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 With Bayesian Estimation One Can Get All That Bayes Factors Offer, and More

2019 ◽  
Author(s):  
Henk Kiers ◽  
Jorge Tendeiro
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Bayesian Estimation ◽  
Density Function ◽  
Null Hypothesis ◽  
Posterior Odds ◽  
Close Link ◽  
Posterior Odds Ratio ◽  
Spike And Slab Prior ◽  
Point Null Hypothesis

Null Hypothesis Bayesian Testing (NHBT) has been proposed as an alternative to Null Hypothesis Significance Testing (NHST). Whereas NHST has a close link to parameter estimation via confidence intervals, such a link of NHBT with Bayesian estimation via a posterior distribution is less straightforward, but does exist, and has recently been reiterated by Rouder, Haaf, and Vandekerckhove (2018). It hinges on a combination of a point mass probability and a probability density function as prior (denoted as the spike-and-slab prior). In the present paper it is first carefully explained how the spike-and-slab prior is defined, and how results can be derived for which proofs were not given in Rouder et al. (2018). Next, it is shown that this spike-and-slab prior can be approximated by a pure probability density function with a rectangular peak around the center towering highly above the remainder of the density function. Finally, we will indicate how this ‘hill-and-chimney’ prior may in turn be approximated by fully continuous priors. In this way it is shown that NHBT results can be approximated well by results from estimation using a strongly peaked prior, and it is noted that the estimation itself offers more than merely the posterior odds ratio on which NHBT is based. Thus, it complies with the strong APA requirement of not just mentioning testing results but also offering effect size information. It also offers a transparent perspective on the NHBT approach employing a prior with a strong peak around the chosen point null hypothesis value.

scholarly journals Bayesian Inference and Testing Any Hypothesis You Can Specify

2018 ◽  
Vol 1 (2) ◽  
pp. 281-295 ◽  
Author(s):  
Alexander Etz ◽  
Julia M. Haaf ◽  
Jeffrey N. Rouder ◽  
Joachim Vandekerckhove
Keyword(s):  
Bayesian Inference ◽  
Model Selection ◽  
Hypothesis Testing ◽  
Likelihood Ratio ◽  
Special Form ◽  
Null Hypothesis ◽  
Bayes Factor ◽  
Alternative Hypotheses ◽  
Competing Models

Hypothesis testing is a special form of model selection. Once a pair of competing models is fully defined, their definition immediately leads to a measure of how strongly each model supports the data. The ratio of their support is often called the likelihood ratio or the Bayes factor. Critical in the model-selection endeavor is the specification of the models. In the case of hypothesis testing, it is of the greatest importance that the researcher specify exactly what is meant by a “null” hypothesis as well as the alternative to which it is contrasted, and that these are suitable instantiations of theoretical positions. Here, we provide an overview of different instantiations of null and alternative hypotheses that can be useful in practice, but in all cases the inferential procedure is based on the same underlying method of likelihood comparison. An associated app can be found at https://osf.io/mvp53/ . This article is the work of the authors and is reformatted from the original, which was published under a CC-By Attribution 4.0 International license and is available at https://psyarxiv.com/wmf3r/ .

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 Significance Tests for Binomial Experiments: Ordering the Sample Space by Bayes Factors and Using Adaptive Significance Levels for Decisions

2017 ◽  
Author(s):  
Carlos A. de B. Pereira ◽  
Adriano Polpo ◽  
Eduardo Y. Nakano
Keyword(s):  
Null Hypothesis ◽  
Bayes Factor ◽  
Adaptive Significance ◽  
Bayes Factors ◽  
P Value ◽  
Significance Tests ◽  
Close Link ◽  
Significance Levels ◽  
Level Of Significance ◽  
Lindley’S Paradox

