Section 1.3 Conditional Probability

Sometimes events can interact with each other. We would like to have the language to talk about a scenario in which evidence that one event has occurred can alter our understanding of the probability of another event occurring. We would also like to develop the mathematical tools to quantify exactly how much that probability changes.

Diagnostic tests for diseases aren’t perfect. When a test comes back positive or negative, a patient will want to understand the (conditional) probability that they have or don’t have the disease based on the evidence (the diagnostic test result).

Introducing event notation, let \(D\) be the event that a patient has the disease, and let \(P\) be the event that they receive a positive test result. Then the sensitivity of the diagnostic test is \(\Pr(P \mid D)\text{,}\) and the specificity is \(\Pr(P^c \mid D^c)\text{.}\) However, when the patient takes a diagnostic test, the conditional probabilities they would be most interested in would be \(\Pr(D \mid P)\) and \(\Pr(D^c \mid P^c)\text{.}\)

Bayes’ Theorem expresses the relationship between a conditional probability \(\Pr(A \mid B)\) and the flipped conditional probability \(\Pr(B \mid A)\text{.}\)

We’re still missing a crucial piece of information: \(\Pr(D)\text{,}\) the probability (not conditioned on any evidence) that the patient has the disease. This is often referred to as the prior, as in, our prior understanding of the probability of something before we gained some new information from evidence. The conditional probability calculated using Bayes’ Theorem is usually called the posterior (i.e., after taking evidence into account).

There isn’t always one single number that’s reasonable to use as the prior probability. For example, in a diagnostic testing situation, the prevalence of the disease—i.e., the proportion of the population who have the disease—might feel like a natural number to use as the prior. However, what prevalence should you use? During the COVID-19 pandemic, the prevalence of COVID in a particular country, state, and city might be different. There’s also the possibility of applying Bayes’ Theorem multiple times to take into account multiple pieces of evidence, using the posterior probability from one application of Bayes’ Theorem to play the role of the prior probability in the next. This idea would apply if, for example, a patient took a second diagnostic test to double-check. (Complicating the issue further, the developers of diagnostic tests often publish two or even more sets of sensitivity and specificity values, depending on whether a patient is already showing certain symptoms.)