Get Started with Bayesian Analysis

Why use the Bayesian Framework?

The Bayesian framework for statistics is quickly gaining in popularity among scientists, associated with the general shift towards open and honest science. Reasons to prefer this approach are:

• reliability (Etz & Vandekerckhove, 2016)
• accuracy (in noisy data and small samples) (Kruschke, Aguinis, & Joo, 2012)
• the possibility of introducing prior knowledge into the analysis (Andrews & Baguley, 2013; Kruschke et al., 2012)
• critically, intuitive nature of results and their straightforward interpretation (Kruschke, 2010; Wagenmakers et al., 2018)

In general, the frequentist approach has been associated with the focus on the null hypothesis testing, and the misuse of p-values has been shown to critically contribute to the reproducibility crisis in social and psychological sciences (Chambers, Feredoes, Muthukumaraswamy, & Etchells, 2014; Szucs & Ioannidis, 2016). There is an emerging consensus that the generalization of the Bayesian approach is one way of overcoming these issues (Benjamin et al., 2018; Etz & Vandekerckhove, 2016).

Once we agree that the Bayesian framework is the right way to go, you might wonder what exactly is this framework.

What’s all the fuss about?

What is the Bayesian Framework?

Adopting the Bayesian framework is more of a shift in the paradigm than a change in the methodology. Indeed, all the common statistical procedures (t-tests, correlations, ANOVAs, regressions, etc.) can be achieved using the Bayesian framework. The key difference is that in the frequentist framework (the “classical” approach to statistics, with p and t values, as well as some weird degrees of freedom), the effects are fixed (but unknown) and data are random. In other words, it assumes that the unknown parameter has a unique value that we are trying to estimate/guess using our sample data. On the other hand, in the Bayesian framework, instead of estimating the “true effect,” the probability of different effects given the observed data is computed, resulting in a distribution of possible values for the parameters, called the posterior distribution.

The uncertainty in Bayesian inference can be summarized, for instance, by the median of the distribution, as well as a range of values of the posterior distribution that includes the 95% most probable values (the 95% credible interval). Cum grano salis, these are considered the counterparts to the point-estimate and confidence interval in a frequentist framework. To illustrate the difference of interpretation, the Bayesian framework allows to say “given the observed data, the effect has 95% probability of falling within this range”, while the frequentist (less intuitive) alternative would be “when repeatedly computing confidence intervals from data of this sort, there is a 95% probability that the effect falls within a given range”. In essence, the Bayesian sampling algorithms (such as MCMC sampling) return a probability distribution (the posterior) of an effect that is compatible with the observed data. Thus, an effect can be described by characterizing its posterior distribution in relation to its centrality (point-estimates), uncertainty, as well as its existence and significance

In other words, putting the maths behind it aside for a moment, we can say that:

• The frequentist approach tries to estimate the real effect. For instance, the “real” value of the correlation between x and y. Hence, the frequentist models return a point-estimate (i.e., a single value and not a distribution) of the “real” correlation (e.g., \(r = 0.42\)) estimated under a number of obscure assumptions (at a minimum, considering that the data is sampled at random from a “parent,” usually normal distribution).

• The Bayesian framework assumes no such thing. The data are what they are. Based on the observed data (and a prior belief about the result), the Bayesian sampling algorithm (MCMC sampling is one example) returns a probability distribution (called the posterior) of the effect that is compatible with the observed data. For the correlation between x and y, it will return a distribution that says, for example, “the most probable effect is 0.42, but this data is also compatible with correlations of 0.12 and 0.74 with certain probabilities.”

• To characterize statistical significance of our effects, we do not need p-values, or any other such indices. We simply describe the posterior distribution of the effect. For example, we can report the median, the 89% Credible Interval or other indices.

Accurate depiction of a regular Bayesian user estimating a credible interval.

Note: Altough the very purpose of this package is to advocate for the use of Bayesian statistics, please note that there are serious arguments supporting frequentist indices (see for instance this thread). As always, the world is not black and white (p < .001).

So… how does it work?

A simple example

install.packages("remotes")
remotes::install_github("easystats/easystats")

install.packages("rstanarm")
library(rstanarm)

model <- lm(Sepal.Length ~ Petal.Length, data = iris)
summary(model)

Call:
lm(formula = Sepal.Length ~ Petal.Length, data = iris)

Residuals:
    Min      1Q  Median      3Q     Max 
-1.2468 -0.2966 -0.0152  0.2768  1.0027 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)    4.3066     0.0784    54.9   <2e-16 ***
Petal.Length   0.4089     0.0189    21.6   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.41 on 148 degrees of freedom
Multiple R-squared:  0.76,  Adjusted R-squared:  0.758 
F-statistic:  469 on 1 and 148 DF,  p-value: <2e-16

model <- stan_glm(Sepal.Length ~ Petal.Length, data = iris)
posteriors <- describe_posterior(model)
# for a nicer table
print_md(posteriors, digits = 2)