binogcp function

Binomial sampling with a general continuous prior

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 $pi$

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

Returns

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 $pi$

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 = ""))

See Also

binobp binodp