21.1: A Simple Example (Section 20.3)

Last updated
Save as PDF

Page ID: 8829

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

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