## 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, a0 and b0, 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, a1 and b1, 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 p*beta(a0,b0)+(1-p)*beta(a1,b1). 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

`binodp` `binogcp` `normmixp`

### 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')

```