## Bayesian inference on a normal mean with a mixture of normal priors

### Description

Evaluates and plots the posterior density for mu, the mean of a normal distribution, with a mixture of normal priors on mu

### Usage

```normmixp(
x,
sigma.x,
prior0,
prior1,
p = 0.5,
mu = NULL,
n.mu = max(100, length(mu)),
...
)
```

### Arguments

 `x` a vector of observations from a normal distribution with unknown mean and known std. deviation. `sigma.x` the population std. deviation of the observations. `prior0` the vector of length 2 which contains the means and standard deviation of your precise prior. `prior1` the vector of length 2 which contains the means and standard deviation of your vague prior. `p` the mixing proportion for the two component normal priors. `mu` a vector of prior possibilities for the mean. If it is `NULL`, then a vector centered on the sample mean is created. `n.mu` the number of possible mu values in the prior. `...` additional arguments that are passed to `Bolstad.control`

### Value

A list will be returned with the following components:

 `mu` the vector of possible mu values used in the prior `prior` the associated probability mass for the values in mu `likelihood` the scaled likelihood function for mu given x and sigma.x `posterior` the posterior probability of mu given x and sigma.x

`binomixp` `normdp` `normgcp`

### Examples

```
## generate a sample of 20 observations from a N(-0.5, 1) population
x = rnorm(20, -0.5, 1)

## find the posterior density with a N(0, 1) prior on mu - a 50:50 mix of
## two N(0, 1) densities
normmixp(x, 1, c(0, 1), c(0, 1))

## find the posterior density with 50:50 mix of a N(0.5, 3) prior and a
## N(0, 1) prior on mu
normmixp(x, 1, c(0.5, 3), c(0, 1))

## Find the posterior density for mu, given a random sample of 4
## observations from N(mu, 1), y = [2.99, 5.56, 2.83, 3.47],
## and a 80:20 mix of a N(3, 2) prior and a N(0, 100) prior for mu
x = c(2.99, 5.56, 2.83, 3.47)
normmixp(x, 1, c(3, 2), c(0, 100), 0.8)

```