## Binomial sampling with a general continuous prior

### Description

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

### Usage

binogcp(
x,
n,
density = c("uniform", "beta", "exp", "normal", "user"),
params = c(0, 1),
n.pi = 1000,
pi = NULL,
pi.prior = NULL,
...
)


### Arguments

 x the number of observed successes in the binomial experiment. n the number of trials in the binomial experiment. density may be one of "beta", "exp", "normal", "student", "uniform" or "user" params if density is one of the parameteric forms then then a vector of parameters must be supplied. beta: a, b exp: rate normal: mean, sd uniform: min, max n.pi the number of possible \pi values in the prior pi a vector of possibilities for the probability of success in a single trial. This must be set if density = "user". pi.prior the associated prior probability mass. This must be set if density = "user". ... additional arguments that are passed to Bolstad.control

### Value

A list will be returned with the following components:

 likelihood the scaled likelihood function for \pi given x and n posterior the posterior probability of \pi given x and n pi the vector of possible \pi values used in the prior pi.prior the associated probability mass for the values in \pi

binobp binodp

### Examples


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

## 6 successes, 8 trials and a Beta(2, 2) prior
binogcp(6, 8,density = "beta", params = c(2, 2))

## 5 successes, 10 trials and a N(0.5, 0.25) prior
binogcp(5, 10, density = "normal", params = c(0.5, 0.25))

## 4 successes, 12 trials with a user specified triangular continuous prior
pi = seq(0, 1,by = 0.001)
pi.prior = rep(0, length(pi))
priorFun = createPrior(x = c(0, 0.5, 1), wt = c(0, 2, 0))
pi.prior = priorFun(pi)
results = binogcp(4, 12, "user", pi = pi, pi.prior = pi.prior)

## find the posterior CDF using the previous example and Simpson's rule
myCdf = cdf(results)
plot(myCdf, type = "l", xlab = expression(pi),
ylab = expression(Pr(pi <= pi)))

## use the quantile function to find the 95% credible region.
qtls = quantile(results, probs = c(0.025, 0.975))
cat(paste("Approximate 95% credible interval : ["
, round(qtls, 4), " ", round(qtls, 4), "]\n", sep = ""))

## find the posterior mean, variance and std. deviation
## using the output from the previous example
post.mean = mean(results)
post.var = var(results)
post.sd = sd(results)

# calculate an approximate 95% credible region using the posterior mean and
# std. deviation
lb = post.mean - qnorm(0.975) * post.sd
ub = post.mean + qnorm(0.975) * post.sd

cat(paste("Approximate 95% credible interval : ["
, round(lb, 4), " ", round(ub, 4), "]\n", sep = ""))