Distributions

Introduction

Commonly encountered distributions are built into BUGS, as described on this page. Further distributions are included in GeoBUGS and ReliaBUGS: see the Spatial distributions, Temporal distributions and Reliability distributions pages for details of these.

If a distribution is not built into BUGS, distributions specified by a log-likelihood can also be used: see Generic sampling distributions for details.

Binomial

The Binomial distribution is defined by the pmf

      

In the BUGS language it is used as

         r ~ dbin(p, n)

Binomial

The Binomial distribution is defined by the pmf

      

In the BUGS language it is used as

         r ~ dbin(p, n)

Categorical

The Categorical distribution is defined by the pmf

      

In the BUGS language it is used as

      r ~ dcat(p[])

Negative Binomial

The Negative Binomial distribution is defined by the pmf

      

In the BUGS language it is used as

      x ~ dnegbin(p, r)

Poisson

The Poisson distribution is defined by the pmf

      
In the BUGS language it is used as
         
         r ~ dpois(lambda)

Non-central hypergeometric

The Non-central hypergeometic distribution is defined by the pmf

      

In the BUGS language it is used as

         x ~ dhyper(n, m, N, psi)

Beta

The Beta distribution is defined by the pdf

      

In the BUGS language it is used as

         p ~ dbeta(a, b)

Chi-squared

The Chi-squared distribution is defined by the pdf

      
      
In the BUGS language it is used as

         x ~ dchisqr(k)

Double Exponential

The Double Exponential distribution is defined by the pdf

      

In the BUGS language it is used as

      x ~ ddexp(mu, tau)

Exponential

The Exponential distribution is defined by the pdf

      

In the BUGS language it is used as
      
      x ~ dexp(lambda)

Flat

The improper Flat distribution has a constant value for all x. It is not a proper distribution.

In the BUGS language it is used as

      x ~ dflat()

Gamma

The Gamma distribution is defined by the pdf

      

In the BUGS language it is used as

      x ~ dgamma(r, mu)

Generalized extreme value

The Generlized extreme value distribution is defined by the pdf

      

In the BUGS language it is used as

      x ~ dgev(mu, sigma, eta)

Generalized F

The Generalized F distribution is defined by the pdf

         

It reduces to the standard F for mu=0, tau=1. In the BUGS language it is used as

      x ~ df(n, m, mu, tau)

Generalized Gamma

The Generalized Gamma distribution is defined by the pdf

      

In the BUGS language it is used as

      x ~ dggamma(r, mu, beta)

Generalized Pareto

The Generalized Pareto distribution is defined by the pdf

      

In the BUGS language it is used as

      x ~ dgpar(mu, sigma, eta)

Generic log-likelihood distribution

The generic log-likelihood distribution is defined by the pdf exp(lambda). It allows generic log-likelihoods to be used in BUGS. See Generic sampling distributions for details. Note it does not depend on x.

In the BUGS language it is used as

      x ~ dloglik(lambda)

Log-normal

The Log-normal distribution is defined by the pdf

      

In the BUGS language it is used as

      x ~ dlnorm(mu, tau)

Logistic

The Logistic distribution is defined by the pdf

      

In the BUGS language it is used as

      x ~ dlogis(mu, tau)

Normal

The Normal distribution is defined by the pdf

      

In the BUGS language it is used as

      x ~ dnorm(mu, tau)

Pareto

The Pareto distribution is defined by the pdf

      

In the BUGS language it is used as

      x ~ dpar(alpha, c)

Student-t

The Student-t distribution is defined by the pdf

      

In the BUGS language it is used as

      x ~ dt(mu, tau, k)

Uniform

The Uniform distribution is defined by the pdf

      

In the BUGS language it is used as

      x ~ dunif(a, b)

Weibull

The Weibull distribution is defined by the pdf

      

In the BUGS language it is used as

      x ~ dweib(v, lambda)

Multinomial

The Multinomial distribution is defined by the pmf

      

In the BUGS language it is used as

      x[] ~ dmulti(p[], N)

Dirichlet

The Dirichlet distribution is defined by the pdf

      

In the BUGS language it is used as

      p[] ~ ddirich(alpha[])

It may also be spelt ddirch as in WinBUGS.

Multivariate normal

The Multivariate Normal distribution is defined by the pdf

      

In the BUGS language it is used as

      x[] ~ dmnorm(mu[], T[,])

Multivariate Student-t

The Multivariate Student-t distribution is defined by the pdf

      

In the BUGS language it is used as

      x[] ~ dmt(mu[], T[,], k)

Wishart

The Wishart distribution is defined by the pdf

      

In the BUGS language it is used as

      x[,] ~ dwish(R[,], k)