limit sets and that the smallest eigenvalue of the laplacian acting on square integrable functions is a quadratic function of either of them when they are sufficiently large a generalization of hopf ergodicity theorem for the geodesic flow with respect to the bowen margulis measure is also proven

a bayesian approach to nonlinear calibration problems abstract in bioassays enzyme assays or radioimmunoassays the concentration response relationships are mostly nonlinear very often because of physical restrictions or for biological reasons the prediction samples cannot be diluted and the determination of very low concentrations is of interest thus one cannot be restricted to the almost linear part of the response curve and standard linear calibration methods are not applicable the nonlinear calibration problem is of great importance in these types of applications to implement the bayesian paradigm in nonlinear calibration problems a substantial amount of numerical integration is necessary in practice when concentrations of hundreds of samples have to be determined routinely even with a highly efficient integration method naylor and smith 1982 the numerical effort becomes prohibitive an approximation method is proposed therefore to reduce the calculation the precision of the approximation method is examined by validation subsamples and

approximate predictive likelihood summary a predictive likelihood is given which approximates both bayes and maximum likeli hood predictive inference by expansion of a posterior likelihood this synthesizes and extends previous results and is widely applicable the approximation usually differs from exact bayes posterior predictive density by op n 2 and from exact predictive likelihood by op n 1 but does not depend on the availability of prior information and is applicable when exact predictive likelihood cannot be found the results are applied to the prediction various treatments of the likelihood function form the basis of parametric inference but until recently few non bayesian attempts have been made to define versions of it suitable for prediction however lauritzen 1974 hinkley 1979 and more recently butler 1986 have proposed rather similar definitions of predictive likelihood based on the idea of conditioning on the value of a sufficient statistic unfortunately they apply only when sufficiency provides a genuine reduction of the data moreover even if for a given problem an exact predictive likelihood exists the calculations needed to derive it can be onerous this paper provides approximations to hinkley lauritzen predictive likelihood accur ate to op n 1 in many cases which are applicable in situations where maximum likelihood estimation is regular they also apply to problems which admit no exact predictive likelihood because sufficiency provides no real reduction of the data the approximations are derived from a posterior predictive density and may if prior informa tion is available be regarded as bayesian procedures their construction requires only the repeated maximization of a likelihood and evaluation of the observed information matrix at the maximum and is fast and accurate using usual numerical maximization methods there is a close connexion with the approximations to posterior moments and marginal densities given by tierney kadane 1986 who suggest the same approxima tion as used here but give no detailed results for prediction section 2 of this paper shows how laplace s method for integrals may be applied to a posterior predictive density to yield approximate predictive likelihood section 3 discusses a connection with conditional inference essentially the same approximation

diagnostic and sampling models in screening summary screening methods are developed which raise the proportion of successes in the retained population to a predetermined level specification regions for the screening variables are derived within both the diagnostic and sampling paradigms dawid 1976 logistic and normal models are used for the two frameworks and the results are illustrated by an example of screening for malnourishment

filtered variate prior distributions for histogram smoothing abstract we develop prior distributions for histogram inference favoring smooth population frequencies that is probability vectors with small differences for neighboring categories we give a theory of prior random probability vectors representable as a linear transform or filter of a standard random probability vector or equivalently a random weighted average of nonrandom smooth probability vectors promising methods of prior assessment are given based on elicitation of a list of typically smooth probability vectors the empirical moments of which can then be matched by the mean vector and variance matrix of a constructed continuous type filtered variate prior distribution