The main objective of this paper is to find a close link between the adaptive level of significance, presented here, and the sample size. We, statisticians, know of the inconsistency, or paradox, in the current classical tests of significance that are based on p-value statistics that is compared to the canonical significance levels (10%, 5% and 1%): "Raise the sample to reject the null hypothesis" is the recommendation of some ill-advised scientists! This paper will show that it is possible to eliminate this problem of significance tests. The Bayesian Lindley's paradox – "increase the sample to accept the hypothesis" – also disappears. Obviously, we present here only the beginning of a possible prominent research. The intention is to extend its use to more complex applications such as survival analysis, reliability tests and other areas. The main tools used here are the Bayes Factor and the extended Neyman-Pearson Lemma.

scholarly journals Conflicts in Bayesian Statistics Between Inference Based on Credible Intervals and Bayes Factors

2020 ◽  
Vol 18 (1) ◽  
pp. 2-27
Author(s):  
Miodrag M. Lovric
Keyword(s):  
Hypothesis Testing ◽  
Null Hypothesis ◽  
Bayes Factor ◽  
Credible Interval ◽  
Test Point ◽  
Type I ◽  
Null Hypothesis Testing ◽  
Frequentist Statistics ◽  
Credible Intervals ◽  
Point Null Hypothesis

In frequentist statistics, point-null hypothesis testing based on significance tests and confidence intervals are harmonious procedures and lead to the same conclusion. This is not the case in the domain of the Bayesian framework. An inference made about the point-null hypothesis using Bayes factor may lead to an opposite conclusion if it is based on the Bayesian credible interval. Bayesian suggestions to test point-nulls using credible intervals are misleading and should be dismissed. A null hypothesized value may be outside a credible interval but supported by Bayes factor (a Type I conflict), or contrariwise, the null value may be inside a credible interval but not supported by the Bayes factor (Type II conflict). Two computer programs in R have been developed that confirm the existence of a countable infinite number of cases, for which Bayes credible intervals are not compatible with Bayesian hypothesis testing.

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 Making 'Null Effects' Informative: Statistical Techniques and Inferential Frameworks

2018 ◽  
Author(s):  
Christopher Harms ◽  
Daniel Lakens
Keyword(s):  
Bayesian Estimation ◽  
Null Hypothesis ◽  
Statistical Approach ◽  
Significance Test ◽  
Bayes Factors ◽  
Knowledge Generation ◽  
Worked Example ◽  
Equivalence Tests ◽  
Online Tools ◽  
Cumulative Knowledge

Being able to interpret `null effects' is important for cumulative knowledge generation in science. To draw informative conclusions from null-effects, researchers need to move beyond the incorrect interpretation of a non-significant result in a null-hypothesis significance test as evidence of the absence of an effect. We explain how to statistically evaluate null-results using equivalence tests, Bayesian estimation, and Bayes factors. A worked example demonstrates how to apply these statistical tools and interpret the results. Finally, we explain how no statistical approach can actually prove that the null-hypothesis is true, and briefly discuss the philosophical differences between statistical approaches to examine null-effects. The increasing availability of easy-to-use software and online tools to perform equivalence tests, Bayesian estimation, and calculate Bayes factors make it timely and feasible to complement or move beyond traditional null-hypothesis tests, and allow researchers to draw more informative conclusions about null-effects.

scholarly journals A tutorial on testing hypotheses using the Bayes factor

2019 ◽  
Author(s):  
Herbert Hoijtink ◽  
Joris Mulder ◽  
Caspar J. Van Lissa ◽  
Xin Gu
Keyword(s):  
Statistical Models ◽  
Null Hypothesis ◽  
Multiple Testing ◽  
Bayes Factor ◽  
Additional Data ◽  
R Package ◽  
Bayes Factors ◽  
Multiple Hypotheses ◽  
Anova Model ◽  
Hypothesis Evaluation

