A conditional probability app

07 Feb 2016

I wrote my first-ever Shiny app! My first-ever app, too. It’s a simple conditional probability calculator.

Say you have two events, \(A\) and \(B\), which occur over two populations \(N_A\) and \(N_B\). Input the number of occurrences of \(A\) and \(B\) and the size of \(N_A\) and \(N_B\), and the app will apply Bayes’ rule to show you $$P(A

B)\(for beliefs you may hold over\)P(B

A)$$.

For example: Suppose you want to know the probability that someone who died in the USA in 2010 was 85+ years old. Event A is taken to be the number of deaths in the USA in 2010, event B is taken to be the population that was 85+ years old in the USA in 2010, and the population size is taken to be the average total population of the USA in 2010 (\(N_A = N_B\)). Three Wolfram Alpha queries later we have

\[\begin{align} P(A) &= \frac{2,550,000}{310,000,000} = 0.008 \cr P(B) &= \frac{5,176,00}{310,000,000} = 0.017 \end{align}\]

By Bayes’ rule, we have

\[P(A|B) = P(B|A) \frac{P(A)}{P(B)}\]

I think \(P(A)/P(B)\) is called the base rate.

Using the app, I can see how the probability someone died given they were 85 or older would change depending on the probability of being 85 or older given they died (in the USA in 2010), or $$P(\text{died in USA in 2010

85 or older in USA in 2010})\(as a function of my beliefs over\)P(\text{85 or older in USA in 2010

died in USA in 2010})$$. In frequency terms, I can calculate the proportion of deaths that were 85+ year olds using the number of 85+ year olds, the number of deaths, the total US population in 2010, and my beliefs over the proportion of 85 year olds who died. I used this example as a “comparison event” for the app.

In the picture below, the red line is $$P(\text{died in USA in 2010

85 or older in USA in 2010})\(, and the black line is\)P(A

B)\(for a hypothetical event where\)P(A) = 15/100\(and\)P(B) = 50/100$$.

I wrote this app to help me think through conditional probabilities more clearly. Often I hold beliefs over some $$P(B

A)\(and use those beliefs to infer something about the corresponding\)P(A

B)$$, not necessarily considering the base rate. Hopefully I’ll neglect base rates a little less going forward.