Reversible jump Markov chain Monte Carlo (RJMCMC) extends ordinary MCMC methods for use in Bayesian multimodel inference. We show that RJMCMC can be implemented as Gibbs sampling with alternating updates of a model indicator and a vector-valued "palette" of parameters denoted $\bm ψ$. Like an artist uses the palette to mix dabs of color for specific needs, we create model-specific parameters from the set available in $\bm ψ$. This description not only removes some of the mystery of RJMCMC, but also provides a basis for fitting models one at a time using ordinary MCMC and computing model weights or Bayes factors by post-processing the Monte Carlo output. We illustrate our procedure using several examples.
Bayesian and other likelihood-based methods require specification of a statistical model and may not be fully satisfactory for inference on quantities, such as quantiles, that are not naturally defined as model parameters. In this paper, we construct a direct and model-free Gibbs posterior distribution for multivariate quantiles. Being model-free means that inferences drawn from the Gibbs posterior are not subject to model misspecification bias, and being direct means that no priors for or marginalization over nuisance parameters are required. We show here that the Gibbs posterior enjoys a roo…
Real-world problems, often couched as machine learning applications, involve quantities of interest that have real-world meaning, independent of any statistical model. To avoid potential model misspecification bias or over-complicating the problem formulation, a direct, model-free approach is desired. The traditional Bayesian framework relies on a model for the data-generating process so, apparently, the desired direct, model-free, posterior-probabilistic inference is out of reach. Fortunately, likelihood functions are not the only means of linking data and quantities of interest. Loss functio…
Duke University（杜克大学）；George Mason University（乔治梅森大学）
In this work, we adopt a general framework based on the Gibbs posterior to update belief distributions for inverse problems governed by partial differential equations (PDEs). The Gibbs posterior formulation is a generalization of standard Bayesian inference that only relies on a loss function connecting the unknown parameters to the data. It is particularly useful when the true data generating mechanism (or noise distribution) is unknown or difficult to specify. The Gibbs posterior coincides with Bayesian updating when a true likelihood function is known and the loss function corresponds to th…
Detection of an image boundary when the pixel intensities are measured with noise is an important problem in image segmentation, with numerous applications in medical imaging and engineering. From a statistical point of view, the challenge is that likelihood-based methods require modeling the pixel intensities inside and outside the image boundary, even though these are typically of no practical interest. Since misspecification of the pixel intensity models can negatively affect inference on the image boundary, it would be desirable to avoid this modeling step altogether. Towards this, we deve…
In the popular approach of "Bayesian variable selection" (BVS), one uses prior and posterior distributions to select a subset of candidate variables to enter the model. A completely new direction will be considered here to study BVS with a Gibbs posterior originating in statistical mechanics. The Gibbs posterior is constructed from a risk function of practical interest (such as the classification error) and aims at minimizing a risk function without modeling the data probabilistically. This can improve the performance over the usual Bayesian approach, which depends on a probability model which…
The current standard Bayesian approach to model calibration, which assigns a Gaussian process prior to the discrepancy term, often suffers from issues of unidentifiability and computational complexity and instability. When the goal is to quantify uncertainty in physical parameters for extrapolative prediction, then there is no need to perform inference on the discrepancy term. With this in mind, we introduce Gibbs posteriors as an alternative Bayesian method for model calibration, which updates the prior with a loss function connecting the data to the parameter. The target of inference is the …
In this paper we consider a Bayesian framework for making inferences about dynamical systems from ergodic observations. The proposed Bayesian procedure is based on the Gibbs posterior, a decision theoretic generalization of standard Bayesian inference. We place a prior over a model class consisting of a parametrized family of Gibbs measures on a mixing shift of finite type. This model class generalizes (hidden) Markov chain models by allowing for long range dependencies, including Markov chains of arbitrarily large orders. We characterize the asymptotic behavior of the Gibbs posterior distribu…
IIt is known that a statistically significant treatment may not be clinically significant. A quantity that can be used to assess clinical significance is called the minimum clinically important difference (MCID), and inference on the MCID is an important and challenging problem. Modeling for the purpose of inference on the MCID is non-trivial, and concerns about bias from a misspecified parametric model or inefficiency from a nonparametric model motivate an alternative approach to balance robustness and efficiency. In particular, a recently proposed representation of the MCID as the minimizer …
In mathematical finance, Levy processes are widely used for their ability to model both continuous variation and abrupt, discontinuous jumps. These jumps are practically relevant, so reliable inference on the feature that controls jump frequencies and magnitudes, namely, the Levy density, is of critical importance. A specific obstacle to carrying out model-based (e.g., Bayesian) inference in such problems is that, for general Levy processes, the likelihood is intractable. To overcome this obstacle, here we adopt a Gibbs posterior framework that updates a prior distribution using a suitable los…
