Reflections on Murray Aitkin's contributions to nonparametric mixture models and Bayes factors

2021 ◽  
pp. 1471082X2098131
Author(s):  
Alan Agresti ◽  
Francesco Bartolucci ◽  
Antonietta Mira
Keyword(s):  
Bayesian Inference ◽  
Mixture Models ◽  
Random Effects ◽  
Bayes Factor ◽  
Bayes Factors ◽  
Posterior Mean ◽  
Bayesian Paradigm ◽  
Research Publications ◽  
Nonparametric Mixture ◽  
Mixed Modelling

We describe two interesting and innovative strands of Murray Aitkin's research publications, dealing with mixture models and with Bayesian inference. Of his considerable publications on mixture models, we focus on a nonparametric random effects approach in generalized linear mixed modelling, which has proven useful in a wide variety of applications. As an early proponent of ways of implementing the Bayesian paradigm, Aitkin proposed an alternative Bayes factor based on a posterior mean likelihood. We discuss these innovative approaches and some research lines motivated by them and also suggest future related methodological implementations.

scholarly journals Bayesian inference for finite mixtures of univariate and multivariate skew-normal and skew-t distributions

Biostatistics ◽  
2010 ◽  
Vol 11 (2) ◽  
pp. 317-336 ◽  
Author(s):  
Sylvia Frühwirth-Schnatter ◽  
Saumyadipta Pyne
Keyword(s):  
Bayesian Inference ◽  
Mixture Models ◽  
Random Effects ◽  
Data Augmentation ◽  
Heterogeneous Data ◽  
High Dimensional ◽  
Finite Mixtures ◽  
Truncated Normal ◽  
T Distribution ◽  
Skew Normal

Abstract Skew-normal and skew-t distributions have proved to be useful for capturing skewness and kurtosis in data directly without transformation. Recently, finite mixtures of such distributions have been considered as a more general tool for handling heterogeneous data involving asymmetric behaviors across subpopulations. We consider such mixture models for both univariate as well as multivariate data. This allows robust modeling of high-dimensional multimodal and asymmetric data generated by popular biotechnological platforms such as flow cytometry. We develop Bayesian inference based on data augmentation and Markov chain Monte Carlo (MCMC) sampling. In addition to the latent allocations, data augmentation is based on a stochastic representation of the skew-normal distribution in terms of a random-effects model with truncated normal random effects. For finite mixtures of skew normals, this leads to a Gibbs sampling scheme that draws from standard densities only. This MCMC scheme is extended to mixtures of skew-t distributions based on representing the skew-t distribution as a scale mixture of skew normals. As an important application of our new method, we demonstrate how it provides a new computational framework for automated analysis of high-dimensional flow cytometric data. Using multivariate skew-normal and skew-t mixture models, we could model non-Gaussian cell populations rigorously and directly without transformation or projection to lower dimensions.

scholarly journals Computing Bayes factors to measure evidence from experiments: An extension of the BIC approximation

2018 ◽  
Vol 55 (1) ◽  
pp. 31-43 ◽  
Author(s):  
Thomas J. Faulkenberry
Keyword(s):  
Bayesian Inference ◽  
Closed Form ◽  
Analysis Of Variance ◽  
Degrees Of Freedom ◽  
Bayes Factor ◽  
T Test ◽  
Bayes Factors ◽  
Computational Tools ◽  
Testing Hypotheses

Summary Bayesian inference affords scientists powerful tools for testing hypotheses. One of these tools is the Bayes factor, which indexes the extent to which support for one hypothesis over another is updated after seeing the data. Part of the hesitance to adopt this approach may stem from an unfamiliarity with the computational tools necessary for computing Bayes factors. Previous work has shown that closed-form approximations of Bayes factors are relatively easy to obtain for between-groups methods, such as an analysis of variance or t-test. In this paper, I extend this approximation to develop a formula for the Bayes factor that directly uses information that is typically reported for ANOVAs (e.g., the F ratio and degrees of freedom). After giving two examples of its use, I report the results of simulations which show that even with minimal input, this approximate Bayes factor produces similar results to existing software solutions.

