R: Binomial sampling with a beta mixture prior

binomixp {Bolstad}

R Documentation

Binomial sampling with a beta mixture prior

Description

Evaluates and plots the posterior density for \pi, the probability of a success in a Bernoulli trial, with binomial sampling when the prior density for \pi is a mixture of two beta distributions, beta(a_0,b_0) and beta(a_1,b_1).

Usage

binomixp(x, n, alpha0 = c(1, 1), alpha1 = c(1, 1), p = 0.5, ...)

Arguments

`x`	the number of observed successes in the binomial experiment.
`n`	the number of trials in the binomial experiment.
`alpha0`	a vector of length two containing the parameters, `a_0` and `b_0`, for the first component beta prior - must be greater than zero. By default the elements of alpha0 are set to 1.
`alpha1`	a vector of length two containing the parameters, `a_1` and `b_1`, for the second component beta prior - must be greater than zero. By default the elements of alpha1 are set to 1.
`p`	The prior mixing proportion for the two component beta priors. That is the prior is `pimes beta(a_0,b_0)+(1-p)imes beta(a_1,b_1)`. `p` is set to 0.5 by default
`...`	additional arguments that are passed to `Bolstad.control`

Value

A list will be returned with the following components:

`pi`	the values of `\pi` for which the posterior density was evaluated
`posterior`	the posterior density of `\pi` given `n` and `x`
`likelihood`	the likelihood function for `\pi` given `x` and `n`, i.e. the `binomial(n,\pi)` density
`prior`	the prior density of `\pi` density

Examples


## simplest call with 6 successes observed in 8 trials and a 50:50 mix
## of two beta(1,1) uniform priors
binomixp(6,8)

## 6 successes observed in 8 trials and a 20:80 mix of a non-uniform
## beta(0.5,6) prior and a uniform beta(1,1) prior
binomixp(6,8,alpha0=c(0.5,6),alpha1=c(1,1),p=0.2)

## 4 successes observed in 12 trials with a 90:10 non uniform beta(3,3) prior
## and a non uniform beta(4,12).
## Plot the stored prior, likelihood and posterior
results = binomixp(4, 12, c(3, 3), c(4, 12), 0.9)$mix

par(mfrow = c(3,1))
y.lims = c(0, 1.1 * max(results$posterior, results$prior))

plot(results$pi,results$prior,ylim=y.lims,type='l'
,xlab=expression(pi),ylab='Density',main='Prior')
polygon(results$pi,results$prior,col='red')

plot(results$pi,results$likelihood,type='l',
     xlab = expression(pi), ylab = 'Density', main = 'Likelihood')
polygon(results$pi,results$likelihood,col='green')

plot(results$pi,results$posterior,ylim=y.lims,type='l'
,xlab=expression(pi),ylab='Density',main='Posterior')
polygon(results$pi,results$posterior,col='blue')