The PAC-Bayesian approach is a powerful set of techniques to derive non- asymptotic risk bounds for random estimators. The corresponding optimal distribution of estimators, usually called the Gibbs posterior, is unfortunately intractable. One may sample from it using Markov chain Monte Carlo, but this is often too slow for big datasets. We consider instead variational approximations of the Gibbs posterior, which are fast to compute. We undertake a general study of the properties of such approximations. Our main finding is that such a variational approximation has often the same rate of converg…
The area under the receiver operating characteristic curve (AUC) serves as a summary of a binary classifier's performance. Methods for estimating the AUC have been developed under a binormality assumption which restricts the distribution of the score produced by the classifier. However, this assumption introduces an infinite-dimensional nuisance parameter and can be inappropriate, especially in the context of machine learning. This motivates us to adopt a model-free Gibbs posterior distribution for the AUC. We present the asymptotic Gibbs posterior concentration rate, and a strategy for tuning…
We consider a Bayesian hierarchical version of the normal theory general linear model which is practically relevant in the sense that it is general enough to have many applications and it is not straightforward to sample directly from the corresponding posterior distribution. Thus we study a block Gibbs sampler that has the posterior as its invariant distribution. In particular, we establish that the Gibbs sampler converges at a geometric rate. This allows us to establish conditions for a central limit theorem for the ergodic averages used to estimate features of the posterior. Geometric ergod…
Youden's index cutoff is a classifier mapping a patient's diagnostic test outcome and available covariate information to a diagnostic category. Typically the cutoff is estimated indirectly by first modeling the conditional distributions of test outcomes given diagnosis and then choosing the optimal cutoff for the estimated distributions. Here we present a Gibbs posterior distribution for direct inference on the cutoff. Our approach makes incorporating prior information about the cutoff much easier compared to existing methods, and does so without specifying probability models for the data, whi…
PAC-Bayesian algorithms and Gibbs posteriors are gaining popularity due to their robustness against model misspecification even when Bayesian inference is inconsistent. The PAC-Bayesian alpha-posterior is a generalization of the standard Bayes posterior which can be tempered with a parameter alpha to handle inconsistency. Data driven methods for tuning alpha have been proposed but are still few, and are often computationally heavy. Additionally, the adequacy of these methods in cases where we use variational approximations instead of exact alpha-posteriors is not clear. This narrows their usag…
The Bayesian lasso is well-known as a Bayesian alternative for Lasso. Although the advantage of the Bayesian lasso is capable of full probabilistic uncertain quantification for parameters, the corresponding posterior distribution can be sensitive to outliers. To overcome such problem, robust Bayesian regression models have been proposed in recent years. In this paper, we consider the robust and efficient estimation for the Bayesian Huberized lasso regression in fully Bayesian perspective. A new posterior computation algorithm for the Bayesian Huberized lasso regression is proposed. The propose…
Recently, the Wasserstein loss function has been proven to be effective when applied to deterministic full-waveform inversion (FWI) problems. We consider the application of this loss function in Bayesian FWI so that the uncertainty can be captured in the solution. Other loss functions that are commonly used in practice are also considered for comparison. Existence and stability of the resulting Gibbs posteriors are shown on function space under weak assumptions on the prior and model. In particular, the distribution arising from the Wasserstein loss is shown to be quite stable with respect to …
The Horseshoe is a widely used and popular continuous shrinkage prior for high-dimensional Bayesian linear regression. Recently, regularized versions of the Horseshoe prior have also been introduced in the literature. Various Gibbs sampling Markov chains have been developed in the literature to generate approximate samples from the corresponding intractable posterior densities. Establishing geometric ergodicity of these Markov chains provides crucial technical justification for the accuracy of asymptotic standard errors for Markov chain based estimates of posterior quantities. In this paper, w…
Use of continuous shrinkage priors -- with a "spike" near zero and heavy-tails towards infinity -- is an increasingly popular approach to induce sparsity in parameter estimates. When the parameters are only weakly identified by the likelihood, however, the posterior may end up with tails as heavy as the prior, jeopardizing robustness of inference. A natural solution is to "shrink the shoulders" of a shrinkage prior by lightening up its tails beyond a reasonable parameter range, yielding a regularized version of the prior. We develop a regularization approach which, unlike previous proposals, p…
The Bayesian posterior probability of the true state is stochastically dominated by that same posterior under the probability law of the true state. This generalizes to notions of "optimism" about posterior probabilities.
We characterise the convergence of the Gibbs sampler which samples from the joint posterior distribution of parameters and missing data in hierarchical linear models with arbitrary symmetric error distributions. We show that the convergence can be uniform, geometric or sub-geometric depending on the relative tail behaviour of the error distributions, and on the parametrisation chosen. Our theory is applied to characterise the convergence of the Gibbs sampler on latent Gaussian process models. We indicate how the theoretical framework we introduce will be useful in analyzing more complex models…
Minimum Bayes factors are commonly used to transform two-sided p-values to lower bounds on the posterior probability of the null hypothesis, as in Pericchi et al. (2017). In this article, we show posterior probabilities of hypothesis by transforming the commonly used -eplog(p), proposed by Vovk (1993) and Sellke et al. (2001). This is achieved after adjusting this minimum Bayes factor with the information to approximate it to an exact Bayes factor, not only when p is a p-value but also when p is a pseudo p-value in the sense of Casella and Berger (2001). Additionally, we show the fit to a refi…
In this study, we present a multi-class graphical Bayesian predictive classifier that incorporates the uncertainty in the model selection into the standard Bayesian formalism. For each class, the dependence structure underlying the observed features is represented by a set of decomposable Gaussian graphical models. Emphasis is then placed on the Bayesian model averaging which takes full account of the class-specific model uncertainty by averaging over the posterior graph model probabilities. An explicit evaluation of the model probabilities is well known to be infeasible. To address this issue…
The focus in this paper is Bayesian system identification based on noisy incomplete modal data where we can impose spatially-sparse stiffness changes when updating a structural model. To this end, based on a similar hierarchical sparse Bayesian learning model from our previous work, we propose two Gibbs sampling algorithms. The algorithms differ in their strategies to deal with the posterior uncertainty of the equation-error precision parameter, but both sample from the conditional posterior probability density functions (PDFs) for the structural stiffness parameters and system modal parameter…
High-dimensional state trajectories of state-space models pose challenges for Bayesian inference. Particle Gibbs (PG) methods have been widely used to sample from the posterior of a state space model. Basically, particle Gibbs is a Particle Markov Chain Monte Carlo (PMCMC) algorithm that mimics the Gibbs sampler by drawing model parameters and states from their conditional distributions. This tutorial provides an introductory view on Particle Gibbs (PG) method and its extensions and variants, and illustrates through several examples of inference in non-linear state space models (SSMs). We also…
An exponential-time exact algorithm is provided for the task of clustering n items of data into k clusters. Instead of seeking one partition, posterior probabilities are computed for summary statistics: the number of clusters, and pairwise co-occurrence. The method is based on subset convolution, and yields the posterior distribution for the number of clusters in O(n * 3^n) operations, or O(n^3 * 2^n) using fast subset convolution. Pairwise co-occurrence probabilities are then obtained in O(n^3 * 2^n) operations. This is considerably faster than exhaustive enumeration of all partitions.
The MC$^3$ (Madigan and York, 1995) and Gibbs (George and McCulloch, 1997) samplers are the most widely implemented algorithms for Bayesian Model Averaging (BMA) in linear regression models. These samplers draw a variable at random in each iteration using uniform selection probabilities and then propose to update that variable. This may be computationally inefficient if the number of variables is large and many variables are redundant. In this work, we introduce adaptive versions of these samplers that retain their simplicity in implementation and reduce the selection probabilities of the many…
Estimating the marginal likelihoods is an essential feature of model selection in the Bayesian context. It is especially crucial to have good estimates when assessing the number of planets orbiting stars when the models explain the noisy data with different numbers of Keplerian signals. We introduce a simple method for approximating the marginal likelihoods in practice when a statistically representative sample from the parameter posterior density is available. We use our truncated posterior mixture estimate to receive accurate model probabilities for models with differing number of Keplerian …
We consider posterior sampling in the very common Bayesian hierarchical model in which observed data depends on high-dimensional latent variables that, in turn, depend on relatively few hyperparameters. When the full conditional over the latent variables has a known form, the marginal posterior distribution over hyperparameters is accessible and can be sampled using a Markov chain Monte Carlo (MCMC) method on a low-dimensional parameter space. This may improve computational efficiency over standard Gibbs sampling since computation is not over the high-dimensional space of latent variables and …
The logistic linear mixed model (LLMM) is one of the most widely used statistical models. Generally, Markov chain Monte Carlo algorithms are used to explore the posterior densities associated with the Bayesian LLMMs. Polson, Scott and Windle's (2013) Polya-Gamma data augmentation (DA) technique can be used to construct full Gibbs (FG) samplers for the LLMMs. Here, we develop efficient block Gibbs (BG) samplers for Bayesian LLMMs using the Polya-Gamma DA method. We compare the FG and BG samplers in the context of a real data example, as the correlation between the fixed effects and the random e…
In this article, we consider Markov chain Monte Carlo(MCMC) algorithms for exploring the intractable posterior density associated with Bayesian probit linear mixed models under improper priors on the regression coefficients and variance components. In particular, we construct the two-block Gibbs sampler using the data augmentation (DA) techniques. Furthermore, we prove geometric ergodicity of the Gibbs sampler, which is the foundation for building central limit theorems for MCMC based estimators and subsequent inferences. The conditions for geometric convergence are similar to those guaranteei…
Gibbs-type random probability measures and the exchangeable random partitions they induce represent the subject of a rich and active literature. They provide a probabilistic framework for a wide range of theoretical and applied problems that are typically referred to as species sampling problems. In this paper, we consider the class of looking-backward species sampling problems introduced in Lijoi et al. (Ann. Appl. Probab. 18 (2008) 1519-1547) in Bayesian nonparametrics. Specifically, given some information on the random partition induced by an initial sample from a Gibbs-type random probabil…
This paper focuses on the influence of a misspecified covariance structure on false discovery rate for the large scale multiple testing problem. Specifically, we evaluate the influence on the marginal distribution of local fdr statistics, which are used in many multiple testing procedures and related to Bayesian posterior probabilities. Explicit forms of the marginal distributions under both correctly specified and incorrectly specified models are derived. The Kullback-Leibler divergence is used to quantify the influence caused by a misspecification. Several numerical examples are provided to …
We give yet another example of the usefulness of working with marginals of multivariate Gibbs distributions (Cerquetti, 2013) in deriving Bayesian nonparametric estimators under Gibbs priors in species sampling problems. Here in particular we substantially reduce length and complexity of the proofs in Bacallado et al. (2013, Th. 1, and Th. 2) for looking backward probabilities under incomplete information.
Possible parameter values in a random sampling model are shown by definition to have uniform base-rate prior probabilities. This allows a frequentist posterior probability distribution to be calculated for such possible parameter values conditional solely on actual study observations. If the likelihood probability distribution of a random selection is modelled with a symmetrical continuous function then the frequentist posterior probability of something equal to or more extreme than the null hypothesis will be equal to the P-value; otherwise the P value would be an approximation. An idealistic…
A Bayesian nonparametric approach to the study of species diversity based on choosing a random discrete distribution as a prior model for the unknown relative abundances of species has been recently introduced in Lijoi et al. (2007, 2008). Explicit posterior predictive estimation of {\it species richness} has been obtained under priors belonging to the $α$-Gibbs class (Gnedin & Pitman, 2006). Here we focus on posterior estimation of {\it species evenness} which accounts for diversity in terms of the proximity to the situation of uniform distribution of the population into different species. We…
We put forward a novel calibration of p values, the "Adaptive Robust Lower Bound" (ARLB) which maps p values into approximations of posterior probabilities taking into account the effect of sample sizes. We build on the Robust Lower Bound proposed by Sellke, Bayarri and Berger (2001), but we incorporate a simple power of the sample size to make it adaptive to different amounts of data. We present several illustrations from where it is apparent that the ARLB closely approximates exact Bayes Factors. In particular, it has the same asymptotics as posterior probabilities but avoiding the problems …
The Gibbs reference posterior distribution provides an objective full-Bayesian solution to the problem of prediction of a stationary Gaussian process with Matérn anisotropic kernel. A full-Bayesian approach is possible, because the posterior distribution is expressed as the invariant distribution of a uniformly ergodic Markovian kernel for which we give an explicit expression. In this paper, we show that it is appropriate for the Universal Kriging framework, that is when an unknown function is added to the stationary Gaussian process. We give sufficient conditions for the existence and proprie…
We study the problem of learning Bayesian network structures from data. Koivisto and Sood (2004) and Koivisto (2006) presented algorithms that can compute the exact marginal posterior probability of a subnetwork, e.g., a single edge, in O(n2n) time and the posterior probabilities for all n(n-1) potential edges in O(n2n) total time, assuming that the number of parents per node or the indegree is bounded by a constant. One main drawback of their algorithms is the requirement of a special structure prior that is non uniform and does not respect Markov equivalence. In this paper, we develop an alg…
It is well known that the Lasso can be interpreted as a Bayesian posterior mode estimate with a Laplacian prior. Obtaining samples from the full posterior distribution, the Bayesian Lasso, confers major advantages in performance as compared to having only the Lasso point estimate. Traditionally, the Bayesian Lasso is implemented via Gibbs sampling methods which suffer from lack of scalability, unknown convergence rates, and generation of samples that are necessarily correlated. We provide a measure transport approach to generate i.i.d samples from the posterior by constructing a transport map …
We study frequentist properties of Bayesian and $L_0$ model selection, with a focus on (potentially non-linear) high-dimensional regression. We propose a construction to study how posterior probabilities and normalized $L_0$ criteria concentrate on the (Kullback-Leibler) optimal model and other subsets of the model space. When such concentration occurs, one also bounds the frequentist probabilities of selecting the correct model, type I and type II errors. These results hold generally, and help validate the use of posterior probabilities and $L_0$ criteria to control frequentist error probabil…
Bayesian analysis of data from the general linear mixed model is challenging because any nontrivial prior leads to an intractable posterior density. However, if a conditionally conjugate prior density is adopted, then there is a simple Gibbs sampler that can be employed to explore the posterior density. A popular default among the conditionally conjugate priors is an improper prior that takes a product form with a flat prior on the regression parameter, and so-called power priors on each of the variance components. In this paper, a convergence rate analysis of the corresponding Gibbs sampler i…
This paper presents a new Markov chain Monte Carlo method to sample from the posterior distribution of conjugate mixture models. This algorithm relies on a flexible split-merge procedure built using the particle Gibbs sampler. Contrary to available split-merge procedures, the resulting so-called Particle Gibbs Split-Merge sampler does not require the computation of a complex acceptance ratio, is simple to implement using existing sequential Monte Carlo libraries and can be parallelized. We investigate its performance experimentally on synthetic problems as well as on geolocation and cancer gen…
In this work, we propose a model for estimating volatility from financial time series, extending the non-Gaussian family of space-state models with exact marginal likelihood proposed by Gamerman, Santos and Franco (2013). On the literature there are models focused on estimating financial assets risk, however, most of them rely on MCMC methods based on Metropolis algorithms, since full conditional posterior distributions are not known. We present an alternative model capable of estimating the volatility, in an automatic way, since all full conditional posterior distributions are known, and it i…
