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 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 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.

scholarly journals Estimating Bayes factors from minimal summary statistics in repeated measures analysis of variance designs

2020 ◽  
Vol 17 (1) ◽  
Author(s):  
Thomas Faulkenberry
Keyword(s):  
Analysis Of Variance ◽  
Repeated Measures ◽  
Bayes Factor ◽  
Linear Models ◽  
Repeated Measurements ◽  
Bayes Factors ◽  
Complex Method ◽  
Summary Statistics ◽  
Minimal Set ◽  
Repeated Measures Analysis

In this paper, I develop a formula for estimating Bayes factors directly from minimal summary statistics produced in repeated measures analysis of variance designs. The formula, which requires knowing only the F-statistic, the number of subjects, and the number of repeated measurements per subject, is based on the BIC approximation of the Bayes factor, a common default method for Bayesian computation with linear models. In addition to providing computational examples, I report a simulation study in which I demonstrate that the formula compares favorably to a recently developed, more complex method that accounts for correlation between repeated measurements. The minimal BIC method provides a simple way for researchers to estimate Bayes factors from a minimal set of summary statistics, giving users a powerful index for estimating the evidential value of not only their own data, but also the data reported in published studies.

scholarly journals Computing Bayes factors from data with missing values

2021 ◽  
Author(s):  
Herbert Hoijtink ◽  
Xin Gu ◽  
Joris Mulder ◽  
Yves Rosseel
Keyword(s):  
Missing Values ◽  
Bayes Factor ◽  
Equality Constraints ◽  
Bayes Factors ◽  
Multiple Imputations

The Bayes factor is increasingly used for the evaluation of hypotheses. These may betraditional hypotheses specified using equality constraints among the parameters of thestatistical model of interest or informative hypotheses specified using equality andinequality constraints. So far no attention has been given to the computation of Bayesfactors from data with missing values. A key property of such a Bayes factor should bethat it is only based on the information in the observed values. This paper will show thatsuch a Bayes factor can be obtained using multiple imputations of the missing values.

scholarly journals A Markov chain representation of the multiple testing problem

2016 ◽  
Vol 27 (2) ◽  
pp. 364-383 ◽  
Author(s):  
Stefano Cabras
Keyword(s):  
Markov Chain ◽  
Multiple Testing ◽  
Bayes Factor ◽  
Alternative Hypothesis ◽  
Multiple Hypothesis Testing ◽  
Bayes Factors ◽  
Rna Seq ◽  
Markov Transition ◽  
Improper Priors ◽  
Alternative Hypotheses

The problem of multiple hypothesis testing can be represented as a Markov process where a new alternative hypothesis is accepted in accordance with its relative evidence to the currently accepted one. This virtual and not formally observed process provides the most probable set of non null hypotheses given the data; it plays the same role as Markov Chain Monte Carlo in approximating a posterior distribution. To apply this representation and obtain the posterior probabilities over all alternative hypotheses, it is enough to have, for each test, barely defined Bayes Factors, e.g. Bayes Factors obtained up to an unknown constant. Such Bayes Factors may either arise from using default and improper priors or from calibrating p-values with respect to their corresponding Bayes Factor lower bound. Both sources of evidence are used to form a Markov transition kernel on the space of hypotheses. The approach leads to easy interpretable results and involves very simple formulas suitable to analyze large datasets as those arising from gene expression data (microarray or RNA-seq experiments).

scholarly journals On the Jensen–Shannon Symmetrization of Distances Relying on Abstract Means

Entropy ◽  
2019 ◽  
Vol 21 (5) ◽  
pp. 485 ◽  
Author(s):  
Frank Nielsen
Keyword(s):  
Closed Form ◽  
Exponential Family ◽  
Geometric Mean ◽  
Mixture Distribution ◽  
Exponential Families ◽  
Leibler Divergence ◽  
Probability Densities ◽  
The Mean ◽  
Jensen Shannon Divergence ◽  
Closed Form Formula

The Jensen–Shannon divergence is a renowned bounded symmetrization of the unbounded Kullback–Leibler divergence which measures the total Kullback–Leibler divergence to the average mixture distribution. However, the Jensen–Shannon divergence between Gaussian distributions is not available in closed form. To bypass this problem, we present a generalization of the Jensen–Shannon (JS) divergence using abstract means which yields closed-form expressions when the mean is chosen according to the parametric family of distributions. More generally, we define the JS-symmetrizations of any distance using parameter mixtures derived from abstract means. In particular, we first show that the geometric mean is well-suited for exponential families, and report two closed-form formula for (i) the geometric Jensen–Shannon divergence between probability densities of the same exponential family; and (ii) the geometric JS-symmetrization of the reverse Kullback–Leibler divergence between probability densities of the same exponential family. As a second illustrating example, we show that the harmonic mean is well-suited for the scale Cauchy distributions, and report a closed-form formula for the harmonic Jensen–Shannon divergence between scale Cauchy distributions. Applications to clustering with respect to these novel Jensen–Shannon divergences are touched upon.

scholarly journals On the Use of Information Criteria for Model Selection in Phylogenetics

2019 ◽  
Vol 37 (2) ◽  
pp. 549-562 ◽  
Author(s):  
Edward Susko ◽  
Andrew J Roger
Keyword(s):  
Model Selection ◽  
Bayes Factor ◽  
Information Criterion ◽  
Information Criteria ◽  
Leibler Divergence ◽  
Incorrect Model ◽  
Substantial Bias ◽  
And Performance ◽  
Phylogenetic Models ◽  
Selection Of

Abstract The information criteria Akaike information criterion (AIC), AICc, and Bayesian information criterion (BIC) are widely used for model selection in phylogenetics, however, their theoretical justification and performance have not been carefully examined in this setting. Here, we investigate these methods under simple and complex phylogenetic models. We show that AIC can give a biased estimate of its intended target, the expected predictive log likelihood (EPLnL) or, equivalently, expected Kullback–Leibler divergence between the estimated model and the true distribution for the data. Reasons for bias include commonly occurring issues such as small edge-lengths or, in mixture models, small weights. The use of partitioned models is another issue that can cause problems with information criteria. We show that for partitioned models, a different BIC correction is required for it to be a valid approximation to a Bayes factor. The commonly used AICc correction is not clearly defined in partitioned models and can actually create a substantial bias when the number of parameters gets large as is the case with larger trees and partitioned models. Bias-corrected cross-validation corrections are shown to provide better approximations to EPLnL than AIC. We also illustrate how EPLnL, the estimation target of AIC, can sometimes favor an incorrect model and give reasons for why selection of incorrectly under-partitioned models might be desirable in partitioned model settings.

Asymmetrical Bending of a Cylindrically Aeolotropic Tapered Disk

1956 ◽  
Vol 23 (1) ◽  
pp. 11-14
Author(s):  
E. S. Baclig ◽  
H. D. Conway
Keyword(s):  
Closed Form ◽  
Isotropic Material ◽  
Numerical Results ◽  
Simple Solution ◽  
Thickness Variation ◽  
Plate Bending ◽  
Small Deflection ◽  
Asymmetrical Bending

Abstract Variations of thickness, anisotropy, and asymmetry of loading usually tend to increase the mathematical difficulty of obtaining solutions to the small-deflection problems of plate bending. However, for the bending of a cylindrically aeolotropic disk twisted about its diameter and having a certain thickness variation, it is possible to obtain a relatively simple solution in closed form. This solution is presented here, numerical results being given for oak and for isotropic material.

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