Bolstad (version 0.2-41)

binomixp: 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

See Also

binodp binogcp normmixp

Examples

# NOT RUN {
## 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')




# }