distributions: probability distributions

In causact: Fast, Easy, and Visual Bayesian Inference

probability distributions

Description

These functions can be used to define random variables in a causact model.

Usage

uniform(min, max, dim = NULL)

normal(mean, sd, dim = NULL, truncation = c(-Inf, Inf))

lognormal(meanlog, sdlog, dim = NULL)

bernoulli(prob, dim = NULL)

binomial(size, prob, dim = NULL)

negative_binomial(size, prob, dim = NULL)

poisson(lambda, dim = NULL)

gamma(shape, rate, dim = NULL)

inverse_gamma(alpha, beta, dim = NULL, truncation = c(0, Inf))

weibull(shape, scale, dim = NULL)

exponential(rate, dim = NULL)

pareto(a, b, dim = NULL)

student(df, mu, sigma, dim = NULL, truncation = c(-Inf, Inf))

laplace(mu, sigma, dim = NULL, truncation = c(-Inf, Inf))

beta(shape1, shape2, dim = NULL)

cauchy(location, scale, dim = NULL, truncation = c(-Inf, Inf))

chi_squared(df, dim = NULL)

logistic(location, scale, dim = NULL, truncation = c(-Inf, Inf))

multivariate_normal(mean, Sigma, dimension = NULL)

lkj_correlation(eta, dimension = 2)

multinomial(size, prob, dimension = NULL)

categorical(prob, dimension = NULL)

dirichlet(alpha, dimension = NULL)

Arguments

min, max	scalar values giving optional limits to uniform variables. Like lower and upper, these must be specified as numerics, unlike lower and upper, they must be finite. min must always be less than max.
dim	Currently ignored. If dag_greta becomes functional again, this specifies the dimensions of the greta array to be returned, either a scalar or a vector of positive integers. See details.
mean, meanlog, location, mu	unconstrained parameters
sd, sdlog, sigma, lambda, shape, rate, df, scale, shape1, shape2, alpha, beta, a, b, eta, size	positive parameters, alpha must be a vector for dirichlet.
truncation	a length-two vector giving values between which to truncate the distribution.
prob	probability parameter (⁠0 < prob < 1⁠), must be a vector for multinomial and categorical
Sigma	positive definite variance-covariance matrix parameter
dimension	Currently ignored. If dag_greta becomes functional again, this specifies, the dimension of a multivariate distribution

Details

The discrete probability distributions (bernoulli, binomial, negative_binomial, poisson, multinomial, categorical) can be used when they have fixed values, but not as unknown variables.

For univariate distributions dim gives the dimensions of the array to create. Each element will be (independently) distributed according to the distribution. dim can also be left at its default of NULL, in which case the dimension will be detected from the dimensions of the parameters (provided they are compatible with one another).

For multivariate distributions (multivariate_normal(), multinomial(), categorical(), and dirichlet() each row of the output and parameters corresponds to an independent realisation. If a single realisation or parameter value is specified, it must therefore be a row vector (see example). n_realisations gives the number of rows/realisations, and dimension gives the dimension of the distribution. I.e. a bivariate normal distribution would be produced with multivariate_normal(..., dimension = 2). The dimension can usually be detected from the parameters.

multinomial() does not check that observed values sum to size, and categorical() does not check that only one of the observed entries is 1. It's the user's responsibility to check their data matches the distribution!

Wherever possible, the parameterizations and argument names of causact distributions match commonly used R functions for distributions, such as those in the stats or extraDistr packages. The following table states the distribution function to which causact's implementation corresponds (this code largely borrowed from the greta package):

causact	reference
uniform	stats::dunif
normal	stats::dnorm
lognormal	stats::dlnorm
bernoulli	extraDistr::dbern
binomial	stats::dbinom
beta_binomial	extraDistr::dbbinom
negative_binomial	stats::dnbinom
hypergeometric	stats::dhyper
poisson	stats::dpois
gamma	stats::dgamma
inverse_gamma	extraDistr::dinvgamma
weibull	stats::dweibull
exponential	stats::dexp
pareto	extraDistr::dpareto
student	extraDistr::dlst
laplace	extraDistr::dlaplace
beta	stats::dbeta
cauchy	stats::dcauchy
chi_squared	stats::dchisq
logistic	stats::dlogis
f	stats::df
multivariate_normal	mvtnorm::dmvnorm
multinomial	stats::dmultinom
categorical	stats::dmultinom (size = 1)
dirichlet	extraDistr::ddirichlet

Examples

## Not run: 

# a uniform parameter constrained to be between 0 and 1
phi <- uniform(min = 0, max = 1)

# a length-three variable, with each element following a standard normal
# distribution
alpha <- normal(0, 1, dim = 3)

# a length-three variable of lognormals
sigma <- lognormal(0, 3, dim = 3)

# a hierarchical uniform, constrained between alpha and alpha + sigma,
eta <- alpha + uniform(0, 1, dim = 3) * sigma

# a hierarchical distribution
mu <- normal(0, 1)
sigma <- lognormal(0, 1)
theta <- normal(mu, sigma)

# a vector of 3 variables drawn from the same hierarchical distribution
thetas <- normal(mu, sigma, dim = 3)

# a matrix of 12 variables drawn from the same hierarchical distribution
thetas <- normal(mu, sigma, dim = c(3, 4))

# a multivariate normal variable, with correlation between two elements
# note that the parameter must be a row vector
Sig <- diag(4)
Sig[3, 4] <- Sig[4, 3] <- 0.6
theta <- multivariate_normal(t(rep(mu, 4)), Sig)

# 10 independent replicates of that
theta <- multivariate_normal(t(rep(mu, 4)), Sig, n_realisations = 10)

# 10 multivariate normal replicates, each with a different mean vector,
# but the same covariance matrix
means <- matrix(rnorm(40), 10, 4)
theta <- multivariate_normal(means, Sig, n_realisations = 10)
dim(theta)

# a Wishart variable with the same covariance parameter
theta <- wishart(df = 5, Sigma = Sig)

## End(Not run)

causact documentation built on Sept. 8, 2023, 5:46 p.m.