There is Nothing Magical about Bayesian Statistics: An Introduction to Epistemic Probabilities in Data Analysis for Psychology Starters

Abstract

This paper is a reader-friendly introduction to Bayesian inference applied to psychological science. We begin by explaining the difference between frequentist and epistemic interpretations of probability that underpin respectively frequentist and Bayesian statistics. We use a concrete example—a student wondering whether s/he carries the virus statisticus malignum—to explain how both approaches are different one from another. We illustrate Bayesian inference with intuitive examples, before introducing the mathematical framework. Different schools of thoughts and recommendations are discussed to illustrate how to use priors in Bayes Factor testing. We discuss how psychology could benefit from a greater reliance on Bayesian methods. Finally, we illustrate how to compute Bayes Factors analyses with real data and provide the R code.

Acknowledgements

We express our sincere gratitude to Julien Diard, Benoît Dompnier, Fabrizio Butera, Jean-Philippe Antonietti and Mathias Schmitz for providing valuable feedbacks on the paper. We wish to thank the two reviewers whose comments helped to improve this paper. We thank Victoria Davoine and Ocyna Rudmann for reading and commenting on the earlier versions of this paper. Finally, we would like to thank Alexandre Charpentier Poncelet for his help in proofreading.

Disclosure statement

The authors have no conflict of interests associated with the publication of this article.

Notes

1 None of the authors advocates for an exclusive use of frequentist or Bayesian statistics. Both advocate for sound statistical practices.

2 Elsewhere in the psychological literature, epistemic probabilities are also referred to as subjective probabilities (e.g., Dienes, 2011)

3 Though partially inspired by the NP approach, current practices for justifying scientific claims in psychology rely almost exclusively on the null hypothesis significance testing (NHST). Here, one rejects the statistical hypothesis of nonexistence of the effect under investigation (H₀), if the probability of observing under this hypothesis one’s set of data or a more extreme set of data (the p-value) is smaller than the conventional threshold of .05. Despite the undisputed popularity in academia, this procedure has no straightforward theoretical justification (Gigerenzer, 2004; Gigerenzer et al., 2004). NHST in its most common use consists both in rejecting and accepting statistical hypotheses on the one hand, and providing evidence against the null hypothesis on the other. This reflects a hybrid logic of competing statistical theories of Neyman and Pearson (1933) and Fisher (1956), respectively (see Gigerenzer et al., 2004). A commonly endorsed practice in NHST, inspired by Fisher’s work, consists in computing p-values to reflect the falsehood of H₀, where lower p-values indicate stronger evidence against H₀ and allow being more confident that it is false, than higher p-values (see Wagenmakers et al., 2008, for why this is a less than an ideal way to quantify evidence against H₀). This is a quasi-Bayesian approach to NHST (Gigerenzer, 1993) that conflicts with a strict frequentist approach to inference, because the level of significance serves, in practice, the same role as epistemic probabilities. This is not warranted within the framework that considers statistical hypotheses only as true or false, hence mentions like probable or improbable do not apply.

4 Assuming the data are being collected with the same sampling intentions as the original study (see Wagenmakers et al., 2008).

5 It may seem counterintuitive that with a 99% chance of obtaining a positive test result when one does really carry the virus, the fact that the test was positive only makes the probability of carrying the virus still relatively as low as less than 35%. This is due to a relatively low prior probability for carrying the virus, namely 5% (see Bayes' Theorem application in Section “The Bayesian approach to statistical inference”).

6 In many realistic applications, the exact value of $p(D)$ can be impossible to calculate due to computational difficulties (see Kruschke, 2014). This explains why Bayesian methods have been historically only seldom used. Scholars have known this approach to inference for centuries, yet only a limited subset of relatively simple models were mathematically solvable under Bayesian analyses (Kruschke, 2014). Without knowing $p(D),$ deriving the exact posterior probability distribution $p(H|D)$ directly from Bayes’ Theorem is not possible. This is not a problem anymore, since recent developments in computational techniques known as Markov Chains Monte Carlo (MCMC; Gamerman & Lopes, 2006; Kruschke, 2014; Lunn et al., 2012; Rouder & Lu, 2005) extended the applicability of Bayesian statistics to a whole range of useful models for daily applications. Their usefulness relies on the fact that they allow to approximate posterior probability distributions without having to calculate$p(D).$

7 A simpler version of Bayes’ Theorem states that the posterior is proportional to likelihood times prior.

8 A Cauchy distribution is a Student t distribution with 1 degree of freedom.