This vignette describes the rvar()
datatype, a multidimensional, samplebased representation of random variables designed to act as much like base R arrays as possible (e.g., by supporting many math operators and functions). This format is also the basis of the draws_rvars()
format.
The rvar()
datatype is inspired by the rv package and Kerman and Gelman (2007), though with a slightly different backing format (multidimensional arrays). It is also designed to interoperate with vectorized distributions in the distributional package, to be able to be used inside data.frame()
s and tibble()
s, and to be used with distribution visualizations in the ggdist package.
rvars
datatypeThe rvar()
datatype is a wrapper around a multidimensional array where the first dimension is the number of draws in the random variable. The most direct way to create a random variable is to pass such an array to the rvar()
function.
For example, to create a “scalar” rvar
, one would pass a onedimensional array or a vector whose length (here 4000
) is the desired number of draws:
## rvar<4000>[1] mean ± sd:
## [1] 1 ± 1
The default display of an rvar
shows the mean and standard deviation of each element of the array.
We can create random vectors by adding an additional dimension beyond just the draws dimension to the input array:
n < 4 # length of output vector
x < rvar(array(rnorm(4000*n, mean = 1, sd = 1), dim = c(4000, n)))
x
## rvar<4000>[4] mean ± sd:
## [1] 1.01 ± 0.99 1.02 ± 0.99 0.98 ± 1.00 0.99 ± 1.02
Or we can create a random matrix:
rows < 4
cols < 3
x < rvar(array(rnorm(4000 * rows * cols, mean = 1, sd = 1), dim = c(4000, rows, cols)))
x
## rvar<4000>[4,3] mean ± sd:
## [,1] [,2] [,3]
## [1,] 1.00 ± 0.98 1.00 ± 1.00 0.97 ± 1.00
## [2,] 1.00 ± 1.01 1.01 ± 1.02 0.99 ± 0.99
## [3,] 1.02 ± 1.01 0.99 ± 1.00 1.00 ± 0.99
## [4,] 1.01 ± 1.01 1.02 ± 1.00 1.00 ± 1.01
Or any array up to an arbitrary number of dimensions. The array backing an rvar
can be accessed (and modified, with caution) via draws_of()
:
## num [1:4000, 1:4, 1:3] 0.6879 0.0448 0.3519 1.261 0.2197 ...
##  attr(*, "dimnames")=List of 3
## ..$ : chr [1:4000] "1" "2" "3" "4" ...
## ..$ : NULL
## ..$ : NULL
While the above examples assume all draws come from a single chain, rvar
s can also contain samples from multiple chains. For example, if your array of draws has iterations as the first dimension and chains as the second dimension, you can use with_chains = TRUE
to create an rvar
that includes chain information:
iterations < 1000
chains < 4
rows < 4
cols < 3
x_array < array(
rnorm(iterations * chains * rows * cols, mean = 1, sd = 1),
dim = c(iterations, chains, rows, cols)
)
x < rvar(x_array, with_chains = TRUE)
x
## rvar<1000,4>[4,3] mean ± sd:
## [,1] [,2] [,3]
## [1,] 0.97 ± 1.00 1.00 ± 0.99 1.02 ± 0.99
## [2,] 1.02 ± 1.00 0.99 ± 1.01 1.01 ± 0.99
## [3,] 1.00 ± 1.00 1.00 ± 1.00 1.01 ± 1.00
## [4,] 1.03 ± 0.99 1.05 ± 1.00 0.98 ± 1.00
Manual construction and modification of rvar
s in this way is not always recommended unless you need it for performance reasons: several other higherlevel interfaces to constructing and manipulating rvar
s are described below.
draws_rvars
datatypeThe draws_rvars()
datatype, like all draws
