The Categorical distribution

Categorical distributions are used to represent events with multiple outcomes, such as what number appears on the roll of a dice. This is also referred to as the 'generalised Bernoulli' or 'multinoulli' distribution. The Cateogorical distribution is a special case of the Multinomial() distribution with n = 1.

dist_categorical(prob, outcomes = NULL)

Arguments

prob

A list of probabilities of observing each outcome category.

outcomes

The values used to represent each outcome.

Details

We recommend reading this documentation on https://pkg.mitchelloharawild.com/distributional/, where the math will render nicely.

In the following, let \(X\) be a Categorical random variable with probability parameters p = \(\{p_1, p_2, \ldots, p_k\}\).

The Categorical probability distribution is widely used to model the occurance of multiple events. A simple example is the roll of a dice, where \(p = \{1/6, 1/6, 1/6, 1/6, 1/6, 1/6\}\) giving equal chance of observing each number on a 6 sided dice.

Support: \(\{1, \ldots, k\}\)

Mean: \(p\)

Variance: \(p \cdot (1 - p) = p \cdot q\)

Probability mass function (p.m.f):

$$ P(X = i) = p_i $$

Cumulative distribution function (c.d.f):

The cdf() of a categorical distribution is undefined as the outcome categories aren't ordered.

Examples

dist <- dist_categorical(prob = list(c(0.05, 0.5, 0.15, 0.2, 0.1), c(0.3, 0.1, 0.6)))

dist
#> <distribution[2]>
#> [1] Categorical[5] Categorical[3]

generate(dist, 10)
#> [[1]]
#>  [1] 3 2 4 3 2 4 2 2 1 2
#> 
#> [[2]]
#>  [1] 1 1 1 1 2 3 3 3 1 3
#> 

density(dist, 2)
#> [1] 0.5 0.1
density(dist, 2, log = TRUE)
#> [1] -0.6931472 -2.3025851

# The outcomes aren't ordered, so many statistics are not applicable.
cdf(dist, 4)
#> [1] NA NA
quantile(dist, 0.7)
#> [1] NA NA
mean(dist)
#> [1] NA NA
variance(dist)
#> [1] NA NA
skewness(dist)
#> [1] NA NA
kurtosis(dist)
#> [1] NA NA

dist <- dist_categorical(
  prob = list(c(0.05, 0.5, 0.15, 0.2, 0.1), c(0.3, 0.1, 0.6)),
  outcomes = list(letters[1:5], letters[24:26])
)

generate(dist, 10)
#> [[1]]
#>  [1] "b" "b" "e" "b" "d" "b" "d" "c" "b" "b"
#> 
#> [[2]]
#>  [1] "z" "z" "x" "z" "x" "x" "z" "y" "z" "z"
#> 

density(dist, "a")
#> [1] 0.05   NA
density(dist, "z", log = TRUE)
#> [1]         NA -0.5108256