Bayesian model selection for group studies — Revisited

The creep deformation process is constructed by complex interactions of multiple factors, and the measurement of creep deformation requires enormous economic costs and a long experimental time, so there is a small amount of measurement data. In such a situation, multiple models are often proposed to explain the same experimental data. The coexistence of multiple models based on different physical assumptions makes it difficult to understand the creep deformation process. The purpose of this study is to construct a framework to compare and evaluate coexistence models based on measurement data using the Bayesian model selection framework. Basically, in the creep deformation model, basis functions, usually the exponential function eat or power function tb, corresponding to the primary creep stage and the tertiary creep stage are set, and the time series of the whole creep deformation process is regressed as the linear sum of these bases. The parameters a and b of the basis functions are parameters that must also be optimized to regress the time series data. Izuno et al. proposed the Bayesian model selection method for the creep deformation process, and it was confirmed that the model with the linear term corresponding to the secondary creep stage, called the modified theta method, is a good model compared to the model without adding the linear term. To obtain the criterion of Bayesian model selection analytically, which can reduce the computational cost greatly, Izuno et al. treated only the regression coefficients as the probabilistic parameter of the creep model. In this method, parameters a and b of the basis functions are optimized by a grid search. Because the grid search requires significant computational costs if multiple basis functions are set for each creep stage, as in the model proposed by Kimura et al., it is difficult to apply the algorithm to models with multiple bases. In addition, if parameters a and b of the basis functions correlate with the regression coefficients, the method of Izuno et al. would offer a biased evaluation of the models. To improve such an existing method, we proposed a Bayesian model selection method that treats the parameters of the basis functions as probabilistic parameters. The large computational cost to treat the parameters of the basis functions as probabilistic parameters was reduced using the exchange Monte Carlo method, which is an efficient sampling method. By applying our proposed method to the evaluation of the creep deformation model, information was obtained to understand the creep deformation process deeply. For example, it was confirmed that there exists a model in which there is a correlation between the parameter of basis function and the regression coefficients.

The aim of this review is to provide a comprehensive survey of statistical challenges in neuroimaging data analysis, from neuroimaging techniques to large-scale neuroimaging studies and statistical learning methods. We briefly review eight popular neuroimaging techniques and their potential applications in neuroscience research and clinical translation. We delineate four themes of neuroimaging data and review major image processing analysis methods for processing neuroimaging data at the individual level. We briefly review four large-scale neuroimaging-related studies and a consortium on imaging genomics and discuss four themes of neuroimaging data analysis at the population level. We review nine major population-based statistical analysis methods and their associated statistical challenges and present recent progress in statistical methodology to address these challenges.

Research on asymptotic model selection in the context of stochastic differential equations (SDEs) is almost nonexistent in the literature. In particular, when a collection of SDEs is considered, the problem of asymptotic model selection has not been hitherto investigated. Indeed, even though the diffusion coefficients may be considered known, questions on appropriate choice of the drift functions constitute a non-trivial model selection problem. In this article, we develop the asymptotic theory for comparisons between collections of SDEs with respect to the choice of drift functions using Bayes factors when the number of equations (individuals) in the collection of SDEs tends to infinity while the time domains remain bounded for each equation. Our asymptotic theory covers situations when the observed processes associated with the SDEs are independently and identically distributed (iid), as well as when they are independently but not identically distributed (non-iid). In particular, we allow incorporation of available time-dependent covariate information into each SDE through a multiplicative factor of the drift function; we also permit different initial values and domains of observations for the SDEs. Our model selection problem thus encompasses selection of a set of appropriate time-dependent covariates from a set of available time-dependent covariates, besides selection of the part of the drift function free of covariates. For both iid and non-iid set-ups, we establish almost sure exponential convergence of the Bayes factor. Furthermore, we demonstrate with simulation studies that even in non-asymptotic scenarios Bayes factor successfully captures the right set of covariates.

This article deals with the problem of testing for the equality of the shape parameters in two inverse Weibull distributions. We propose Bayesian hypothesis testing procedures for the equality of the shape parameters under the noninformative prior. The noninformative prior is usually improper which yields a calibration problem that makes the Bayes factor to be defined up to a multiplicative constant. So we propose the default Bayesian hypothesis testing procedures based on the fractional Bayes factor and the intrinsic Bayes factors under the reference priors. Simulation study and an example are provided.

Intrinsic priors for testing exponential means

Variable selection can be naturally seen as a model selection problem where the entertained models differ in which subset of variables explains the outcome of interest. Posterior model probabilities are a simple function of Bayes factors, a key inferential tool in Bayesian analysis. This approach to variable selection automatically provides sparse answers along with probabilistic assessments regarding their credibility. This methodology is, however, not exempt from difficulties including prior elicitation and numerical challenges related with its practical implementation. Particularly in the context of linear and generalized linear models, the so-called g-priors are often used due to their appealing properties formally described in Bayarri et al. (2012). In this chapter we review the implementation of variable selection through Bayes factors in linear and generalized linear models, using g-priors. Emphasis is placed on providing: i) practical guides for implementation, including documentation for the use of R packages and ii) the analysis of real examples which illustrate the enormous potential of this approach to variable selection.