datatypes in posterior, contains multiple variables in a joint sample from some distribution (e.g. a posterior or prior distribution).
You can construct draws_rvars()
objects directly using the draws_rvars()
function. The input rvar
s must have the same number of chains and iterations, but can otherwise have different shapes:
d < draws_rvars(x = x, y = rvar(rnorm(iterations * chains), nchains = 4))
d
## # A draws_rvars: 1000 iterations, 4 chains, and 2 variables
## $x: rvar<1000,4>[4,3] mean ± sd:
## [,1] [,2] [,3]
## [1,] 0.97 ± 1.00 1.00 ± 0.99 1.02 ± 0.99
## [2,] 1.02 ± 1.00 0.99 ± 1.01 1.01 ± 0.99
## [3,] 1.00 ± 1.00 1.00 ± 1.00 1.01 ± 1.00
## [4,] 1.03 ± 0.99 1.05 ± 1.00 0.98 ± 1.00
##
## $y: rvar<1000,4>[1] mean ± sd:
## [1] 0.0034 ± 1
Existing objects can also be converted to the draws_rvars()
format using as_draws_rvars()
. Below is the example_draws("multi_normal")
dataset converted into the draws_rvars()
format. This dataset has 100 iterations from 4 chains from the posterior of a a 3dimensional multivariate normal model. The mu
variable is a mean vector of length 3 and the Sigma
variable is a \(3 \times 3\) covariance matrix:
post < as_draws_rvars(example_draws("multi_normal"))
post
## # A draws_rvars: 100 iterations, 4 chains, and 2 variables
## $mu: rvar<100,4>[3] mean ± sd:
## [1] 0.051 ± 0.11 0.111 ± 0.20 0.186 ± 0.31
##
## $Sigma: rvar<100,4>[3,3] mean ± sd:
## [,1] [,2] [,3]
## [1,] 1.28 ± 0.17 0.53 ± 0.20 0.40 ± 0.28
## [2,] 0.53 ± 0.20 3.67 ± 0.45 2.10 ± 0.48
## [3,] 0.40 ± 0.28 2.10 ± 0.48 8.12 ± 0.95
The draws_rvars()
datatype works much the same way that other draws
formats do; see the main package vignette at vignette("posterior")
for an introduction to draws
objects. One difference is that draws_rvars
counts variables differently, because it allows variables to be multidimensional. For example, the post
object above contains two variables, mu
and Sigma
:
variables(post)
## [1] "mu" "Sigma"
But converted to a draws_list()
, it contains one variable for each combination of the dimensions of its variables:
variables(as_draws_list(post))
## [1] "mu[1]" "mu[2]" "mu[3]" "Sigma[1,1]" "Sigma[2,1]"
## [6] "Sigma[3,1]" "Sigma[1,2]" "Sigma[2,2]" "Sigma[3,2]" "Sigma[1,3]"
## [11] "Sigma[2,3]" "Sigma[3,3]"
rvar
sThe rvar()
datatype implements most math operations, including basic arithmetic, functions in the Math and Summary groups, like log()
and exp()
(see help("groupGeneric")
for a list), and more. Binary operators can be performed between multiple rvar
s or between rvar
s and numeric
s. A simple example:
mu < post$mu
Sigma < post$Sigma
mu + 1
## rvar<100,4>[3] mean ± sd:
## [1] 1.1 ± 0.11 1.1 ± 0.20 1.2 ± 0.31
Matrix multiplication is also implemented (using a tensor product under the hood). Because the normal matrix multiplication operator in R (%*%
) cannot be properly implemented for S3 datatypes, rvar
uses %**%
instead. A trivial example:
Sigma %**% diag(1:3)
## rvar<100,4>[3,3] mean ± sd:
## [,1] [,2] [,3]
## [1,] 1.28 ± 0.17 1.05 ± 0.40 1.21 ± 0.85
## [2,] 0.53 ± 0.20 7.33 ± 0.89 6.30 ± 1.44
## [3,] 0.40 ± 0.28 4.20 ± 0.96 24.35 ± 2.84
The set of mathematical functions and operators supported by rvar
s includes:
Group  Functions and operators 

