Skip to main content
Statistics LibreTexts

21.1: A Simple Example (Section 20.3)

  • 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

    21.1: A Simple Example (Section 20.3) is shared under a not declared license and was authored, remixed, and/or curated by Russell A. Poldrack via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.