The **posterior** R package is intended to provide useful tools for both users and developers of packages for fitting Bayesian models or working with output from Bayesian models. The primary goals of the package are to:

- Efficiently convert between many different useful formats of draws (samples) from posterior or prior distributions.
- Provide consistent methods for operations commonly performed on draws, for example, subsetting, binding, or mutating draws.
- Provide various summaries of draws in convenient formats.
- Provide lightweight implementations of state of the art posterior inference diagnostics.

If you are new to **posterior** we recommend starting with these vignettes:

*The posterior R package*: an introduction to the package and its main functionality*rvar: The Random Variable Datatype*: an overview of the new random variable datatype

You can install the latest official release version via

or build the developmental version directly from GitHub via

Here we offer a few examples of using the package. For a more detailed overview see the vignette *The posterior R package*.

```
library("posterior")
#> This is posterior version 1.2.0
#>
#> Attaching package: 'posterior'
#> The following objects are masked from 'package:stats':
#>
#> mad, sd, var
```

To demonstrate how to work with the **posterior** package, we will use example posterior draws obtained from the eight schools hierarchical meta-analysis model described in Gelman et al. (2013). Essentially, we have an estimate per school (`theta[1]`

through `theta[8]`

) as well as an overall mean (`mu`

) and standard deviation across schools (`tau`

).

```
eight_schools_array <- example_draws("eight_schools")
print(eight_schools_array, max_variables = 3)
#> # A draws_array: 100 iterations, 4 chains, and 10 variables
#> , , variable = mu
#>
#> chain
#> iteration 1 2 3 4
#> 1 2.0 3.0 1.79 6.5
#> 2 1.5 8.2 5.99 9.1
#> 3 5.8 -1.2 2.56 0.2
#> 4 6.8 10.9 2.79 3.7
#> 5 1.8 9.8 -0.03 5.5
#>
#> , , variable = tau
#>
#> chain
#> iteration 1 2 3 4
#> 1 2.8 2.80 8.7 3.8
#> 2 7.0 2.76 2.9 6.8
#> 3 9.7 0.57 8.4 5.3
#> 4 4.8 2.45 4.4 1.6
#> 5 2.8 2.80 11.0 3.0
#>
#> , , variable = theta[1]
#>
#> chain
#> iteration 1 2 3 4
#> 1 3.96 6.26 13.3 5.78
#> 2 0.12 9.32 6.3 2.09
#> 3 21.25 -0.97 10.6 15.72
#> 4 14.70 12.45 5.4 2.69
#> 5 5.96 9.75 8.2 -0.91
#>
#> # ... with 95 more iterations, and 7 more variables
```

The draws for this example come as a `draws_array`

object, that is, an array with dimensions iterations x chains x variables. We can easily transform it to another format, for instance, a data frame with additional meta information.

```
eight_schools_df <- as_draws_df(eight_schools_array)
print(eight_schools_df)
#> # A draws_df: 100 iterations, 4 chains, and 10 variables
#> mu tau theta[1] theta[2] theta[3] theta[4] theta[5] theta[6]
#> 1 2.01 2.8 3.96 0.271 -0.74 2.1 0.923 1.7
#> 2 1.46 7.0 0.12 -0.069 0.95 7.3 -0.062 11.3
#> 3 5.81 9.7 21.25 14.931 1.83 1.4 0.531 7.2
#> 4 6.85 4.8 14.70 8.586 2.67 4.4 4.758 8.1
#> 5 1.81 2.8 5.96 1.156 3.11 2.0 0.769 4.7
#> 6 3.84 4.1 5.76 9.909 -1.00 5.3 5.889 -1.7
#> 7 5.47 4.0 4.03 4.151 10.15 6.6 3.741 -2.2
#> 8 1.20 1.5 -0.28 1.846 0.47 4.3 1.467 3.3
#> 9 0.15 3.9 1.81 0.661 0.86 4.5 -1.025 1.1
#> 10 7.17 1.8 6.08 8.102 7.68 5.6 7.106 8.5
#> # ... with 390 more draws, and 2 more variables
#> # ... hidden reserved variables {'.chain', '.iteration', '.draw'}
```

Different formats are preferable in different situations and hence posterior supports multiple formats and easy conversion between them. For more details on the available formats see `help("draws")`

. All of the formats are essentially base R object classes and can be used as such. For example, a `draws_matrix`

object is just a `matrix`

with a little more consistency and additional methods.

