dist_categorical: The Categorical distribution

In distributional: Vectorised Probability Distributions

The Categorical distribution

Description

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.

Usage

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

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

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

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)

density(dist, "a")
density(dist, "z", log = TRUE)

distributional documentation built on March 31, 2023, 7:12 p.m.