variational bayesian multinomial probit regression with gaussian process priors it is well known in the statistics literature that augmenting binary and polychotomous response models with gaussian latent variables enables exact bayesian analysis via gibbs sampling from the parameter posterior by adopting such a data augmentation strategy dispensing with priors over regression coefficients in favor of gaussian process gp priors over functions and employing variational approximations to the full posterior we obtain efficient computational methods for gp classification in the multiclass setting the model augmentation with additional latent variables ensures full a posteriori class coupling while retaining the simple a priori independent gp covariance structure from which sparse approximations such as multiclass informative vector machines ivm emerge in a natural and straightforward manner this is the first time that a fully variational bayesian treatment for multiclass gp classification has been developed without having to resort to additional explicit approximations to the nongaussian likelihood term empirical comparisons with exact analysis use markov chain monte carlo mcmc and laplace approximations illustrate the utility of the variational approximation as a computationally economic alternative to full mcmc and it is shown to be more accurate than the laplace approximation

logarithm laws for equilibrium states in negative curvature let m be a pinched negatively curved riemannian manifold whose unit tangent bundle is endowed with a gibbs measure mf associated with a potential f we compute the hausdorff dimension of the conditional measures of mf we study the mf almost sure asymptotic penetration behaviour of locally geodesic lines of m into small neighbourhoods of closed geodesics and of other compact locally convex subsets of m we prove khintchine type and logarithm law type results for the spiraling of geodesic lines around these objects as an arithmetic consequence we give almost sure diophantine approximation results of real numbers by quadratic irrationals with respect to general holder quasi invariant measures

integration of multimodal functions by monte carlo importance sampling abstract numerical integration of a multimodal integrand f is approached by monte carlo integration via importance sampling a mixture of multivariate t density functions is suggested as an importance function g for its easy random variate generation thick tails and high flexibility the number of components in the mixture is determined by the number of modes of f and the mixing weights and location and scale parameters of the component distributions are determined by numerical minimization of a monte carlo estimate of the squared variation coefficient of the weight function f g stratified importance sampling and control variates are shown to be particularly effective variance reduction techniques in this case the algorithm is applied to a 10 dimensional example and shown to yield significant improvement over usual integration schemes

composite likelihood bayesian information criteria for model selection in high dimensional data for high dimensional data sets with complicated dependency structures the full likelihood approach often leads to intractable computational complexity this imposes difficulty on model selection given that most traditionally used information criteria require evaluation of the full likelihood we propose a composite likelihood version of the bayes information criterion bic and establish its consistency property for the selection of the true underlying marginal model our proposed bic is shown to be selection consistent under some mild regularity conditions where the number of potential model parameters is allowed to increase to infinity at a certain rate of the sample size simulation studies demonstrate the empirical performance of this new bic especially for the scenario where the number of parameters increases with sample size technical proofs of our theoretical results are provided in the online supplemental materials

a bayesian model averaging approach for estimating the relative risk of mortality associated with heat waves in 105 u s cities estimating the risks heat waves pose to human health is a critical part of assessing the future impact of climate change in this article we propose a flexible class of time series models to estimate the relative risk of mortality associated with heat waves and conduct bayesian model averaging bma to account for the multiplicity of potential models applying these methods to data from 105 u s cities for the period 1987 2005 we identify those cities having a high posterior probability of increased mortality risk during heat waves examine the heterogeneity of the posterior distributions of mortality risk across cities assess sensitivity of the results to the selection of prior distributions and compare our bma results to a model selection approach our results show that no single model best predicts risk across the majority of cities and that for some cities heat wave risk estimation is sensitive to model choice although model averaging leads to posterior distributions with increased variance as compared to statistical inference conditional on a model obtained through model selection we find that the posterior mean of heat wave mortality risk is robust to accounting for model uncertainty over a broad class of models

diagonal update method for a quadratic matrix equation the quadratic matrix equation a x 2 b x c 0 has been studied in many researches the sufficient condition for the existence of the primary solution is already provided where the equation arose from an overdamped vibrating system with m matrix coefficients the bernoulli s method can not be modified by relaxation in this equation in this paper we will introduce the diagonal update method and modify the bernoulli s method

