Bayesian statistics has gone from being a minority and almost exotic approach in psychology to becoming an increasingly present alternative in leading journals. This shift is due in part to the replicability crisis, which has highlighted the limitations of frequentist inference based exclusively on p-values (what some call the tyranny of the p-value), and in part to the fact that software for Bayesian analysis has become much more accessible in recent years. However, many researchers in psychology feel intimidated by Bayesian statistics because they perceive it as mathematically complex or conceptually abstract. The reality is that Bayesian intuition is, in many respects, more natural than the frequentist one.
The fundamental difference between the two approaches has to do with how they interpret probability. In the frequentist framework, probability refers to long-run relative frequencies: if you repeat an experiment infinitely many times, 95% of the confidence intervals will contain the true parameter. In the Bayesian framework, probability is a degree of belief or certainty about a hypothesis given the available data. This means that a 95% Bayesian credible interval can be directly interpreted as "there is a 95% probability that the parameter lies within this interval," which is what most researchers believe (erroneously) that a frequentist confidence interval means.
Bayes' theorem in practice
The entire Bayesian edifice is built upon Bayes' theorem, which is a formula for updating beliefs in light of new evidence. The idea is simple: you start with a prior belief (the prior distribution) about the parameter of interest, collect data, and Bayes' theorem tells you how to modify your prior belief to incorporate the information from the data. The result is the posterior distribution, which combines your prior knowledge with the empirical evidence.
The prior distribution is the most controversial aspect of Bayesian statistics, because it introduces an apparently subjective component into the analysis. Critics argue that different researchers can choose different priors and arrive at different conclusions. This is true, but in practice the impact of the prior distribution diminishes as the amount of data increases: with sufficiently large samples, the data dominate over the prior and different reasonable priors converge on similar conclusions. Moreover, there are non-informative or weakly informative priors that express maximum uncertainty and that minimize the influence of prior beliefs on the results.
The Bayes factor
One of the most useful Bayesian tools for researchers in psychology is the Bayes factor (BF). The Bayes factor quantifies the relative evidence in favor of one hypothesis versus another. A BF10 of 5 means that the data are 5 times more likely under the alternative hypothesis than under the null hypothesis. A BF10 of 0.2 (equivalent to a BF01 of 5) means that the data are 5 times more likely under the null than under the alternative. And a BF10 close to 1 means that the data are equally compatible with both hypotheses, indicating that the evidence is inconclusive.
This ability to quantify the evidence in favor of the null hypothesis is an important advantage over frequentist tests, which can only reject or fail to reject the null but can never positively affirm that the null is true. In psychology, where many theoretically interesting questions concern the absence of differences (for example, whether two treatments are equivalent), the Bayes factor provides a tool that the frequentist framework simply does not offer.
Jeffreys' benchmarks for interpreting the Bayes factor are the most widely used: a BF of 1 to 3 indicates anecdotal evidence, 3 to 10 moderate evidence, 10 to 30 strong evidence, 30 to 100 very strong evidence, and above 100 extreme evidence. These cutoff points are guidelines and should not be applied mechanically, just as the p < 0.05 threshold should not be the sole criterion for making decisions in the frequentist framework.
How to get started with Bayesian statistics
If you want to start using Bayesian statistics without an excessive learning curve, JASP is probably your best option. JASP offers Bayesian analyses (Bayesian t-tests, Bayesian ANOVA, Bayesian correlations, Bayesian regression) with a graphical interface practically identical to that of its frequentist counterparts. You simply check the "Bayes Factor" option and you obtain the Bayes factor alongside the usual frequentist results. This allows you to compare both approaches with the same data and see how they complement each other.
For R users, the BayesFactor package allows you to calculate Bayes factors for the most common designs in psychology (comparison of means, ANOVA, correlation, regression). For more complex models, brms (Bayesian Regression Models using Stan) offers extraordinary flexibility with a syntax similar to that of the lme4 package. The learning curve for brms is moderate if you are already familiar with regression models in R, and online learning resources have multiplied in recent years. The important thing is to start with simple analyses, understand the Bayesian logic through concrete examples, and gradually incorporate these tools into your analytical repertoire. If you are not yet sure which test you need, our statistical test selector can help you decide.