binogcp {Bolstad}R Documentation

Binomial sampling with a general continuous prior

Description

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

Usage

binogcp(x, n, density = "uniform", params = c(0,1), n.theta = 1000,
        theta = NULL, theta.prior = NULL, ret = FALSE)

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.theta the number of possible theta values in the prior
theta a vector of possibilities for the probability of success in a single trial. This must be set if density="user".
theta.prior the associated prior probability mass. This must be set if density="user".
ret if true then the likelihood and posterior are returned as a list.

Value

If ret is true, then a list will be returned with the following components:

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

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
theta<-seq(0,1,by=0.001)
theta.prior<-rep(0,length(theta))
theta.prior[theta<=0.5]<-4*theta[theta<=0.5]
theta.prior[theta>0.5]<-4-4*theta[theta>0.5]
results<-binogcp(4,12,"user",theta=theta,theta.prior=theta.prior,ret=TRUE)

## find the posterior CDF using the previous example and Simpson's rule
cdf<-sintegral(theta,results$posterior,n.pts=length(theta),ret=TRUE)
plot(cdf,type="l",xlab=expression(theta[0])
        ,ylab=expression(Pr(theta<=theta[0])))

## use the cdf to find the 95% credible region. Thanks to John Wilkinson for this simplified code.
lcb<-cdf$x[with(cdf,which.max(x[y<=0.025]))]
ucb<-cdf$x[with(cdf,which.max(x[y<=0.975]))]
cat(paste("Approximate 95% credible interval : ["
        ,round(lcb,4)," ",round(ucb,4),"]\n",sep=""))

## find the posterior mean, variance and std. deviation
## using Simpson's rule and the output from the previous example
dens<-theta*results$posterior # calculate theta*f(theta | x, n)
post.mean<-sintegral(theta,dens)

dens<-(theta-post.mean)^2*results$posterior
post.var<-sintegral(theta,dens)
post.sd<-sqrt(post.var)

# 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=""))

[Package Bolstad version 0.1-9 Index]