scholarly journals Benefits of Bayesian Model Selection and Averaging for Mixed-Effects Modeling

2021 ◽  
Author(s):  
Daniel W. Heck ◽  
Florence Bockting
Keyword(s):  
Model Selection ◽  
Random Effects ◽  
Mixed Models ◽  
Bayesian Model ◽  
Fixed Effects ◽  
Bayes Factor ◽  
Mixed Effects ◽  
Bayes Factors ◽  
Mixed Effects Models ◽  
Within Subjects

Bayes factors allow researchers to test the effects of experimental manipulations in within-subjects designs using mixed-effects models. van Doorn et al. (2021) showed that such hypothesis tests can be performed by comparing different pairs of models which vary in the specification of the fixed- and random-effect structure for the within-subjects factor. To discuss the question of which of these model comparisons is most appropriate, van Doorn et al. used a case study to compare the corresponding Bayes factors. We argue that researchers should not only focus on pairwise comparisons of two nested models but rather use the Bayes factor for performing model selection among a larger set of mixed models that represent different auxiliary assumptions. In a standard one-factorial, repeated-measures design, the comparison should include four mixed-effects models: fixed-effects H0, fixed-effects H1, random-effects H0, and random-effects H1. Thereby, the Bayes factor enables testing both the average effect of condition and the heterogeneity of effect sizes across individuals. Bayesian model averaging provides an inclusion Bayes factor which quantifies the evidence for or against the presence of an effect of condition while taking model-selection uncertainty about the heterogeneity of individual effects into account. We present a simulation study showing that model selection among a larger set of mixed models performs well in recovering the true, data-generating model.

scholarly journals The Bayes factor: A bridge to Bayesian inference

2017 ◽  
Author(s):  
Matt Williams ◽  
Rasmus A. Bååth ◽  
Michael Carl Philipp
Keyword(s):  
Bayesian Inference ◽  
Bayesian Estimation ◽  
Null Hypothesis ◽  
General Framework ◽  
Bayes Factor ◽  
A Priori ◽  
Bayes Factors ◽  
Point Null Hypothesis

This paper will discuss the concept of Bayes factors as inferential tools that can directly replace NHST in the day-to-day work of developmental researchers. A Bayes factor indicates the degree which data observed should increase (or decrease) our support for one hypothesis in comparison to another. This framework allows researchers to not just reject but also produce evidence in favor of null hypotheses. Bayes factor alternatives to common tests used by developmental psychologists are available in easy-to-use software. However, we note that Bayesian estimation (rather than Bayes factors) may be a more appealing and general framework when a point null hypothesis is a priori implausible.

scholarly journals A Bayesian solution to the Behrens–Fisher problem

2021 ◽  
Vol 115 (4) ◽  
Author(s):  
Fco. Javier Girón ◽  
Carmen del Castillo
Keyword(s):  
Closed Form ◽  
Hierarchical Model ◽  
Bayes Factor ◽  
Bayes Factors ◽  
Simple Solution ◽  
Asymptotic Approximations ◽  
Normal Distributions ◽  
Fisher Distribution ◽  
Leibler Divergence

AbstractA simple solution to the Behrens–Fisher problem based on Bayes factors is presented, and its relation with the Behrens–Fisher distribution is explored. The construction of the Bayes factor is based on a simple hierarchical model, and has a closed form based on the densities of general Behrens–Fisher distributions. Simple asymptotic approximations of the Bayes factor, which are functions of the Kullback–Leibler divergence between normal distributions, are given, and it is also proved to be consistent. Some examples and comparisons are also presented.

