BDA3 Chapter 3 Exercise 4

Posted on 5 October, 2018 by Brian
Tags: bda chapter 3, solutions, bayes, odds ratio, noninformative priors
Category: bda3

Here’s my solution to exercise 4, 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 two independent trials where the likelihood of death is binomial, \(y_i \mid p_0, p_1 \sim \dbinomial(n_i, p_i)\), \(i = 0, 1\). We will compare two different non-informative priors on the odds ratio

\[ \theta := \frac{p_0}{1 - p_0} / \frac{p_1}{1 - p_1} . \]

Here are the given data.

df <- tibble(
    cohort = c('control', 'treatment'),
    patients = c(674, 680),
    deaths = c(39, 22)
  ) %>% 
  mutate(survived = patients - deaths)

cohort	patients	deaths	survived
control	674	39	635
treatment	680	22	658

We’ll need a couple of functions for drawing random samples for \(\theta\).

odds <- function(p) p / (1 - p)

simulate <- function(n, k, a, b, draws = 10000)
  # draws from posterior
  # n bernouille trials, k successes with beta(a, b) prior
  tibble(
    draw = 1:draws, 
    value = rbeta(draws, k + a, n - k + b)
  )

posterior <- function(a, b, draws = 10000)
  # random samples from theta posterior with beta(a, b) prior
  df %>% 
    transmute(
      cohort,
      draws = map2(patients, deaths, simulate, a, b, draws)
    ) %>% 
    unnest(draws) %>% 
    spread(cohort, value) %>% 
    mutate(theta = odds(control) / odds(treatment))

Let’s compare a uniform prior to a prior close to \(\dbeta(0, 0)\).

uni <- posterior(1, 1) %>% 
  mutate(prior = 'uniform')

zero <- posterior(0.000000001, 0.000000001) %>% 
  mutate(prior = 'zero')

posteriors <- bind_rows(uni, zero)

Here are the 95% posterior credible intervals for \(\theta\).

cis <- posteriors %>% 
  group_by(prior) %>% 
  summarise(
    q05 = quantile(theta, 0.05),
    q50 = quantile(theta, 0.5),
    q95 = quantile(theta, 0.95)
  )

prior	q05	q50	q95
uniform	1.173093	1.807827	2.845203
zero	1.189055	1.842631	2.961087

The estimates with the “zero” prior are slightly higher than those from the uniform prior, especially in the tails.