Learning about hypothesis evaluation using the Bayes factor could enhance psychologicalresearch. In contrast to null-hypothesis significance testing: it renders the evidence in favorof each of the hypotheses under consideration (it can be used to quantify support for thenull-hypothesis) instead of a dichotomous reject/do-not-reject decision; it canstraightforwardly be used for the evaluation of multiple hypotheses without having tobother about the proper manner to account for multiple testing; and, it allows continuousre-evaluation of hypotheses after additional data have been collected (Bayesian updating).This tutorial addresses researchers considering to evaluate their hypotheses by meansof the Bayes factor. The focus is completely applied and each topic discussed is illustratedusing Bayes factors for the evaluation of hypotheses in the context of an ANOVA model,obtained using the R package bain. Readers can execute all the analyses presented whilereading this tutorial if they download bain and the R-codes used. It will be elaborated in acompletely non-technical manner: what the Bayes factor is, how it can be obtained, howBayes factors should be interpreted, and what can be done with Bayes factors. Afterreading this tutorial and executing the associated code, researchers will be able to use theirown data for the evaluation of hypotheses by means of the Bayes factor, not only in thecontext of ANOVA models, but also in the context of other statistical models.

scholarly journals A Bayesian bird's eye view of ‘Replications of important results in social psychology’

2017 ◽  
Vol 4 (1) ◽  
pp. 160426 ◽  
Author(s):  
Maarten Marsman ◽  
Felix D. Schönbrodt ◽  
Richard D. Morey ◽  
Yuling Yao ◽  
Andrew Gelman ◽  
...  
Keyword(s):  
Social Psychology ◽  
Effect Size ◽  
Null Hypothesis ◽  
Bayes Factor ◽  
Alternative Hypothesis ◽  
Hypothesis Test ◽  
Meta Analysis ◽  
Bayes Factors ◽  
Credible Intervals ◽  
Complementary Techniques

We applied three Bayesian methods to reanalyse the preregistered contributions to the Social Psychology special issue ‘Replications of Important Results in Social Psychology’ (Nosek & Lakens. 2014 Registered reports: a method to increase the credibility of published results. Soc. Psychol. 45 , 137–141. ( doi:10.1027/1864-9335/a000192 )). First, individual-experiment Bayesian parameter estimation revealed that for directed effect size measures, only three out of 44 central 95% credible intervals did not overlap with zero and fell in the expected direction. For undirected effect size measures, only four out of 59 credible intervals contained values greater than 0.10 (10% of variance explained) and only 19 intervals contained values larger than 0.05 . Second, a Bayesian random-effects meta-analysis for all 38 t -tests showed that only one out of the 38 hierarchically estimated credible intervals did not overlap with zero and fell in the expected direction. Third, a Bayes factor hypothesis test was used to quantify the evidence for the null hypothesis against a default one-sided alternative. Only seven out of 60 Bayes factors indicated non-anecdotal support in favour of the alternative hypothesis ( BF 10 > 3 ), whereas 51 Bayes factors indicated at least some support for the null hypothesis. We hope that future analyses of replication success will embrace a more inclusive statistical approach by adopting a wider range of complementary techniques.

scholarly journals Bayesian inference and testing any hypothesis you can specify

2018 ◽  
Author(s):  
Alexander Etz ◽  
Julia M. Haaf ◽  
Jeffrey N. Rouder ◽  
Joachim Vandekerckhove
Keyword(s):  
Bayesian Inference ◽  
Model Selection ◽  
Hypothesis Testing ◽  
Likelihood Ratio ◽  
Special Form ◽  
Null Hypothesis ◽  
Bayes Factor ◽  
Alternative Hypotheses ◽  
Competing Models

Hypothesis testing is a special form of model selection. Once a pair of competing models is fully defined, their definition immediately leads to a measure of how strongly each model supports the data. The ratio of their support is often called the likelihood ratio or the Bayes factor. Critical in the model selection endeavor is the specification of the models. In the case of hypothesis testing, it is of the greatest importance that we specify exactly what is meant by a "null" hypothesis as well as the alternative to which it is contrasted, and that these are suitable instantiations of theoretical positions. Here, we provide an overview of different instantiations of null and alternative hypotheses that can be useful in practice, while the underlying method of likelihood comparison is universal and identical in all cases. An associated app can be found via https://osf.io/mvp53/.

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