binogcp {Bolstad}

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

See Also

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[0]),
	   ylab = expression(Pr(pi <= pi[0])))

## 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[1], 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 = ""))