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# bayestestR <img src='man/figures/logo.png' align="right" height="139" />


***Become a Bayesian master you will***

Existing R packages allow users to easily fit a large variety of models
and extract and visualize the posterior draws. However, most of these
packages only return a limited set of indices (e.g., point-estimates and
CIs). **bayestestR** provides a comprehensive and consistent set of
functions to analyze and describe posterior distributions generated by a
variety of models objects, including popular modeling packages such as
**rstanarm**, **brms** or **BayesFactor**.

## Installation

Run the following:

``` r

``` r

## Documentation


Click on the buttons above to access the package
[**documentation**](https://easystats.github.io/bayestestR/) and the
[**easystats blog**](https://easystats.github.io/blog/posts/), and
check-out these vignettes:

#### Tutorials

  - [Get Started with Bayesian
  - [Example 1: Initiation to Bayesian
  - [Example 2: Confirmation of Bayesian
  - [Example 3: Become a Bayesian

#### Articles

  - [Credible Intervals
  - [Probability of Direction
  - [Region of Practical Equivalence
  - [Bayes Factors
  - [Comparison of
  - [Comparison of Indices of Effect
  - [Reporting

# Features

The following figures are meant to illustrate the (statistical) concepts
behind the functions. However, for most functions, `plot()`-methods are
available from the [see-package](http://easystats.github.io/see).

## Describing the Posterior Distribution

is the master function with which you can compute all of the indices
cited below at once.

``` r
##   Parameter Median CI CI_low CI_high   pd ROPE_CI ROPE_low ROPE_high
## 1 Posterior  0.036 89   -1.6     1.5 0.51      89     -0.1       0.1
##   ROPE_Percentage
## 1           0.091

## Point-estimates

### MAP Estimate

find the **Highest Maximum A Posteriori (MAP)** estimate of a posterior,
*i.e.*, the value associated with the highest probability density (the
“peak” of the posterior distribution). In other words, it is an
estimation of the *mode* for continuous parameters.

``` r
posterior <- distribution_normal(100, 0.4, 0.2)
## MAP = 0.40

![](man/figures/unnamed-chunk-6-1.png)<!-- -->

## Uncertainty

### Highest Density Interval (HDI) and Equal-Tailed Interval (ETI)

computes the **Highest Density Interval (HDI)** of a posterior
distribution, i.e., the interval which contains all points within the
interval have a higher probability density than points outside the
interval. The HDI can be used in the context of Bayesian posterior
characterisation as **Credible Interval (CI)**.

Unlike equal-tailed intervals (see
that typically exclude 2.5% from each tail of the distribution, the HDI
is *not* equal-tailed and therefore always includes the mode(s) of
posterior distributions.

By default, `hdi()` returns the 89% intervals (`ci = 0.89`), deemed to
be more stable than, for instance, 95% intervals. An effective sample
size of at least 10.000 is recommended if 95% intervals should be
computed (Kruschke, 2015). Moreover, 89 indicates the arbitrariness of
interval limits - its only remarkable property is being the highest
prime number that does not exceed the already unstable 95% threshold
(McElreath, 2018).

``` r
posterior <- distribution_chisquared(100, 3)

hdi(posterior, ci = .89)
## # Highest Density Interval
##       89% HDI
##  [0.11, 6.05]

eti(posterior, ci = .89)
## # Equal-Tailed Interval
##       89% ETI
##  [0.42, 7.27]

![](man/figures/unnamed-chunk-8-1.png)<!-- -->

## Null-Hypothesis Significance Testing (NHST)

### ROPE

computes the proportion (in percentage) of the HDI (default to the 89%
HDI) of a posterior distribution that lies within a region of practical

Statistically, the probability of a posterior distribution of being
different from 0 does not make much sense (the probability of it being
different from a single point being infinite). Therefore, the idea
underlining ROPE is to let the user define an area around the null value
enclosing values that are *equivalent to the null* value for practical
purposes (Kruschke & Liddell, 2018, p. @kruschke2018rejecting).

Kruschke suggests that such null value could be set, by default, to the
-0.1 to 0.1 range of a standardized parameter (negligible effect size
according to Cohen, 1988). This could be generalized: For instance, for
linear models, the ROPE could be set as `0 +/- .1 * sd(y)`. This ROPE
range can be automatically computed for models using the

Kruschke suggests using the proportion of the 95% (or 90%, considered
more stable) HDI that falls within the ROPE as an index for
“null-hypothesis” testing (as understood under the Bayesian framework,

``` r
posterior <- distribution_normal(100, 0.4, 0.2)
rope(posterior, range = c(-0.1, 0.1))
## # Proportion of samples inside the ROPE [-0.10, 0.10]:
##  inside ROPE
##       1.11 %

![](man/figures/unnamed-chunk-10-1.png)<!-- -->

### Equivalence test

is a **Test for Practical Equivalence** based on the *“HDI+ROPE decision
rule”* (Kruschke, 2018) to check whether parameter values should be
accepted or rejected against an explicitly formulated “null hypothesis”
(*i.e.*, a

``` r
posterior <- distribution_normal(100, 0.4, 0.2)
equivalence_test(posterior, range = c(-0.1, 0.1))
## # Test for Practical Equivalence
##   ROPE: [-0.10 0.10]
##         H0 inside ROPE     89% HDI
##  Undecided      0.01 % [0.09 0.71]

### Probability of Direction (*p*d)

computes the **Probability of Direction** (***p*d**, also known as the
Maximum Probability of Effect - *MPE*). It varies between 50% and 100%
(*i.e.*, `0.5` and `1`) and can be interpreted as the probability
(expressed in percentage) that a parameter (described by its posterior
distribution) is strictly positive or negative (whichever is the most
probable). It is mathematically defined as the proportion of the
posterior distribution that is of the median’s sign. Although
differently expressed, this index is fairly similar (*i.e.*, is strongly
correlated) to the frequentist ***p*-value**.

**Relationship with the p-value**: In most cases, it seems that the *pd*
corresponds to the frequentist one-sided *p*-value through the formula
`p-value = (1-pd/100)` and to the two-sided *p*-value (the most commonly
reported) through the formula `p-value = 2*(1-pd/100)`. Thus, a `pd` of
`95%`, `97.5%` `99.5%` and `99.95%` corresponds approximately to a
two-sided *p*-value of respectively `.1`, `.05`, `.01` and `.001`. See
the [*reporting