Computing summaries of posterior or prior draws and convergence diagnostics for posterior draws is one of the most common tasks when working with Bayesian models fit using Markov Chain Monte Carlo (MCMC) methods. The **posterior** package provides a flexible interface for this purpose via `summarise_draws()`

:

```
# summarise_draws or summarize_draws
summarise_draws(eight_schools_df)
#> # A tibble: 10 × 10
#> variable mean median sd mad q5 q95 rhat ess_bulk ess_tail
#> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 mu 4.18 4.16 3.40 3.57 -0.854 9.39 1.02 558. 322.
#> 2 tau 4.16 3.07 3.58 2.89 0.309 11.0 1.01 246. 202.
#> 3 theta[1] 6.75 5.97 6.30 4.87 -1.23 18.9 1.01 400. 254.
#> 4 theta[2] 5.25 5.13 4.63 4.25 -1.97 12.5 1.02 564. 372.
#> 5 theta[3] 3.04 3.99 6.80 4.94 -10.3 11.9 1.01 312. 205.
#> 6 theta[4] 4.86 4.99 4.92 4.51 -3.57 12.2 1.02 695. 252.
#> 7 theta[5] 3.22 3.72 5.08 4.38 -5.93 10.8 1.01 523. 306.
#> 8 theta[6] 3.99 4.14 5.16 4.81 -4.32 11.5 1.02 548. 205.
#> 9 theta[7] 6.50 5.90 5.26 4.54 -1.19 15.4 1.00 434. 308.
#> 10 theta[8] 4.57 4.64 5.25 4.89 -3.79 12.2 1.02 355. 146.
```

Basically, we get a data frame with one row per variable and one column per summary statistic or convergence diagnostic. The summaries `rhat`

, `ess_bulk`

, and `ess_tail`

are described in Vehtari et al. (2020). We can choose which summaries to compute by passing additional arguments, either functions or names of functions. For instance, if we only wanted the mean and its corresponding Monte Carlo Standard Error (MCSE) we would use:

```
summarise_draws(eight_schools_df, "mean", "mcse_mean")
#> # A tibble: 10 × 3
#> variable mean mcse_mean
#> <chr> <dbl> <dbl>
#> 1 mu 4.18 0.150
#> 2 tau 4.16 0.213
#> 3 theta[1] 6.75 0.319
#> 4 theta[2] 5.25 0.202
#> 5 theta[3] 3.04 0.447
#> 6 theta[4] 4.86 0.189
#> 7 theta[5] 3.22 0.232
#> 8 theta[6] 3.99 0.222
#> 9 theta[7] 6.50 0.250
#> 10 theta[8] 4.57 0.273
```

For a function to work with `summarise_draws`

, it needs to take a vector or matrix of numeric values and returns a single numeric value or a named vector of numeric values.

Another common task when working with posterior (or prior) draws, is subsetting according to various aspects of the draws (iterations, chains, or variables). **posterior** provides a convenient interface for this purpose via the `subset_draws()`

method. For example, here is the code to extract the first five iterations of the first two chains of the variable `mu`

:

```
subset_draws(eight_schools_df, variable = "mu", chain = 1:2, iteration = 1:5)
#> # A draws_df: 5 iterations, 2 chains, and 1 variables
#> mu
#> 1 2.0
#> 2 1.5
#> 3 5.8
#> 4 6.8
#> 5 1.8
#> 6 3.0
#> 7 8.2
#> 8 -1.2
#> 9 10.9
#> 10 9.8
#> # ... hidden reserved variables {'.chain', '.iteration', '.draw'}
```

The same call to `subset_draws()`

can be used regardless of whether the object is a `draws_df`

, `draws_array`

, `draws_list`

, etc.

The magic of having obtained draws from the joint posterior (or prior) distribution of a set of variables is that these draws can also be used to obtain draws from any other variable that is a function of the original variables. That is, if are interested in the posterior distribution of, say, `phi = (mu + tau)^2`

all we have to do is to perform the transformation for each of the individual draws to obtain draws from the posterior distribution of the transformed variable. This procedure is automated in the `mutate_variables`

method:

```
x <- mutate_variables(eight_schools_df, phi = (mu + tau)^2)
x <- subset_draws(x, c("mu", "tau", "phi"))
print(x)
#> # A draws_df: 100 iterations, 4 chains, and 3 variables
#> mu tau phi
#> 1 2.01 2.8 22.8
#> 2 1.46 7.0 71.2
#> 3 5.81 9.7 240.0
#> 4 6.85 4.8 135.4
#> 5 1.81 2.8 21.7
#> 6 3.84 4.1 62.8
#> 7 5.47 4.0 88.8
#> 8 1.20 1.5 7.1
#> 9 0.15 3.9 16.6
#> 10 7.17 1.8 79.9
#> # ... with 390 more draws
#> # ... hidden reserved variables {'.chain', '.iteration', '.draw'}
```

When we do the math ourselves, we see that indeed for each draw, `phi`

is equal to `(mu + tau)^2`

(up to rounding two 2 digits for the purpose of printing).

We may also easily rename variables, or even entire vectors of variables via `rename_variables`

, for example:

```
x <- rename_variables(eight_schools_df, mean = mu, alpha = theta)
variables(x)
#> [1] "mean" "tau" "alpha[1]" "alpha[2]" "alpha[3]" "alpha[4]" "alpha[5]"
#> [8] "alpha[6]" "alpha[7]" "alpha[8]"
```

As with all **posterior** methods, `mutate_variables`

and `rename_variables`

can be used with all draws formats.

