dist_mixture: Create a mixture of distributions

In distributional: Vectorised Probability Distributions

Create a mixture of distributions

Description

A mixture distribution combines multiple component distributions with specified weights. Two types of mixture are supported:

• Probability mixture (type = "probability", the default): the CDF is a weighted average of the component CDFs. Also known as linear pooling.

• Quantile mixture (type = "quantile"): the quantile function is a weighted average of the component quantile functions. Also known as Vincentization.

Usage

dist_mixture(..., weights = numeric(), type = c("probability", "quantile"))

Arguments

...	Distributions to be used in the mixture. Can be any distributional objects.
weights	A numeric vector of non-negative weights that sum to 1. The length must match the number of distributions passed to .... Each weight w_i represents the probability that a random draw comes from the i-th component distribution.
type	"probability" (default) for a probability mixture or "quantile" for a quantile mixture (Vincentization). Quantile mixtures only support univariate component distributions.

Details

In the following, let X be a mixture random variable composed of K component distributions F_1, F_2, \ldots, F_K with corresponding weights w_1, w_2, \ldots, w_K where \sum_{i=1}^K w_i = 1 and w_i \geq 0 for all i.

Probability mixture (type = "probability")

Support: The union of the supports of all component distributions

Mean:

For univariate mixtures:

E(X) = \sum_{i=1}^K w_i \mu_i

where \mu_i is the mean of the i-th component distribution.

For multivariate mixtures:

E(\mathbf{X}) = \sum_{i=1}^K w_i \boldsymbol{\mu}_i

where \boldsymbol{\mu}_i is the mean vector of the i-th component distribution.

Variance:

For univariate mixtures:

\text{Var}(X) = \sum_{i=1}^K w_i (\mu_i^2 + \sigma_i^2) - \left(\sum_{i=1}^K w_i \mu_i\right)^2

where \sigma_i^2 is the variance of the i-th component distribution.

Covariance:

For multivariate mixtures:

\text{Cov}(\mathbf{X}) = \sum_{i=1}^K w_i \left[ (\boldsymbol{\mu}_i - \bar{\boldsymbol{\mu}})(\boldsymbol{\mu}_i - \bar{\boldsymbol{\mu}})^T + \boldsymbol{\Sigma}_i \right]

where \bar{\boldsymbol{\mu}} = \sum_{i=1}^K w_i \boldsymbol{\mu}_i is the overall mean vector and \boldsymbol{\Sigma}_i is the covariance matrix of the i-th component distribution.

Probability density/mass function (p.d.f/p.m.f):

f(x) = \sum_{i=1}^K w_i f_i(x)

where f_i(x) is the density or mass function of the i-th component distribution.

Cumulative distribution function (c.d.f):

For univariate mixtures:

F(x) = \sum_{i=1}^K w_i F_i(x)

where F_i(x) is the c.d.f. of the i-th component distribution.

For multivariate mixtures, the c.d.f. is approximated numerically.

Quantile function:

For univariate probability mixtures, the quantile function has no closed form and is computed numerically by inverting the c.d.f. using root-finding (stats::uniroot()).

For multivariate mixtures, quantiles are not yet implemented.

Quantile mixture (type = "quantile")

Also known as a Vincent average or Vincentization, only univariate component distributions are supported.

Quantile function (closed form):

Q(p) = \sum_{i=1}^K w_i Q_i(p)

where Q_i(p) is the quantile function of the i-th component distribution.

Cumulative distribution function: computed numerically by inverting Q(p) via stats::uniroot().

Probability density function: derived analytically from the quantile function. For p = F(x):

f(x) = \frac{1}{Q'(p)} = \frac{1}{\sum_{i=1}^K w_i / f_i(Q_i(p))}

Mean: E(X) = \sum_{i=1}^K w_i \mu_i (identical to the probability mixture mean).

Variance: computed numerically as \int_0^1 Q(p)^2 \, dp - \left(E(X)\right)^2.

See Also

stats::uniroot(), vctrs::vec_unique_count()

Examples

# Probability mixture of two normal distributions (default)
dist <- dist_mixture(dist_normal(0, 1), dist_normal(5, 2), weights = c(0.3, 0.7))
dist

mean(dist)
variance(dist)

density(dist, 2)
cdf(dist, 2)
quantile(dist, 0.5)

generate(dist, 10)

# Quantile mixture (Vincentization) of two normal distributions
vdist <- dist_mixture(
  dist_normal(0, 1), dist_normal(5, 2),
  weights = c(0.3, 0.7), type = "quantile"
)
vdist

mean(vdist)
variance(vdist)

density(vdist, 2)
cdf(vdist, 2)
quantile(vdist, 0.5)

distributional documentation built on June 27, 2026, 5:06 p.m.