# Example: Bayesian Probability Illustration Diagram

Published 2012-01-15 | Author: John Henderson

The following was created in response to reading about “the mammography problem,” a commonly used example illustrating the use of Bayesian Probability, on Eliezer Yudkowsky’s page, “An Intuitive Explanation of Bayes’ Theorem”. The visualization presents the problem as involving “sieves” which behave differently depending on whether the individual passing through has or does not have cancer, illustrating the split probability created by the reliability of a mammography (chance of producing true positives and false positives). The illustration was posted on LessWrong.com, a site devoted to rationality, created by Yudkowsky.

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% Mammography problem from 'Intro to Bayes'
% Author: John Henderson
\documentclass[10pt]{article}
\usepackage{tikz}
\usetikzlibrary{positioning,decorations.pathreplacing,shapes}
\usepackage[english]{babel}
\usepackage{microtype}
\usepackage[hmargin=1.5cm,vmargin=1cm]{geometry}
\usepackage{amsmath}
\DeclareMathOperator{\p}{p}
\newcommand*{\cancer}{\text{cancer}}
\newcommand*{\testp}{\text{test}+}
\begin{document}
\begin{tikzpicture}[%
% common options for blocks:
block/.style = {draw, fill=blue!30, align=center, anchor=west,
minimum height=0.65cm, inner sep=0},
% common options for the circles:
ball/.style = {circle, draw, align=center, anchor=north, inner sep=0}]

% circle illustrating all women
\node[ball,text width=3cm,fill=purple!20] (all) at (6,0) {All women};

% two circles showing split of p{cancer} and p{~cancer}
\node[ball,fill=red!70,text width=0.1cm,anchor=base] (pcan) at (3.5,-5.5) {};
\node[ball,fill=blue!40,text width=2.9cm,anchor=base] (pncan) at (8.5,-6)
{Women without cancer\\
$\p({\sim}\cancer) = 99\%$};

% arrows showing split from all women to cancer and ~cancer
\draw[->,thick,draw=red!50] (all.south) to [out=270,in=90] (pcan.north);
\draw[->,thick,draw=blue!80] (all.south) to [out=270,in=110] (pncan.100);

% transition from all women to actual cancer rates
\node[anchor=north,text width=10cm,inner sep=.05cm,align=center,fill=white]
(why1) at (6,-3.7) {In measuring, we find:};

% note illustration the p{cancer} circle (text won't fit inside)
\node[inner sep=0,anchor=east,text width=3.3cm] (note1) at (3.2,-5.5) {
Women with cancer $\p(\cancer) = 1\%$};

% draw the sieves
\node[block,anchor=north,text width=4.4cm,fill=green!50] (tray1) at
(3.5,-8.8) {\small{$\p(\testp\mid\cancer)=0.8$}};

\node[block,anchor=north,text width=4.4cm,fill=green!50] (tray2) at
(8.5,-8.8) {$\p(\testp\mid{\sim}\cancer)=0.096$};

% text explaining how p{cancer} and p{~cancer} behave as they
% pass through the sieves
\node[anchor=west,text width=6cm] (note1) at (-6,-9.1) {
Now we pass both groups through the sieve; note that both
sieves are \emph{the same}; they just behave differently
depending on which group is passing through. \\
Let $\testp=$ a positve mammography.};

% arrows showing the circles passing through the seives
\draw[->,thick,draw=red!80] (3.5,-5.9) -- (3.5,-8.6);
\draw[->,thick,draw=blue!50] (8.5,-8.1) -- (8.5,-8.6);

% numerator
\node[ball,text width=0.05cm,fill=red!70] (can) at (6,-10.5) {};

% dividing line
\draw[thick] (5,-11) -- (7,-11);

% demoniator
\node[ball,text width=0.39cm,fill=blue!40,anchor=base] (ncan) at (6.5,-11.5) {};
\node[ball,text width=0.05cm,fill=red!70,anchor=base] (can2) at (5.5,-11.5) {};

\draw[thick] (5.9,-11.4) -- (5.9,-11.6);
\draw[thick] (5.8,-11.5) -- (6,-11.5);

% arrows showing the output of the sieves formed the fraction
\draw[->,thick,draw=red!80] (tray1.south) to [out=280,in=180] (can);
\draw[->,thick,draw=red!80] (tray1.south) to [out=280,in=180] (can2);
\node[anchor=north,inner sep=.1cm,align=center,fill=white] (why2) at
(3.8,-9.8) {$1\% * 80\%$};

\draw[->,thick,draw=blue!50] (tray2.south) to [out=265,in=0] (ncan);
\node[anchor=north,inner sep=.1cm,align=center,fill=white] (why2) at
(8.4,-9.8) {$99\% * 9.6\%$};

% explanation of final formula
\node[anchor=north west,text width=6.5cm] (note2) at (-6,-12.5)
{Finally, to find the probability that a positive test
\emph{actually means cancer}, we look at those who passed
through the sieve \emph{with cancer}, and divide by all who
received a positive test, cancer or not.};

% illustrated fraction turned into math
\node[anchor=north,text width=10cm] (solution) at (6,-12.5) {
\begin{align*}
\frac{\p(\testp\mid\cancer)}{\p(\testp\mid\cancer)
+ \p(\testp\mid{\sim}\cancer)} &= \\
\frac{1\% * 80\%}{(1\% * 80\%) + (99\% * 9.6\%)} &= 7.8\%
= \p(\cancer\mid\testp)
\end{align*}};
\end{tikzpicture}
\end{document}


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