Suppose we have multiple draws objects that we want to bind together:

```
x1 <- draws_matrix(alpha = rnorm(5), beta = 1)
x2 <- draws_matrix(alpha = rnorm(5), beta = 2)
x3 <- draws_matrix(theta = rexp(5))
```

Then, we can use the `bind_draws`

method to bind them along different dimensions. For example, we can bind `x1`

and `x3`

together along the `'variable'`

dimension:

```
x4 <- bind_draws(x1, x3, along = "variable")
print(x4)
#> # A draws_matrix: 5 iterations, 1 chains, and 3 variables
#> variable
#> draw alpha beta theta
#> 1 0.39 1 2.27
#> 2 -0.80 1 0.86
#> 3 0.95 1 1.93
#> 4 0.38 1 0.67
#> 5 0.18 1 2.04
```

Or, we can bind `x1`

and `x2`

together along the `'draw'`

dimension:

```
x5 <- bind_draws(x1, x2, along = "draw")
print(x5)
#> # A draws_matrix: 10 iterations, 1 chains, and 2 variables
#> variable
#> draw alpha beta
#> 1 0.39 1
#> 2 -0.80 1
#> 3 0.95 1
#> 4 0.38 1
#> 5 0.18 1
#> 6 0.13 2
#> 7 0.10 2
#> 8 -0.61 2
#> 9 0.12 2
#> 10 1.48 2
```

As with all **posterior** methods, `bind_draws`

can be used with all draws formats.

The `eight_schools`

example already comes in a format natively supported by posterior but we could of course also import the draws from other sources, for example, from common base R objects:

```
x <- matrix(rnorm(50), nrow = 10, ncol = 5)
colnames(x) <- paste0("V", 1:5)
x <- as_draws_matrix(x)
print(x)
#> # A draws_matrix: 10 iterations, 1 chains, and 5 variables
#> variable
#> draw V1 V2 V3 V4 V5
#> 1 -0.89 0.37 -0.25 -0.57 -2.85
#> 2 1.84 0.19 0.39 -0.52 1.26
#> 3 0.79 -0.74 -1.61 0.99 -0.11
#> 4 -2.25 0.28 -0.19 -0.33 0.92
#> 5 0.58 0.35 -0.92 0.56 0.82
#> 6 -1.38 -0.12 -0.40 -1.23 -0.60
#> 7 -0.18 1.18 -1.27 0.51 0.78
#> 8 0.17 1.50 -2.12 -0.45 -0.73
#> 9 -0.60 0.69 -0.43 -1.40 1.14
#> 10 0.18 0.96 -1.37 -0.58 -0.63
summarise_draws(x, "mean", "sd", "median", "mad")
#> # A tibble: 5 × 5
#> variable mean sd median mad
#> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 V1 -0.174 1.16 -0.00490 1.03
#> 2 V2 0.467 0.651 0.358 0.596
#> 3 V3 -0.817 0.770 -0.671 0.798
#> 4 V4 -0.301 0.773 -0.486 0.672
#> 5 V5 -0.000826 1.27 0.338 1.27
```

Instead of `as_draws_matrix()`

we also could have just used `as_draws()`

, which attempts to find the closest available format to the input object. In this case this would result in a `draws_matrix`

object either way.

We welcome contributions! The **posterior** package is under active development. If you find bugs or have ideas for new features (for us or yourself to implement) please open an issue on GitHub (https://github.com/stan-dev/posterior/issues).

Developing and maintaining open source software is an important yet often underappreciated contribution to scientific progress. Thus, whenever you are using open source software (or software in general), please make sure to cite it appropriately so that developers get credit for their work.

When using **posterior**, please cite it as follows:

- Bürkner P. C., Gabry J., Kay M., & Vehtari A. (2020). “posterior: Tools for Working with Posterior Distributions.” R package version XXX, <URL: https://mc-stan.org/posterior/>.

When using the MCMC convergence diagnostics `rhat`

, `ess_bulk`

, or `ess_tail`

, please also cite

- Vehtari A., Gelman A., Simpson D., Carpenter B., & Bürkner P. C. (2021). Rank-normalization, folding, and localization: An improved Rhat for assessing convergence of MCMC (with discussion).
*Bayesian Analysis*. 16(2), 667–718. doi.org/10.1214/20-BA1221

The same information can be obtained by running `citation("posterior")`

.

Gelman A., Carlin J. B., Stern H. S., David B. Dunson D. B., Aki Vehtari A., & Rubin D. B. (2013). *Bayesian Data Analysis, Third Edition*. Chapman and Hall/CRC.

Vehtari A., Gelman A., Simpson D., Carpenter B., & Bürkner P. C. (2021). Rank-normalization, folding, and localization: An improved Rhat for assessing convergence of MCMC (with discussion). *Bayesian Analysis*. 16(2), 667–718. doi.org/10.1214/20-BA1221

The **posterior** package is licensed under the following licenses:

- Code: BSD 3-clause (https://opensource.org/licenses/BSD-3-Clause)
- Documentation: CC-BY 4.0 (https://creativecommons.org/licenses/by/4.0/)