Bayesian model selection and parameter estimation is attracting a lot of interest in the astronomical community because of its power and logical consistency. Markov chain Monte Carlo provides the computational power for Bayesian parameter estimation problems in large parameter spaces but needs to be supported with other numerical techniques for efficient exploration of multi-modal probability distributions. Bayesian model selection is easy in concept but remains a difficult challenge for large parameter spaces. My comments here on the paper by Roberto Trotta are based on lessons learned from developing a controlled statistical fusion approach to some of these issues.KeywordsMarkov Chain Monte CarloMarginal LikelihoodProposal DistributionMarkov Chain Monte Carlo AlgorithmMarkov Chain Monte Carlo SampleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the signal processing literature, many methods have been proposed for solving the important model comparison and selection problem. However, most of these methods only find the most likely model or only work well under particular circumstances such as a large number of data points or a high signal-to-noise ratio (SNR). One of the most successful classes of methods is the Bayesian information criteria (BIC) and in this paper, we extend some of the recent work on the BIC. In particular, we develop methods in a full Bayesian framework which work well across a large/small number of data points and high/low SNR for either real- or complex-valued data originating from a regression model. Aside from selecting the most probable model, these rules can also be used for model averaging as they assign a probability to each candidate model. Through simulations on a polynomial trend model, we demonstrate that the proposed rules outperform other rules in terms of detecting the true model order, de-noising the noisy signal, and making predictions of unobserved data points. The simulation code is available online.

The Analysis of Variance technique estimates variance components by comparing their mean squares to their expected values. Nevertheless, this method could give variance component estimates that are found outside the parameter space, i.e. negative estimates. In a bid to overcome this deficiency, alternate approaches are essential, and likelihood-based approaches have become common over time. Bayesian techniques have also been proposed and Bayes factors developed for examining various models. We applied the Bayes factor proposed by Faulkenberry (2018) to a Balanced Two Way ANOVA under three (3) cases, namely Case 1: the levels of the two factors are fixed; Case 2: the levels of the two factors are random; and Case 3: the levels of one factor are considered as fixed, while the levels of the other factor are considered as random. We realized that when the levels of the two factors are fixed, the Bayesian conclusion about the variability in the effects is in line with that of a frequentist. But when the same data set was considered to be wholly or partly as sample observations drawn randomly from a given population of interest, the Bayesian conclusion differed slightly from that of the frequentist. Keywords: Bayes Factor; Bayesian; Frequentist, Fixed; Random.

기존의 frequentist 추론에 비해 Bayesian 추론에서의 가설 검정 및 모형 선택 문제는 학자들 간에 일치된 견해를 보이지 못하고 있으며 아직도 논란이 되는 것들이 많다. Bayesian 추론에서 가설 검정 및 모형 선택의 기준으로 널리 쓰이는 Bayes factor는 이해하기 쉬우나 여러 경우에 구하기 어려운 단점이 존재한다. 그 외에 다른 기준으로 Spiegelhalter 등 (2002)가 제시한 DIC(Deviance Information Criterion)과 frequentist 추론에서의 P-value에 대비되는 Bayesian P-value가 있다. 본 논문에서는 Swiss banknote 자료를 Bayesian 로지스틱 회귀모형으로 분석하고 관련 기준들을 구하여 각 기준들이 일관성 있는 결론을 보이는지 확인하고자 한다. Model selection and hypothesis testing problems in Bayesian inference are still debated between scholars. Bayesian factors traditionally used as a criterion in Bayesian hypothesis testing and model selection, are easy to understand but sometimes hard to compute. In addition, there are other model selection criterions such as DIC(Deviance Information Criterion) by Spiegelhalter et al. (2002) and Bayesian P-values for testing. In this paper, we briefly introduce the Bayesian hypothesis testing and model selection procedure. In addition we have applied a Bayesian inference to Swiss banknote data by a fitting logistic regression model and computing several test statistics to see if they provide consistent results.

Zellner's $g$-prior is a popular prior choice for the model selection problems in the context of normal regression models. Wang and Sun [J. Statist. Plann. Inference 147 (2014) 95-105] recently adopt this prior and put a special hyper-prior for $g$, which results in a closed-form expression of Bayes factor for nested linear model comparisons. They have shown that under very general conditions, the Bayes factor is consistent when two competing models are of order $O(n^{\tau})$ for $\tau <1$ and for $\tau=1$ is almost consistent except a small inconsistency region around the null hypothesis. In this paper, we study Bayes factor consistency for nonnested linear models with a growing number of parameters. Some of the proposed results generalize the ones of the Bayes factor for the case of nested linear models. Specifically, we compare the asymptotic behaviors between the proposed Bayes factor and the intrinsic Bayes factor in the literature.

Equality and inequality constrained multivariate linear models: Objective model selection using constrained posterior priors

MEG source localization under multiple constraints: An extended Bayesian framework

Posterior simulation and Bayes factors in panel count data models

Schizophrenia is a chronic neurobiological disorder whose early detection has attracted significant attention from the clinical, psychiatric, and also artificial intelligence communities. This latter approach has been mainly focused on the analysis of neuroimaging and genetic data. A less explored strategy consists in exploiting the power of natural language processing (NLP) algorithms applied over narrative texts produced by schizophrenic subjects. In this paper, a novel dataset collected from a proper field study is presented. Also, grammatical traits discovered in narrative documents are used to build computational representations of texts, allowing an automatic classification of discourses generated by schizophrenic and non-schizophrenic subjects. The attained results showed that the use of the proposed computational representations along with machine learning techniques enables a novel and precise strategy to automatically detect texts produced by schizophrenic subjects.