on hydrological model complexity its geometrical interpretations and prediction uncertainty knowledge of hydrological model complexity can aid selection of an optimal prediction model out of a set of available models optimal model selection is formalized as selection of the least complex model out of a subset of models that have lower empirical risk this may be considered equivalent to minimizing an upper bound on prediction error defined here as the mathematical expectation of empirical risk in this paper we derive an upper bound that is free from assumptions on data and underlying process distribution as well as on independence of model predictions over time we demonstrate that hydrological model complexity as defined in the presented theoretical framework plays an important role in determining the upper bound the model complexity also acts as a stabilizer to a hydrological model selection problem if it is deemed ill posed we provide an algorithm for computing complexity of any arbitrary hydrological model we also demonstrate that hydrological model complexity has a geometric interpretation as the size of model output space the presented theory is applied to quantify complexities of two hydrological model structures sac sma and sixpar it detects that sac sma is indeed more complex than sixpar we also develop an algorithm to estimate the upper bound on prediction error which is applied on five different rainfall runoff model structures that vary in complexity we show that a model selection problem is stabilized by regularizing it with model complexity complexity regularized model selection yields models that are robust in predicting future but yet unseen data

variational multinomial logit gaussian process gaussian process prior with an appropriate likelihood function is a flexible non parametric model for a variety of learning tasks one important and standard task is multi class classification which is the categorization of an item into one of several fixed classes a usual likelihood function for this is the multinomial logistic likelihood function however exact inference with this model has proved to be difficult because high dimensional integrations are required in this paper we propose a variational approximation to this model and we describe the optimization of the variational parameters experiments have shown our approximation to be tight in addition we provide data independent bounds on the marginal likelihood of the model one of which is shown to be much tighter than the existing variational mean field bound in the experiments we also derive a proper lower bound on the predictive likelihood that involves the kullback leibler divergence between the approximating and the true posterior we combine our approach with a recently proposed sparse approximation to give a variational sparse approximation to the gaussian process multi class model we also derive criteria which can be used to select the inducing set and we show the effectiveness of these criteria over random selection in an experiment

a sample reuse method for accurate parametric empirical bayes confidence intervals abstract parametric empirical bayes methods of point estimation for a vector of unknown parameters date to the landmark paper of james and stein 1961 the usual approach is to use the mean of the estimated posterior distribution of each parameter where the estimation of the prior parameters hyperparameters is accomplished through the marginal distribution of the data while point estimates computed this way usually perform well interval estimates based on the estimated posterior called naive eb intervals are not they fail to account for the variability in the estimation of the hyperparameters generally resulting in sub nominal coverage probability in the eb sense defined in morris 1983a in this paper we extend the work of carlin and gelfand 1989 who proposed a conditional bias correction method for developing eb intervals which corrects the deficiencies in the naive intervals we show how bias correction can be implemented in general via a type iii parametric bootstrap procedure a sample reuse method first employed by laird and louis 1987

multi parameter estimation in compound gaussian clutter by variational bayesian in this paper we consider the problem of multi parameter estimation in the presence of compound gaussian clutter for cognitive radar by the variational bayesian method the advantage of variational bayesian is that the estimation of multi variate parameters is decomposed to problems of estimation of univariate parameters by variational approximation thus enabling analytically tractable approximate posterior densities in complex statistical models consisting of observed data unknown parameters and hidden variables we derive the asymptotic bayesian cramer rao bounds and demonstrate by numerical simulations that the proposed approach leads to improved estimation accuracy than the expectation maximization method and the exact bayesian method in the case of non gaussian nonlinear signal models and small data sample size

approximate predictive pivots and densities summary this paper suggests two predictive likelihoods that can be applied in almost any parametric model setting the first can sometimes be interpreted as an approximate predictive pivot barnard 1986 while the second is often an approximation to a bayesian predictive density with a flat prior the issue of calibrating various predictive likelihoods in terms of long run predictive coverage is also discussed and a specific criterion by which these likelihoods can be compared is proposed