Arithmetic operators 
+ ,  , * , / , ^ , %% , %/%

Logical operators 
& ,  , !

Comparison operators 
== , != , < , <= , >= , >

Matrix multiplication  %**% 
Basic functions 
abs() , sign() sqrt() floor() , ceiling() , trunc() , round() , signif()

Logarithms and exponentials 
exp() , expm1() log() , log10() , log2() , log1p()

Trigonometric functions 
cos() , sin() , tan() cospi() , sinpi() , tanpi() acos() , asin() , atan()

Hyperbolic functions 
cosh() , sinh() , tanh() acosh() , asinh() , atanh()

Special functions 
lgamma() , gamma() , digamma() , trigamma()

Cumulative functions 
cumsum() , cumprod() , cummax() , cummin()

Array transposition 
t() , aperm()

Matrix decomposition  chol() 
The E()
function is an alias of mean()
, producing means within each cell of an rvar
. For example, given mu
:
mu
## rvar<100,4>[3] mean ± sd:
## [1] 0.051 ± 0.11 0.111 ± 0.20 0.186 ± 0.31
We can get the expectation of each cell of mu
:
E(mu)
## [1] 0.05139284 0.11132363 0.18581977
Expectations of logical expressions are probabilities, and can be computed either with E()
/ mean()
or with Pr()
. Pr()
is provided as notational sugar, but also checks that the input is a logical variable before taking the mean:
Pr(mu > 0)
## [1] 0.6600 0.6900 0.7025
More generally, the rvar
data type provides two types of summary functions:
Summary functions that mimic baseR vector summary functions, except applied to rvar
vectors. These apply their summaries over elements of the input vectors within each draw, generally returning an rvar
of length 1. These functions are prefixed with rvar_
as a reminder that they return rvar
s. Here is an example of rvar_mean()
:
rvar_mean(mu)
## rvar<100,4>[1] mean ± sd:
## [1] 0.12 ± 0.11
Summary functions that summarise within elements of input vectors and over draws. These summary functions generally return base arrays (numeric
or logical
) of the same shape as the input rvar
, and are especially useful for diagnostic summaries. These summary functions are not prefixed with rvar_
as they do not return rvar
s. Here is an example of mean()
:
mean(mu)
## [1] 0.05139284 0.11132363 0.18581977
You should expect the same values from these functions (though in a different shape) when you use them with summarise_draws()
, for example:
summarise_draws(mu, mean)
## # A tibble: 3 x 2
## variable mean
## <chr> <dbl>
## 1 mu[1] 0.0514
## 2 mu[2] 0.111
## 3 mu[3] 0.186
Here is a table of both types of summary functions:
1. Summarise within draws, over elements 
2. Summarise over draws, within elements 


Output format of res = f(x)

rvar of length 1 
array of same shape as input rvar

Help page  help("rvarsummarieswithindraws") 
help("rvarsummariesoverdraws") 
Numeric summaries 
rvar_median() rvar_sum() , rvar_prod() rvar_min() , rvar_max()

median() sum() , prod() min() , max()

Mean 
rvar_mean() N/A 
mean() , E() Pr() : enforces that input is logical

Spread 
rvar_sd() rvar_var() rvar_mad()

sd() var() , variance() mad()

Range 
rvar_range() Note: length(res) == 2

range() Note: dim(res) == c(2, dim(x))

Quantiles 
rvar_quantile() Note: length(res) == length(probs)

quantile() Note: dim(res) == c(length(probs), dim(x))

Logical summaries 
rvar_all() , rvar_any()

all() , any()

Special value predicates 
rvar_is_finite() rvar_is_infinite() rvar_is_nan() rvar_is_na() Note: dim(res) == dim(x) . These functions act within draws but do not summarise over elements. 
is.finite() is.infinite() is.nan() is.na() Note: res[i] == TRUE if x[i] has any draws matching predicate (except for is.finite() , where all draws in x[i] must match) 
Diagnostics  N/A 
ess_basic() , ess_bulk() , ess_quantile() , ess_sd() , ess_tail() ,mcse_mean() , mcse_quantile() , mcse_sd() rhat() , rhat_basic()

