bcwr/bcwr05.R at master · rghan/bcwr · GitHub

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# Bayesian Computation with R (Second Edition)
# by Jim Albert

library(LearnBayes)

# 5 Introduction to Bayesian Computation

# 5.1 Introduction

# 5.2 Computing Integrals

# 5.3 Setting Up a Problem in R

# 5.4 A Beta-Binomial Model for Overdispersion

# Table 5.1. Cancer mortality data. Each ordered pair represents the number of
# cancer deaths yj and the number at risk nj for an individual city in Missouri.

str(cancermortality)

mycontour(betabinexch0, c(0.0001, 0.003, 1, 20000), cancermortality, xlab = "eta", ylab = "K")

# Fig. 5.1. Contour plot of parameters η and K in the beta-binomial model problem.

mycontour(betabinexch, c(-8, -4.5, 3, 16.5), cancermortality, xlab = "logit eta", ylab = "log K")

# Fig. 5.2. Contour plot of transformed parameters logit(η) and log K in the beta- binomial model problem.

# 5.5 Approximations Based on Posterior Modes

# 5.6 The Example

fit <- laplace(betabinexch, c(-7, 6), cancermortality)
fit

npar <- list(m = fit$mode, v = fit$var)
mycontour(lbinorm, c(-8, -4.5, 3, 16.5), npar, xlab = "logit eta", ylab = "log K")

# Fig. 5.3. Contour plot of normal approximation of logit(η) and log K in the beta-binomial model problem.

se <- sqrt(diag(fit$var))
fit$mode - 1.645*se
fit$mode + 1.645*se

# 5.7 Monte Carlo Method for Computing Integrals

p <- rbeta(1000, 14.26, 23.19)
est <- mean(p^2)
se <- sd(p^2)/sqrt(1000)
c(est, se)

# 5.8 Rejection Sampling

betabinT <- function(theta, datapar) {
  data <- datapar$data
  tpar <- datapar$par
  d <- betabinexch(theta,data) - dmt(theta, mean = c(tpar$m), S = tpar$var, df = tpar$df, log = TRUE)
  return(d)
}

tpar <- list(m = fit$mode, var = 2*fit$var, df = 4)
datapar <- list(data = cancermortality, par = tpar)

start <- c(-6.9, 12.4)
fit1 <- laplace(betabinT, start, datapar)
fit1$mode

betabinT(fit1$mode, datapar)

theta <- rejectsampling(betabinexch, tpar, -569.2813, 10000, cancermortality)
dim(theta)

mycontour(betabinexch, c(-8, -4.5, 3, 16.5), cancermortality, xlab = "logit eta", ylab = "log K")
points(theta[ , 1], theta[ , 2])

# Fig. 5.4. Contour plot of logit(η) and log K in the beta-binomial model problem
# together with simulated draws from the rejection algorithm.

# 5.9 Importance Sampling

# 5.9.1 Introduction

betabinexch.cond <- function (log.K, data) {
  eta <- exp(-6.818793)/(1 + exp(-6.818793))
  K <- exp(log.K)
  y <- data[ , 1]
  n <- data[ , 2]
  N <- length(y)
  logf <- 0*log.K
  for (j in 1:length(y))
    logf <- logf + lbeta(K*eta + y[j], K*(1 - eta) + n[j] - y[j]) - lbeta(K*eta, K*(1 - eta))
  val <- logf + log.K - 2*log(1 + K)
  return(exp(val - max(val)))
}

I <- integrate(betabinexch.cond, 2, 16, cancermortality)
par(mfrow = c(2, 2))
curve(betabinexch.cond(x, cancermortality)/I$value, from = 3, to = 16,
      ylab = "Density", xlab = "log K", lwd = 3, main = "Densities")
curve(dnorm(x, 8, 2), add = TRUE)
legend("topright", legend = c("Exact", "Normal"), lwd = c(3, 1))
curve(betabinexch.cond(x, cancermortality)/I$value/dnorm(x,8,2), from = 3, to = 16,
      ylab = "Weight", xlab = "log K", main = "Weight = g/p")
curve(betabinexch.cond(x, cancermortality)/I$value, from = 3, to = 16,
      ylab = "Density", xlab = "log K", lwd = 3, main = "Densities")
curve(1/2*dt(x - 8, df = 2), add = TRUE)
legend("topright", legend = c("Exact", "T(2)"), lwd = c(3, 1))
curve(betabinexch.cond(x, cancermortality)/I$value/(1/2*dt(x - 8, df = 2)), from = 3, to = 16,
      ylab = "Weight", xlab = "log K", main = "Weight = g/p")

# Fig. 5.5. Graph of the posterior density of log K and weight function using a normal
# proposal density (top) and a t(2) proposal density (bottom). By using a t proposal
# density, the weight function appears to be bounded from above.

# 5.9.2 Using a Multivariate t as a Proposal Density

tpar <- list(m = fit$mode, var = 2*fit$var, df = 4)
myfunc <- function(theta)
  return(theta[2])
s <- impsampling(betabinexch, tpar, myfunc, 10000, cancermortality)
cbind(s$est, s$se)

# 5.10 Sampling Importance Resampling

theta.s <- sir(betabinexch, tpar, 10000, cancermortality)

S <- bayes.influence(theta.s, cancermortality)

par(mfrow = c(1, 1))
plot(c(0, 0, 0), S$summary, type = "b", lwd = 3, xlim = c(-1, 21), ylim = c(5, 11), xlab = "Observation removed", ylab = "log K")
for (i in 1:20)
  lines(c(i, i, i), S$summary.obs[i, ], type = "b")

# Fig. 5.6. Ninety percent interval estimates for log K for the full dataset (thick line)
# and interval estimates for datasets with each individual observation removed.

# 5.11 Further Reading

# 5.12 Summary of R Functions

# 5.13 Exercises