on iterative methods for the quadratic matrix equation with m matrix in this paper we consider newton s method and bernoulli s method for a quadratic matrix equation arising from an overdamped vibrating system by introducing m matrix to this equation we provide a sufficient condition for the existence of the primary solution moreover we show that newton s method and bernoulli s method with an initial zero matrix converge to the primary solvent under the proposed sufficient condition

approximate marginals in latent gaussian models we consider the problem of improving the gaussian approximate posterior marginals computed by expectation propagation and the laplace method in latent gaussian models and propose methods that are similar in spirit to the laplace approximation of tierney and kadane 1986 we show that in the case of sparse gaussian models the computational complexity of expectation propagation can be made comparable to that of the laplace method by using a parallel updating scheme in some cases expectation propagation gives excellent estimates where the laplace approximation fails inspired by bounds on the correct marginals we arrive at factorized approximations which can be applied on top of both expectation propagation and the laplace method the factorized approximations can give nearly indistinguishable results from the non factorized approximations and their computational complexity scales linearly with the number of variables we experienced that the expectation propagation based marginal approximations we introduce are typically more accurate than the methods of similar complexity proposed by rue et al 2009

a bayesian analysis of record statistics from the gompertz model the gompertz distribution has been used as a growth model and it can be used to fit tumor growth record values can be viewed as order statistics from a sample whose size is determined by the values and the order of occurrence of observations based on record values from the two parameter gompertz distribution bayes estimators for the two unknown parameters are obtained by using laplace approximation these estimates are obtained based on the squared error and linex loss functions predictions for future upper record values from the gompertz model are obtained from a bayesian approach the maximum likelihood and bayes estimates are compared via monte carlo simulation study and a numerical example is given to illustrate the results of prediction

approximate bayesian inference for latent gaussian models by using integrated nested laplace approximations summary structured additive regression models are perhaps the most commonly used class of models in statistical applications it includes among others generalized linear models generalized additive models smoothing spline models state space models semiparametric regression spatial and spatiotemporal models log gaussian cox processes and geostatistical and geoadditive models we consider approximate bayesian inference in a popular subset of structured additive regression models latent gaussian models where the latent field is gaussian controlled by a few hyperparameters and with non gaussian response variables the posterior marginals are not available in closed form owing to the non gaussian response variables for such models markov chain monte carlo methods can be implemented but they are not without problems in terms of both convergence and computational time in some practical applications the extent of these problems is such that markov chain monte carlo sampling is simply not an appropriate tool for routine analysis we show that by using an integrated nested laplace approximation and its simplified version we can directly compute very accurate approximations to the posterior marginals the main benefit of these approximations is computational where markov chain monte carlo algorithms need hours or days to run our approximations provide more precise estimates in seconds or minutes another advantage with our approach is its generality which makes it possible to perform bayesian analysis in an automatic streamlined way and to compute model comparison criteria and various predictive measures so that models can be compared and the model under study can be challenged

the ergodic theory of discrete isometry groups on manifolds of variable negative curvature this paper studies the ergodic theory at infinity of an arbitrary discrete isometry group acting on any hadamard manifold h of pinched variable negative curvature most of the results obtained by sullivan in the constant curvature case are generalized to the case of variable curvature we describe connections between measures supported on the limit set of dynamics of the geodesic flow and the geometry of m h we explore the relationship between the growth exponent of the group the hausdorff dimension of the limit set and the topological entropy of the geodesic flow the equivalence of various descriptions of an analogue of the hopf dichotomy is proved as applications we settle a question of j feldman and m ratner about the horocycle flow on a finite volume surface of negative curvature and obtain an asymptotic formula for the counting function of lattice points at the end of this paper we apply our results to the study of some rigidity problems more applications to mostow rigidity for discrete subgroups of rank 1 noncompact semisimple lie groups with infinite covolume will be published in subsequent papers by the author notations h is a hadamard manifold with pinched sectional curvature k 2 k k 1 k2 k1 0 is a non elementary discrete isometry group acting on h freely and properly discontinuously which is called fuchsian h is the ideal boundary of h m h is a complete riemannian manifold sh or sm is the unit tangent bundle g is the geodesic flow on sh or sm v t is the unique geodesic in sh resp sm with the initial velocity v 0 v sh resp sm v resp v is the asymptotic class of the geodesic v t resp v t l is the limit set of on h l is the radical limit set of in l is the nonwandering set of the geodesic flow on sm received by the editors may 7 1995 1991 mathematics subject classification primary 58f17 secondary 58f11 58f15 20h10 research at msri supported by nsf grant dms 8505550 also partially supported by nsf grant dms 9403870 and sfb 170 c 1996 american mathematical society

