## Binomial sampling with a discrete prior

### Description

Evaluates and plots the posterior density for pi, the probability of a success in a Bernoulli trial, with binomial sampling and a discrete prior on pi

### Usage

```binodp(x, n, pi = NULL, pi.prior = NULL, n.pi = 10, ...)
```

### Arguments

 `x` the number of observed successes in the binomial experiment. `n` the number of trials in the binomial experiment. `pi` a vector of possibilities for the probability of success in a single trial. if `pi` is `NULL` then a discrete uniform prior for pi will be used. `pi.prior` the associated prior probability mass. `n.pi` the number of possible pi values in the prior `...` additional arguments that are passed to `Bolstad.control`

### Value

A list will be returned with the following components:

 `pi` the vector of possible pi values used in the prior `pi.prior` the associated probability mass for the values in pi `likelihood` the scaled likelihood function for pi given x and n `posterior` the posterior probability of pi given x and n `f.cond` the conditional distribution of x given pi and n `f.joint` the joint distribution of x and pi given n `f.marg` the marginal distribution of x

`binobp` `binogcp`

### Examples

```
## simplest call with 6 successes observed in 8 trials and a uniform prior
binodp(6,8)

## same as previous example but with more possibilities for pi
binodp(6, 8, n.pi = 100)

## 6 successes, 8 trials and a non-uniform discrete prior
pi = seq(0, 1, by = 0.01)
pi.prior = runif(101)
pi.prior = sort(pi.prior / sum(pi.prior))
binodp(6, 8, pi, pi.prior)

## 5 successes, 6 trials, non-uniform prior
pi = c(0.3, 0.4, 0.5)
pi.prior = c(0.2, 0.3, 0.5)
results = binodp(5, 6, pi, pi.prior)

## plot the results from the previous example using a side-by-side barplot
results.matrix = rbind(results\$pi.prior,results\$posterior)
colnames(results.matrix) = pi
barplot(results.matrix, col = c("red", "blue"), beside = TRUE,
xlab = expression(pi), ylab=expression(Probability(pi)))
box()
legend("topleft", bty = "n", cex = 0.7,
legend = c("Prior", "Posterior"), fill = c("red", "blue"))

```