## Bayesian inference on a normal mean with a normal prior

### Description

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

### Usage

```normnp(
x,
m.x = 0,
s.x = 1,
sigma.x = NULL,
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. `m.x` the mean of the normal prior `s.x` the standard deviation of the normal prior `sigma.x` the population std. deviation of the normal distribution. If this value is NULL, which it is by default, then a flat prior is used and m.x and s.x are ignored `mu` a vector of prior possibilities for the true mean. If this is `null`, then a set of values centered on the sample mean is used. `n.mu` the number of possible mu values in the prior `...` optional control arguments. See `Bolstad.control`

### Value

A list will be returned with the following components:

 `mu` the vector of possible mu values used in the prior `mu.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 `mean` the posterior mean `sd` the posterior standard deviation `qtls` a selection of quantiles from the posterior density

`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
normnp(x,sigma=1)

## find the posterior density with N(0.5,3) prior on mu
normnp(x,0.5,3,1)

## Find the posterior density for mu, given a random sample of 4
## observations from N(mu,sigma^2=1), y = [2.99, 5.56, 2.83, 3.47],
## and a N(3,sd=2)\$ prior for mu
y = c(2.99,5.56,2.83,3.47)
normnp(y,3,2,1)

```