a mixed effects model for overdispersed count data in animal breeding count data models are developed for animal breeding applications to account for more variability than in a poisson mixed effects model a gamma distribution is assigned to poisson parameters thereby leading to a negative binomial model the natural log of the expected value of the poisson parameter is expressed as a linear function of fixed and random polygenic effects the negative binomial and poisson mixed models were compared in two simulations in the first marginal maximum likelihood mml estimates of genetic variances obtained under a poisson sire model psm and under a poisson animal model pam accounting for half sib relationships were different contrary to what occurs in a gaussian mixed linear model mml estimates of genetic variance under a negative binomial sire model were less biased than estimates under a psm and had a slightly smaller mean squared error mse the second simulation compared animal models in which the variance of the residuals was larger than the genetic variance empirical relative bias and mse of mml estimates of genetic variance were larger under a pam that ignored the residuals than under a negative binomial animal model differences in performance widened as genetic variance increased an application to the analysis of number of artificial inseminations until conception in dairy heifers is presented to illustrate potential differences in genetic variance estimates under the two animal models

x tmcmc adaptive kriging for bayesian inverse modeling abstract the bayesian inference of models associated with large scale simulations is prohibitively expensive even for massively parallel architectures we demonstrate that we can drastically reduce this cost by combining adaptive kriging with the population based transitional markov chain monte carlo tmcmc techniques for uni modal posterior probability distribution functions pdf the proposed hybrid method can reduce the computational cost by an order of magnitude with the same computational resources for complex posterior pdf landscapes we show that it is necessary to further extend the tmcmc by langevin adjusted proposals the proposed hybrid method exhibits high parallel efficiency we demonstrate the capabilities of our method on test bed problems and on high fidelity simulations in structural dynamics

on the equivalence of prospective and retrospective likelihood methods in case control studies summary we present new approaches to analyzing case control studies using prospective likelihood methods in the classical framework we extend the equality of the profile likelihoods to the barndorff nielsen modified profile likelihoods for prospective and retrospective models this enables simple and accurate approximate conditional inference for stratified case control studies of moderate stratum size in the bayesian framework we provide sufficient conditions on priors for the prospective model parameters to yield a prospective marginal posterior density equal to its retrospective counterpart our results extend the prospective retrospective equivalence in the bayesian paradigm with a more general class of priors than has previously been investigated

accurate approximations for posterior moments and marginal densities abstract this article describes approximations to the posterior means and variances of positive functions of a real or vector valued parameter and to the marginal posterior densities of arbitrary i e not necessarily positive parameters these approximations can also be used to compute approximate predictive densities to apply the proposed method one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima nevertheless the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third order expansions of the likelihood and requiring the evaluation of third derivatives the approximate marginal posterior densities behave very much like saddle point approximations for sampling distributions the principal regularity condition required is that the likelihood times prior be unimodal

approximations for binary gaussian process classification we provide a comprehensive overview of many recent algorithms for approximate inference in gaussian process models for probabilistic binary classification the relationships between several approaches are elucidated theoretically and the properties of the different algorithms are corroborated by experimental results we examine both 1 the quality of the predictive distributions and 2 the suitability of the different marginal likelihood approximations for model selection selecting hyperparameters and compare to a gold standard based on mcmc interestingly some methods produce good predictive distributions although their marginal likelihood approximations are poor strong conclusions are drawn about the methods the expectation propagation algorithm is almost always the method of choice unless the computational budget is very tight we also extend existing methods in various ways and provide unifying code implementing all approaches