Constant rvar
s can be constructed by converting numeric vectors or arrays into rvar
s using as_rvar()
, which will return an rvar
with one draw and the same dimensions as its input:
const < as_rvar(1:3)
const
## rvar<1>[3] mean ± sd:
## [1] 1 ± NA 2 ± NA 3 ± NA
While normally rvar
s must have the same number of draws to be used in the same expression, rvar
s with one draw are treated like constants, and can be combined with other rvar
s:
mu + const
## rvar<100,4>[3] mean ± sd:
## [1] 1.1 ± 0.11 2.1 ± 0.20 3.2 ± 0.31
rvar
sWhile rvar
s attempt to emulate as much of the functionality of base R arrays as possible, there are situations in which an existing R function may not work directly with an rvar
. There are several approaches to solving this problem.
For example, say you wish to generate samples from the following expression for \(\mu\), \(\sigma\), and \(x\):
\[ \begin{align} \left[\begin{matrix}\mu_1 \\ \vdots \\ \mu_4 \end{matrix}\right] &\sim \textrm{Normal}\left(\left[\begin{matrix}1 \\ \vdots \\ 4 \end{matrix}\right],1\right)\\ \sigma &\sim \textrm{Gamma}(1,1)\\ \left[\begin{matrix}x_1 \\ \vdots \\ x_4 \end{matrix}\right] &\sim \textrm{Normal}\left(\left[\begin{matrix}\mu_1 \\ \vdots \\ \mu_4 \end{matrix}\right], \sigma\right) \end{align} \]
There are three different approaches you might take to doing this: converting existing R functions with rfun()
, executing expressions of random variables with rdo()
, or evaluating random number generator functions using rvar_rng()
.
rfun()
The rfun()
wrapper converts an existing R function into a new function that rvar
s can be passed to it as arguments, and which will return rvar
s. We can use rfun()
to convert the base rnorm()
and rgamma()
random number generating functions into functions that accept and return rvar
s:
Then we can translate the above example into code using those functions:
mu < rvar_norm(4, mean = 1:4, sd = 1)
sigma < rvar_gamma(1, shape = 1, rate = 1)
x < rvar_norm(4, mu, sigma)
x
## rvar<4000>[4] mean ± sd:
## [1] 0.99 ± 1.7 1.98 ± 1.7 2.99 ± 1.8 4.01 ± 1.7
While rfun()
converted functions work well for prototyping, they will generally speaking be slower than functions designed specifically for rvar
s. Thus, you may find you need to adopt other strategies (like rvar_rng()
, described below; or rewriting functions to support rvar
directly using math operators and/or the draws_of()
function).
rdo()
An alternative to rfun()
is to use rdo()
, which can be passed nearlyarbitrary R expressions. The expression will be executed multiple times to construct an rvar
. E.g., we can write an expression for mu
like in the above example:
## rvar<4000>[4] mean ± sd:
## [1] 1 ± 1.01 2 ± 1.01 3 ± 0.99 4 ± 1.02
We can also control the number of draws using the ndraws
argument:
## rvar<1000>[4] mean ± sd:
## [1] 0.98 ± 0.98 2.03 ± 1.03 2.98 ± 0.98 4.03 ± 1.02
rdo()
expressions can also contain other rvar
s, so long as all rvar
s in the expression have the same number of draws. Thus, we can rewrite the example above that used rfun()
as follows:
mu < rdo(rnorm(4, mean = 1:4, sd = 1))
sigma < rdo(rgamma(1, shape = 1, rate = 1))
x < rdo(rnorm(4, mu, sigma))
x
## rvar<4000>[4] mean ± sd:
## [1] 1 ± 1.7 2 ± 1.7 3 ± 1.7 4 ± 1.7
Like rfun()
, rdo()
is not necessarily fast, so you may find it more useful for prototyping than production code.
rvar_rng()
rvar_rng()
is an alternative to rfun()
/rdo()
designed specifically to work with random number generating functions that follow the typical API of such functions in base R. Such functions, like rnorm()
, rgamma()
, rbinom()
, etc all following this interface:
n
, giving the number of draws to take from the distribution.mean
, sd
, shape
, rate
, etc.) are vectorized.n
, representing n
draws from the distribution.You can use any function with this interface with rvar_rng()
, and it will adapt it to be able to take rvar
arguments and return an rvar
, as follows:
mu < rvar_rng(rnorm, 4, mean = 1:4, sd = 1)
sigma < rvar_rng(rgamma, 1, shape = 1, rate = 1)
x < rvar_rng(rnorm, 4, mu, sigma)
x
## rvar<4000>[4] mean ± sd:
## [1] 1 ± 1.7 2 ± 1.8 3 ± 1.7 4 ± 1.7
In contrast to the rfun()
and rdo()
examples above, rvar_rng()
takes advantage of the existing vectorization of the underlying random number generating function to execute quickly.
Broadcasting for rvar