scholarly journals Simulation Studies as a Tool to Understand Bayes Factors

2021 ◽  
Vol 4 (1) ◽  
pp. 251524592097262
Author(s):  
Don van Ravenzwaaij ◽  
Alexander Etz
Keyword(s):  
Simulation Study ◽  
Prior Distribution ◽  
Bayes Factor ◽  
R Package ◽  
Bayes Factors ◽  
Simulation Studies ◽  
Social Scientists ◽  
Simple Simulation ◽  
Set Up

When social scientists wish to learn about an empirical phenomenon, they perform an experiment. When they wish to learn about a complex numerical phenomenon, they can perform a simulation study. The goal of this Tutorial is twofold. First, it introduces how to set up a simulation study using the relatively simple example of simulating from the prior. Second, it demonstrates how simulation can be used to learn about the Jeffreys-Zellner-Siow (JZS) Bayes factor, a currently popular implementation of the Bayes factor employed in the BayesFactor R package and freeware program JASP. Many technical expositions on Bayes factors exist, but these may be somewhat inaccessible to researchers who are not specialized in statistics. In a step-by-step approach, this Tutorial shows how a simple simulation script can be used to approximate the calculation of the Bayes factor. We explain how a researcher can write such a sampler to approximate Bayes factors in a few lines of code, what the logic is behind the Savage-Dickey method used to visualize Bayes factors, and what the practical differences are for different choices of the prior distribution used to calculate Bayes factors.

Do sexist jokes increase rape proclivity among males high in hostile sexism? Evidence from two pre-registered direct replications of Thomae & Viki (2013)

2021 ◽  
Author(s):  
Neil McLatchie ◽  
Manuela Thomae
Keyword(s):  
Bayes Factor ◽  
Alternative Hypothesis ◽  
Meta Analysis ◽  
Statistical Significance ◽  
Original Study ◽  
Bayes Factors ◽  
Hostile Sexism ◽  
Replication Studies ◽  
Open Research ◽  
Rape Proclivity

Thomae and Viki (2013) reported that increased exposure to sexist humour can increase rape proclivity among males, specifically those who score high on measures of Hostile Sexism. Here we report two pre-registered direct replications (N = 530) of Study 2 from Thomae and Viki (2013) and assess replicability via (i) statistical significance, (ii) Bayes factors, (iii) the small-telescope approach, and (iv) an internal meta-analysis across the original and replication studies. The original results were not supported by any of the approaches. Combining the original study and the replications yielded moderate evidence in support of the null over the alternative hypothesis with a Bayes factor of B = 0.13. In light of the combined evidence, we encourage researchers to exercise caution before claiming that brief exposure to sexist humour increases male’s proclivity towards rape, until further pre-registered and open research demonstrates the effect is reliably reproducible.

Uncertainty of prior and posterior model probability: Implications for interpreting Bayes factors

2021 ◽  
Author(s):  
John K. Kruschke
Keyword(s):  
Posterior Distribution ◽  
Bayesian Model ◽  
Model Comparison ◽  
Bayes Factor ◽  
Bayes Factors ◽  
Prior Model ◽  
Bayesian Hypothesis Testing ◽  
Bayesian Model Comparison ◽  
Posterior Model Probability ◽  
Model Probability

In most applications of Bayesian model comparison or Bayesian hypothesis testing, the results are reported in terms of the Bayes factor only, not in terms of the posterior probabilities of the models. Posterior model probabilities are not reported because researchers are reluctant to declare prior model probabilities, which in turn stems from uncertainty in the prior. Fortunately, Bayesian formalisms are designed to embrace prior uncertainty, not ignore it. This article provides a novel derivation of the posterior distribution of model probability, and shows many examples. The posterior distribution is useful for making decisions taking into account the uncertainty of the posterior model probability. Benchmark Bayes factors are provided for a spectrum of priors on model probability. R code is posted at https://osf.io/36527/. This framework and tools will improve interpretation and usefulness of Bayes factors in all their applications.

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