# BDA3 Chapter 3 Exercise 3

Here’s my solution to exercise 3, chapter 3, of Gelman’s Bayesian Data Analysis (BDA), 3rd edition. There are solutions to some of the exercises on the book’s webpage.

Suppose we have $$n$$ measurements $$y \mid \mu, \sigma \sim \dnorm(\mu, \sigma)$$, where the prior $$p(\mu, \log \sigma) \propto 1$$ is uniform. The calculations on page 66 show that the marginal posterior distribution of $$\mu$$ is $$\mu \mid y \sim \dt(\bar y, s / n)$$, where $$s$$ is the sample standard deviation. The measurements are as follows.

control <- list(
n = 32,
mean = 1.013,
sd = 0.24
)

treatment <- list(
n = 36,
mean = 1.173,
sd = 0.2
)

The t-distribution in base-R is a standardised t-distribution. For a more general t-distribution (with arbitrary location and scale), we’ll use the LaplacesDemon package.

library(LaplacesDemon)

This allows us to plot the marginal posterior means.

mp <- tibble(value = seq(0, 2, 0.01)) %>%
mutate(
ctrl = dst(value, control$mean, control$sd / sqrt(control$n), control$n - 1),
trt = dst(value, treatment$mean, treatment$sd / sqrt(treatment$n), treatment$n - 1)
) %>%
gather(cohort, density, ctrl, trt) 

The 95% credible interval of the marginal posterior means is:

draws <- 10000

difference <- tibble(draw = 1:draws) %>%
mutate(
ctrl = rst(n(), control$mean, control$sd / sqrt(control$n), control$n - 1),
trt = rst(n(), treatment$mean, treatment$sd / sqrt(treatment$n), treatment$n - 1),
difference = trt - ctrl
)

ci <- difference\$difference %>%
quantile(probs = c(0.05, 0.95))

ci
        5%        95%
0.06752309 0.25173035 

We can also plot the distribution of the difference.