s does not follow R’s vector recycling rules. Instead, when two variables with different dimensions are being used with basic arithmetic functions, dimensions are added until both variables have the same number of dimensions. If two variables \(x\) and \(y\) differ on the length of dimension \(d\), they can be broadcast to the same size so long as one of the variables has dimension \(d\) of size 1. Then that variable will be broadcast up to the same size as the other variable along that dimension. If two variables disagree on the size of a dimension and neither has size 1, it is an error.
For example, consider this random matrix:
## rvar<4000>[4,3] mean ± sd:
## [,1] [,2] [,3]
## [1,] 1 ± 0.99 5 ± 1.00 9 ± 1.02
## [2,] 2 ± 1.01 6 ± 0.99 10 ± 1.01
## [3,] 3 ± 1.00 7 ± 1.01 11 ± 1.00
## [4,] 4 ± 1.01 8 ± 1.02 12 ± 1.01
And this vector of length 3:
## rvar<4000>[3] mean ± sd:
## [1] 3 ± 1 2 ± 1 1 ± 1
If we attempt to add X
and y
, it will produce an error as vectors are by default treated as column vectors, and y
has length 3 while columns of X
have length 4:
X + y
## Error: Cannot broadcast array of shape [4000,3,1] to array of shape [4000,4,3]:
## All dimensions must be 1 or equal.
By contrast, R arrays of the same shape will simply recycle y
until it is the same length as X
(regardless of the dimensions). Thus will produce a result, though likely not the intended result:
## [,1] [,2] [,3]
## [1,] 4.010833 6.970957 10.01651
## [2,] 3.972172 7.033491 12.97111
## [3,] 4.019509 9.979396 12.97034
## [4,] 6.990685 9.978003 13.01769
On the other hand, if y were a row vector…
row_y = t(y)
row_y
## rvar<4000>[1,3] mean ± sd:
## [,1] [,2] [,3]
## [1,] 3 ± 1 2 ± 1 1 ± 1
…it would have the same number of columns as X
and contain only one row, so it can be broadcast along rows of X
:
X + row_y
## rvar<4000>[4,3] mean ± sd:
## [,1] [,2] [,3]
## [1,] 4 ± 1.4 7 ± 1.4 10 ± 1.4
## [2,] 5 ± 1.4 8 ± 1.4 11 ± 1.4
## [3,] 6 ± 1.4 9 ± 1.4 12 ± 1.4
## [4,] 7 ± 1.4 10 ± 1.5 13 ± 1.4
rvar
sThe rvar
data type supplies an implementation of as.list()
, which should give compatibility with the base R family of functions for applying functions over arrays: apply()
, lapply()
, vapply()
, sapply()
, etc. You can also manually use as.list()
to convert an rvar
into a list along its first dimension, which may be necessary for compatibility with some functions (like purrr:map()
).
For example, given this multidimensional rvar
…
## rvar<4000>[2,3,4] mean ± sd:
## , , 1
##
## [,1] [,2] [,3]
## [1,] 1 ± 1.00 3 ± 0.98 5 ± 1.00
## [2,] 2 ± 1.00 4 ± 1.00 6 ± 1.01
##
## , , 2
##
## [,1] [,2] [,3]
## [1,] 7 ± 1.00 9 ± 1.00 11 ± 1.03
## [2,] 8 ± 0.98 10 ± 0.99 12 ± 1.00
##
## , , 3
##
## [,1] [,2] [,3]
## [1,] 13 ± 1.00 15 ± 1.00 17 ± 0.99
## [2,] 14 ± 1.01 16 ± 1.00 18 ± 1.00
##
## , , 4
##
## [,1] [,2] [,3]
## [1,] 19 ± 0.99 21 ± 0.99 23 ± 0.99
## [2,] 20 ± 1.00 22 ± 1.00 24 ± 1.00
… you can apply functions along the margins using apply()
(here, a silly example):
## [,1] [,2] [,3]
## [1,] 4 4 4
## [2,] 4 4 4
One exception is that while apply()
will work with an rvar
input if your function returns base data types (like numerics), it will not give you simplified rvar
arrays if your function returns an rvar
. Thus, we supply the rvar_apply()
function, which takes in either base arrays or rvar
arrays and returns rvar
arrays, and which also uses the rvar
broadcasting rules to combine the results of the applied function.
For example, you can use rvar_apply()
with rvar_mean()
to compute the distributions of means along one margin of an array:
rvar_apply(x, 1, rvar_mean)
## rvar<4000>[2] mean ± sd:
## [1] 12 ± 0.29 13 ± 0.29
Or along multiple dimensions:
rvar_apply(x, c(2,3), rvar_mean)
## rvar<4000>[3,4] mean ± sd:
## [,1] [,2] [,3] [,4]
## [1,] 1.5 ± 0.70 7.5 ± 0.69 13.5 ± 0.71 19.5 ± 0.70
## [2,] 3.5 ± 0.70 9.5 ± 0.71 15.5 ± 0.72 21.5 ± 0.70
## [3,] 5.5 ± 0.71 11.5 ± 0.72 17.5 ± 0.71 23.5 ± 0.70
rvar
s in data frames and in ggplot2rvar
s can be used as columns in data.frame()
or tibble()
objects:
data.frame(x = c("a","b","c"), y)
## x y
## 1 a 2.997187 ± 1.0017769
## 2 b 1.983319 ± 0.9998891
## 3 c 1.028725 ± 1.0021676
This makes them convenient for adding predictions to a data frame alongside the data used to generate the predictions. rvar
s can then be visualized with ggplot2 using the stat_dist_...
family of geometries in the ggdist package.