Get Started with Bayesian Analysis

• Why use the Bayesian Framework?
• What is the Bayesian Framework?
• A simple example
  • BayestestR Installation
  • Traditional linear regression
  • Bayesian linear regression
• References

This vignette can be referred to by citing the package:

• Makowski, D., Ben-Shachar, M. S., & Lüdecke, D. (2019). bayestestR: Describing Effects and their Uncertainty, Existence and Significance within the Bayesian Framework. Journal of Open Source Software, 4(40), 1541. https://doi.org/10.21105/joss.01541

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

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

Once we agreed that the Bayesian framework is the right way to go, you might wonder what is the Bayesian 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, …) can be achieved using the Bayesian framework. One of the core difference is that in the frequentist view (the “classic” statistics, with p and t values, as well as some weird degrees of freedom), the effects are fixed (but unknown) and data are random. On the other hand, in the Bayesian inference process, instead of having estimates of 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 straightforward 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 existence and significance

In other words, omitting the maths behind it, we can say that:

• The frequentist bloke tries to estimate “the real effect”. For instance, the “real” value of the correlation between x and y. Hence, frequentist models return a “point-estimate” (i.e., a single value) 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 master assumes no such thing. The data are what they are. Based on this observed data (and a prior belief about the result), the Bayesian sampling algorithm (sometimes referred to for example as MCMC sampling) 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”.
• To characterize our effects, no need of p values or other cryptic 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

BayestestR Installation

You can install bayestestR along with the whole easystats suite by running the following:

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

Let’s also install and load the rstanarm, that allows fitting Bayesian models, as well as bayestestR, to describe them.

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

Traditional linear regression

Let’s start by fitting a simple frequentist linear regression (the lm() function stands for linear model) between two numeric variables, Sepal.Length and Petal.Length from the famous iris dataset, included by default in R.

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

This analysis suggests that there is a significant (whatever that means) and positive (with a coefficient of 0.41) linear relationship between the two variables.

Fitting and interpreting frequentist models is so easy that it is obvious that people use it instead of the Bayesian framework… right?

Not anymore.

Bayesian linear regression

model <- stan_glm(Sepal.Length ~ Petal.Length, data=iris)
describe_posterior(model)

That’s it! You fitted a Bayesian version of the model by simply using stan_glm() instead of lm() and described the posterior distributions of the parameters. The conclusion that we can drawn, for this example, are very similar. The effect (the median of the effect’s posterior distribution) is about 0.41, and it can be also be considered as significant in the Bayesian sense (more on that later).

So, ready to learn more? Check out the next tutorial!

References