21.1: A Simple Example (Section 20.3)

bayes_df = data.frame(prior=NA, 
                      likelihood=NA, 
                      marginal_likelihood=NA, 
                      posterior=NA)

bayes_df$prior <- 1/1000000 


nTests <- 3
nPositives <- 3
sensitivity <- 0.99
specificity <- 0.99

bayes_df$likelihood <- dbinom(nPositives, nTests, 0.99)

bayes_df$marginal_likelihood <- 
  dbinom(
    x = nPositives, 
    size = nTests, 
    prob = sensitivity
  ) * bayes_df$prior + 
  dbinom(
    x = nPositives, 
    size = nTests, 
    prob = 1 - specificity
  ) * 
  (1 - bayes_df$prior)

bayes_df$posterior <- 
  (bayes_df$likelihood * bayes_df$prior) / 
  bayes_df$marginal_likelihood