BDA3 Chapter 3 Exercise 2

Here’s my solution to exercise 2, chapter 3, of Gelman’s Bayesian Data Analysis (BDA), 3rd edition. There are solutions to some of the exercises on the book’s webpage.

Let θpre∼Dirichlet3(αpre) and θpost∼Dirichlet3(αpost) be the pre- and post-debate priors. Then the posteriors are

θpre∣y∼Dirichlet(294+α(pre,b),307+α(pre,d),38+α(pre,o))θpost∣y∼Dirichlet(288+α(post,b),332+α(post,d),19+α(post,o)).

Denote the proportion of support for Bush amongst Bush/Dukakis supporters by ϕ∙:=θ(∙,b)θ(∙,b)+θ(∙,d). The results of the previous exercise show that we can treat this as a beta-binomial problem by simply ignoring the results for other. More precisely,

ϕpre∣y∼Beta(294+α(pre,b),307+α(pre,d))ϕpost∣y∼Beta(288+α(post,b),332+α(post,d)).

Let’s plot these posteriors and their difference δ:=(ϕpost∣y)−(ϕpre∣y).

alpha_bush <- 1
alpha_dukakis <- 1

posteriors <- expand.grid(
    value = seq(0, 1, 0.001),
    survey = df$survey
  ) %>% 
  as_tibble() %>% 
  inner_join(df, by = 'survey') %>% 
  mutate(
    post_bush = bush + alpha_bush,
    post_dukakis = dukakis + alpha_dukakis,
    density = dbeta(value, post_bush, post_dukakis)
  ) 

N <- 10000

shift <- expand.grid(
    draw = 1:N, 
    survey = df$survey
  ) %>% 
  as_tibble() %>% 
  inner_join(df, by = 'survey') %>% 
  mutate(
    post_bush = bush + alpha_bush,
    post_dukakis = dukakis + alpha_dukakis,
    value = rbeta(n(), post_bush, post_dukakis)
  ) %>% 
  select(draw, survey, value) %>% 
  spread(survey, value) %>% 
  mutate(difference = `post-debate` - `pre-debate`) 

There is a 19% probability (the area above 0) that there was a positive shift towards Bush.