---
title: "Get Started with Bayesian Analysis"
output: 
  rmarkdown::html_vignette:
    toc: true
    fig_width: 10.08
    fig_height: 6
tags: [r, bayesian, posterior, test]
vignette: >
  \usepackage[utf8]{inputenc}
  %\VignetteIndexEntry{Get Started with Bayesian Analysis}
  %\VignetteEngine{knitr::rmarkdown}
editor_options: 
  chunk_output_type: console
bibliography: bibliography.bib
csl: apa.csl
---

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

---

```{r message=FALSE, warning=FALSE, include=FALSE}
library(knitr)

knitr::opts_chunk$set(comment = ">")
options(knitr.kable.NA = "")
options(digits = 2)

if (!requireNamespace("rstanarm", quietly = TRUE)) {
  knitr::opts_chunk$set(eval = FALSE)
} else {
  library(rstanarm)
  library(bayestestR)
}
```


## 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** [@etz2016bayesian]
- **accuracy** (in noisy data and small samples) [@kruschke2012time]
- the possibility of introducing **prior knowledge** into the analysis [@andrews2013prior; @kruschke2012time]
- critically, **intuitive nature of results** and their **straightforward interpretation** [@kruschke2010believe; @wagenmakers2018bayesian]

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 [@chambers2014instead; @szucs2016empirical]. There is an emerging
consensus that the generalization of the Bayesian approach is *one* way of
overcoming these issues [@benjamin2018redefine; @etz2016bayesian].

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](https://easystats.github.io/bayestestR/articles/guidelines.html) 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](https://easystats.github.io/bayestestR/articles/credible_interval.html)
or [other indices](https://easystats.github.io/bayestestR/articles/guidelines.html).

```{r echo=FALSE, fig.cap="Accurate depiction of a regular Bayesian user estimating a credible interval.", fig.align='center', out.width="50%"}
knitr::include_graphics("https://github.com/easystats/easystats/raw/master/man/figures/bayestestR/bayesianMaster.jpg")
```

*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](https://discourse.datamethods.org/t/language-for-communicating-frequentist-results-about-treatment-effects/934/16)). 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**](https://github.com/easystats/easystats) suite by running the following:

```{r eval=FALSE, message=FALSE, warning=FALSE}
install.packages("remotes")
remotes::install_github("easystats/easystats")
```

<!-- Now that it's done, we can *load* the packages we need, in this case [`report`](https://github.com/easystats/report), which combines `bayestestR` and other packages together. -->

<!-- ```{r message=FALSE, warning=FALSE} -->
<!-- library(report)  # Calling library(...) enables you to use the package's functions -->
<!-- ``` -->


Let's also install and load the [`rstanarm`](https://mc-stan.org/rstanarm/), that allows fitting Bayesian models, as well as [`bayestestR`](https://github.com/easystats/bayestestR), to describe them.

```{r message=FALSE, warning=FALSE, eval=FALSE}
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`](https://en.wikipedia.org/wiki/Iris_flower_data_set) dataset, included by default in R.

```{r message=FALSE, warning=FALSE, eval=FALSE}
model <- lm(Sepal.Length ~ Petal.Length, data = iris)
summary(model)
```

```{r echo=FALSE, message=FALSE, warning=FALSE, comment=NA}
model <- lm(Sepal.Length ~ Petal.Length, data = iris)
summary(model)
```

<!-- ```{r message=FALSE, warning=FALSE, eval=FALSE} -->
<!-- model <- lm(Sepal.Length ~ Petal.Length, data=iris) -->
<!-- report(model) -->
<!-- ``` -->
<!-- ```{r echo=FALSE, message=FALSE, warning=FALSE, comment=NA} -->
<!-- library(dplyr) -->

<!-- lm(Sepal.Length ~ Petal.Length, data=iris) %>%  -->
<!--   report() %>%  -->
<!--   to_text(width=100) -->
<!-- ``` -->

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

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

**Not anymore.**

### Bayesian linear regression

<!-- ```{r message=FALSE, warning=FALSE, eval=FALSE} -->
<!-- model <- stan_glm(Sepal.Length ~ Petal.Length, data=iris) -->
<!-- report(model) -->
<!-- ``` -->
<!-- ```{r echo=FALSE, message=FALSE, warning=FALSE, comment=NA, results='hide'} -->
<!-- library(rstanarm) -->
<!-- set.seed(333) -->

<!-- model <- stan_glm(Sepal.Length ~ Petal.Length, data=iris) -->
<!-- ``` -->
<!-- ```{r echo=FALSE, message=FALSE, warning=FALSE, comment=NA} -->
<!-- model %>%  -->
<!--   report() %>%  -->
<!--   to_text(width=100) -->
<!-- ``` -->

```{r message=FALSE, warning=FALSE, eval=FALSE}
model <- stan_glm(Sepal.Length ~ Petal.Length, data = iris)
posteriors <- describe_posterior(model)
# for a nicer table
print_md(posteriors, digits = 2)
```

```{r echo=FALSE, message=FALSE, warning=FALSE, comment=NA}
set.seed(333)
model <- stan_glm(Sepal.Length ~ Petal.Length, data = iris, refresh = 0)
posteriors <- describe_posterior(model)
# for a nicer table
print_md(posteriors, digits = 2)
```


**That's it!** 

You just fitted a Bayesian version of the model by simply using the
[`stan_glm()`](https://mc-stan.org/rstanarm/reference/stan_glm.html) function
instead of `lm()` and described the posterior distributions of the parameters!

The conclusion we draw, 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**](https://easystats.github.io/bayestestR/articles/example1.html)!

And, if you want even more, you can check out other articles describing all the functionality the package has to offer!

<https://easystats.github.io/bayestestR/articles/>

## References