bayes inference in the tobit censored regression model we consider the bayes estimation of the tobit censored regression model with normally distributed errors a simple condition for the existence of posterior moments is provided suitable versions of monte carlo procedures based on symmetric multivariate t distributions and laplacian approximations in a certain parametrization are developed and illustrated ideas involving data augmentation and gibbs sampling cf tanner and wong 1987 and gelfand and smith 1990 are also developed the methods are compared in two examples with diffuse priors and various combinations of sample sizes and degrees of censoring

on prediction intervals based on predictive likelihood or bootstrap methods we argue that prediction intervals based on predictive likelihood do not correct for curvature with respect to the parameter value when they implicitly approximate an unknown probability density partly as a result of this difficulty the order of coverage error associated with predictive intervals and predictive limits is equal to only the inverse of sample size in this respect those methods do not improve on the simpler naive or estimative approach moreover in cases of practical importance the latter can be preferable in terms of both the size and sign of coverage error we show that bootstrap calibration of both naive and predictive likelihood approaches increases coverage accuracy of prediction intervals by an order of magnitude and in the case of naive intervals preserves that method s numerical and analytical simplicity therefore we argue the bootstrap calibrated naive approach is a particularly competitive alternative to more conventional but more complex techniques based on predictive likelihood

counting and equidistribution in heisenberg groups we strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications that arises from the action of arithmetic groups on complex hyperbolic spaces especially in dimension 2 we prove a mertens formula for the integer points over a quadratic imaginary number fields k in the light cone of hermitian forms as well as an equidistribution theorem of the set of rational points over k in heisenberg groups we give a counting formula for the cubic points over k in the complex projective plane whose galois conjugates are orthogonal and isotropic for a given hermitian form over k and a counting and equidistribution result for arithmetic chains in the heisenberg group when their cygan diameter tends to 0

bayesian leave one out cross validation approximations for gaussian latent variable models the future predictive performance of a bayesian model can be estimated using bayesian cross validation in this article we consider gaussian latent variable models where the integration over the latent values is approximated using the laplace method or expectation propagation ep we study the properties of several bayesian leave one out loo cross validation approximations that in most cases can be computed with a small additional cost after forming the posterior approximation given the full data our main objective is to assess the accuracy of the approximative loo cross validation estimators that is for each method laplace and ep we compare the approximate fast computation with the exact brute force loo computation secondarily we evaluate the accuracy of the laplace and ep approximations themselves against a ground truth established through extensive markov chain monte carlo simulation our empirical results show that the approach based upon a gaussian approximation to the loo marginal distribution the so called cavity distribution gives the most accurate and reliable results among the fast methods

application of nonlinear models with random coefficients to growth data summary the application of nonlinear random coefficient models to the analysis of growth curve data is described the approach is further developed for the estimation of mean growth curves and their variability from mark recapture data when the age of an animal at first capture is unknown but the time between successive captures is known these methods are of wide applicability as shown by the analysis of data on the growth of emus dromaius novaehollandiae noisy scrub birds atrichornis clamosus and whelks dicathais aegaota simulations of a mark recapture experiment on the rock spiny lobster panulirus argus in mexico using a known growth curve showed the effectiveness of the approach in estimating both the population s growth curve and the variability in individual growth

bayesian classification with gaussian processes we consider the problem of assigning an input vector to one of m classes by predicting p c x for c 1 m for a two class problem the probability of class one given x is estimated by spl sigma y x where spl sigma y 1 1 e sup y a gaussian process prior is placed on y x and is combined with the training data to obtain predictions for new x points we provide a bayesian treatment integrating over uncertainty in y and in the parameters that control the gaussian process prior the necessary integration over y is carried out using laplace s approximation the method is generalized to multiclass problems m 2 using the softmax function we demonstrate the effectiveness of the method on a number of datasets

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea