Approximate Bayesian Inference

This is the Editorial article summarizing the scope of the Special Issue: Approximate Bayesian Inference.     

野生中药材供给与稽查的不完全信息动态博弈分析

针对中药材供给中的信息不对称性,建立了三个中药材供给方与政府稽查部门之间的三个不完全信息动态博弈模型,得到了相应的子博弈精炼贝叶斯均衡解的五个结论.这五个结论合理刻画了中药材供给方与政府稽查部门的博弈行为并揭示了非法采挖野生中药材现象难以杜绝的原因.结果表明,政府对稽查行为的条件激励措施并不能有效杜绝非法采挖野生中药材现象,只有政府对所有的稽查行为实施普遍的强激励机制才可从根本上消除非法采挖野生中药材的现象.     

Comments on “Likelihood-based belief function: Justification and some extensions to low-quality data” by Thierry Denœux

The paper by Denœux justifies the use of a consonant belief function to represent the information provided by a likelihood function and proposes some extensions to low-quality data. In my comments I consider the point of view of imprecise probabilities for the representation of likelihood information and the relationships with the proposal in the paper. I also argue for some alternatives to the use of consonant belief functions. Finally, I add some clarifications about the comparison with the Bayesian approach.     

Divide-and-Conquer With Sequential Monte Carlo

We propose a novel class of Sequential Monte Carlo (SMC) algorithms, appropriate for inference in probabilistic graphical models. This class of algorithms adopts a divide-and-conquer approach based upon an auxiliary tree-structured decomposition of the model of interest, turning the overall inferential task into a collection of recursively solved subproblems. The proposed method is applicable to a broad class of probabilistic graphical models, including models with loops. Unlike a standard SMC sampler, the proposed divide-and-conquer SMC employs multiple independent populations of weighted particles, which are resampled, merged, and propagated as the method progresses. We illustrate empirically that this approach can outperform standard methods in terms of the accuracy of the posterior expectation and marginal likelihood approximations. Divide-and-conquer SMC also opens up novel parallel implementation options and the possibility of concentrating the computational effort on the most challenging subproblems. We demonstrate its performance on a Markov random field and on a hierarchical logistic regression problem. Supplementary materials including proofs and additional numerical results are available online.     

A Marginal Sampler for σ-Stable Poisson–Kingman Mixture Models

We investigate the class of σ-stable Poisson–Kingman random probability measures (RPMs) in the context of Bayesian nonparametric mixture modeling. This is a large class of discrete RPMs, which encompasses most of the popular discrete RPMs used in Bayesian nonparametrics, such as the Dirichlet process, Pitman–Yor process, the normalized inverse Gaussian process, and the normalized generalized Gamma process. We show how certain sampling properties and marginal characterizations of σ-stable Poisson–Kingman RPMs can be usefully exploited for devising a Markov chain Monte Carlo (MCMC) algorithm for performing posterior inference with a Bayesian nonparametric mixture model. Specifically, we introduce a novel and efficient MCMC sampling scheme in an augmented space that has a small number of auxiliary variables per iteration. We apply our sampling scheme to a density estimation and clustering tasks with unidimensional and multidimensional datasets, and compare it against competing MCMC sampling schemes. Supplementary materials for this article are available online.     

Efficient Computation of Bayesian Optimal Discriminating Designs

An efficient algorithm for the determination of Bayesian optimal discriminating designs for competing regression models is developed, where the main focus is on models with general distributional assumptions beyond the “classical” case of normally distributed homoscedastic errors. For this purpose, we consider a Bayesian version of the Kullback–Leibler (KL). Discretizing the prior distribution leads to local KL-optimal discriminating design problems for a large number of competing models. All currently available methods either require a large amount of computation time or fail to calculate the optimal discriminating design, because they can only deal efficiently with a few model comparisons. In this article, we develop a new algorithm for the determination of Bayesian optimal discriminating designs with respect to the Kullback–Leibler criterion. It is demonstrated that the new algorithm is able to calculate the optimal discriminating designs with reasonable accuracy and computational time in situations where all currently available procedures are either slow or fail.     

Reliability‐targeted HPLC–UV method validation—A protocol enrichment perspective

Method validation is important in analytical chemistry to obtain the reliability of an analytical method. Guidelines provided by the regulatory bodies can be used as a general framework to assess the validity of a method. Since these guidelines do not focus on the reliability of analytical results exclusively, this study was aimed to combine a few recently evolved strategies that may render analytical method validation more reliable and trustworthy. In this research, the analytical error function was determined by appropriate polynomial regression statistics that determine the range of analyte concentration that may lead to more accurate measurements by producing the least possible total error in the assay and can be regarded as a reliable weighting method. The reliability of the analytical results over a particular concentration range has been proposed by a Bayesian probability study. In order to ensure the applicability of this approach, it was applied for the validation of an HPLC–UV assay method dedicated to the quantification of cefepime and tazobactam in human plasma. A comparison between the newer approach and the usual method validation revealed that the application of analytical error function and Bayesian analysis at the end of the validation process can produce significant improvements in the analytical results.     

BFF: Bayesian,Fiducial, and Frequentist Analysis of Cognitive Engagement among Cognitively Impaired Older Adults

Engagement in cognitively demanding activities is beneficial to preserving cognitive health. Our goal was to demonstrate the utility of frequentist, Bayesian, and fiducial statistical methods for evaluating the robustness of effects in identifying factors that contribute to cognitive engagement for older adults experiencing cognitive decline. We collected a total of 504 observations across two longitudinal waves of data from 28 cognitively impaired older adults. Participants’ systolic blood pressure responsivity, an index of cognitive engagement, was continuously sampled during cognitive testing. Participants reported on physical and mental health challenges and provided hair samples to assess chronic stress at each wave. Using the three statistical paradigms, we compared results from six model testing levels and longitudinal changes in health and stress predicting changes in cognitive engagement. Findings were mostly consistent across the three paradigms, providing additional confidence in determining effects. We extend selective engagement theory to cognitive impairment, noting that health challenges and stress appear to be important moderators. Further, we emphasize the utility of the Bayesian and fiducial paradigms for use with relatively small sample sizes because they are not based on asymptotic distributions. In particular, the fiducial paradigm is a useful tool because it provides more information than p values without the need to specify prior distributions, which may unduly influence the results based on a small sample. We provide the R code used to develop and implement all models.     

Variational Message Passing and Local Constraint Manipulation in Factor Graphs

Accurate evaluation of Bayesian model evidence for a given data set is a fundamental problem in model development. Since evidence evaluations are usually intractable, in practice variational free energy (VFE) minimization provides an attractive alternative, as the VFE is an upper bound on negative model log-evidence (NLE). In order to improve tractability of the VFE, it is common to manipulate the constraints in the search space for the posterior distribution of the latent variables. Unfortunately, constraint manipulation may also lead to a less accurate estimate of the NLE. Thus, constraint manipulation implies an engineering trade-off between tractability and accuracy of model evidence estimation. In this paper, we develop a unifying account of constraint manipulation for variational inference in models that can be represented by a (Forney-style) factor graph, for which we identify the Bethe Free Energy as an approximation to the VFE. We derive well-known message passing algorithms from first principles, as the result of minimizing the constrained Bethe Free Energy (BFE). The proposed method supports evaluation of the BFE in factor graphs for model scoring and development of new message passing-based inference algorithms that potentially improve evidence estimation accuracy.