``` r
posterior <- distribution_normal(100, 0.4, 0.2)
## # Probability of Direction (pd)
## pd = 98.00%

![](man/figures/unnamed-chunk-13-1.png)<!-- -->

### Bayes Factor

computes Bayes factors against the null (either a point or an interval),
bases on prior and posterior samples of a single parameter. This Bayes
factor indicates the degree by which the mass of the posterior
distribution has shifted further away from or closer to the null
value(s) (relative to the prior distribution), thus indicating if the
null value has become less or more likely given the observed data.

When the null is an interval, the Bayes factor is computed by comparing
the prior and posterior odds of the parameter falling within or outside
the null; When the null is a point, a Savage-Dickey density ratio is
computed, which is also an approximation of a Bayes factor comparing the
marginal likelihoods of the model against a model in which the tested
parameter has been restricted to the point null (Wagenmakers, Lodewyckx,
Kuriyal, & Grasman, 2010).

``` r
prior <- rnorm(1000, mean = 0, sd = 1)
posterior <- rnorm(1000, mean = 1, sd = 0.7)

bayesfactor_parameters(posterior, prior, direction = "two-sided", null = 0)
## # Bayes Factor (Savage-Dickey density ratio)
##  Bayes Factor
##          2.16
## * Evidence Against The Null: [0]

![](man/figures/unnamed-chunk-15-1.png)<!-- -->

<sup>*The lollipops represent the density of a point-null on the prior
distribution (the blue lollipop on the dotted distribution) and on the
posterior distribution (the red lollipop on the yellow distribution).
The ratio between the two - the Savage-Dickey ratio - indicates the
degree by which the mass of the parameter distribution has shifted away
from or closer to the null.*</sup>

For more info, see [the Bayes factors

### MAP-based *p*-value

computes a Bayesian equivalent of the p-value, related to the odds that
a parameter (described by its posterior distribution) has against the
null hypothesis (*h0*) using Mills’ (2014, 2017) *Objective Bayesian
Hypothesis Testing* framework. It corresponds to the density value at 0
divided by the density at the Maximum A Posteriori (MAP).

``` r
posterior <- distribution_normal(100, 0.4, 0.2)
## # MAP-based p-value
## p (MAP) = 0.193

![](man/figures/unnamed-chunk-17-1.png)<!-- -->

## Utilities

### Find ROPE’s appropriate range

This function attempts at automatically finding suitable “default”
values for the Region Of Practical Equivalence (ROPE). Kruschke (2018)
suggests that such null value could be set, by default, to a range from
`-0.1` to `0.1` of a standardized parameter (negligible effect size
according to Cohen, 1988), which can be generalised for linear models to
`-0.1 * sd(y), 0.1 * sd(y)`. For logistic models, the parameters
expressed in log odds ratio can be converted to standardized difference
through the formula `sqrt(3)/pi`, resulting in a range of `-0.05` to

``` r

### Density Estimation

This function is a wrapper over different methods of density estimation.
By default, it uses the base R `density` with by default uses a
different smoothing bandwidth (`"SJ"`) from the legacy default
implemented the base R `density` function (`"nrd0"`). However, Deng &
Wickham suggest that `method = "KernSmooth"` is the fastest and the most

### Perfect Distributions

Generate a sample of size n with near-perfect distributions.

``` r
distribution(n = 10)
##  [1] -1.28 -0.88 -0.59 -0.34 -0.11  0.11  0.34  0.59  0.88  1.28

### Probability of a Value

Compute the density of a given point of a distribution.

``` r
density_at(rnorm(1000, 1, 1), 1)
## [1] 0.37

## Credits

You can cite the package as following:

  - Makowski, D., Ben-Shachar M. S., & Lüdecke, D. (2019). *Understand
    and Describe Bayesian Models and Posterior Distributions using
    bayestestR*. Available from

# References

<div id="refs" class="references">

<div id="ref-kruschke2015doing">

Kruschke, J. K. (2015). *Doing Bayesian data analysis: A tutorial with
R, JAGS, and Stan* (2. ed). Amsterdam: Elsevier, Academic Press.


<div id="ref-kruschke2018rejecting">

Kruschke, J. K. (2018). Rejecting or accepting parameter values in
bayesian estimation. *Advances in Methods and Practices in Psychological
Science*, *1*(2), 270–280. <https://doi.org/10.1177/2515245918771304>


<div id="ref-kruschke2018bayesian">

Kruschke, J. K., & Liddell, T. M. (2018). The bayesian new statistics:
Hypothesis testing, estimation, meta-analysis, and power analysis from a
bayesian perspective. *Psychonomic Bulletin & Review*, *25*(1), 178–206.


<div id="ref-mcelreath2018statistical">

McElreath, R. (2018). *Statistical rethinking*.


<div id="ref-wagenmakers2010bayesian">

Wagenmakers, E.-J., Lodewyckx, T., Kuriyal, H., & Grasman, R. (2010).
Bayesian hypothesis testing for psychologists: A tutorial on the
savageDickey method. *Cognitive Psychology*, *60*(3), 158–189.


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