Note: This article is translated from my Japanese article.
What is Bayes’ theorem?
In this article, I would like to explain the importance of Bayes’ theorem, one of the most famous theorems in statistics, using the PCR testing that became widespread during the COVID-19 pandemic as an example.
Bayes’ theorem can be expressed as shown in Equation 1.
\[ P(A|B) \propto P(B|A) P(A) \tag{1}\]
Equation 1 shows the following:
- The probability \(P(A|B)\) that an observed result \(B\) was caused by \(A\)
- is proportional (\(\propto\)) to the product of the probability \(P(A)\) that cause \(A\) occurs and the probability \(P(B|A)\) that result \(B\) occurs given cause \(A\)
Let’s use a more familiar example. Suppose we define:
- Cause \(A\): whether or not someone is infected with COVID-19
- Result \(B\): the diagnostic result from a PCR test or similar test
With this in mind, Equation 1 can be rewritten as Equation 2.
\[ P(\text{infection status}|\text{diagnosis result}) \propto P(\text{diagnosis result}|\text{infection status}) P(\text{infection status}) \tag{2}\]
Why is Bayes’ theorem important?
Let’s think about the importance of Bayes’ theorem based on Equation 2. In fact, all three probabilities that appear in the version of Bayes’ theorem shown in Equation 2 correspond to very important indicators (see the table below). In other words, Equation 2 tells us that even if a test’s accuracy (sensitivity and specificity) is high, the positive predictive value will be low if the pre-test probability is low.
| Probability | Corresponding indicator in the medical field |
|---|---|
| \(P(\text{diagnosis result}|\text{infection status})\) |
Post-test probability: called the posterior probability in statistics
|
| \(P(\text{diagnosis result}|\text{infection status})\) |
|
| \(P(\text{infection status})\) |
Pre-test probability: called the prior probability in statistics
|
For example, let’s consider the following situation:
- Both sensitivity and specificity are 99% (false negatives and false positives each occur with a 1% probability)
- The pre-test probability is 1% (1 out of every 100 people tested is infected, and the remaining 99 are uninfected)
In this case, the probability of being infected given a positive diagnosis, \(P(\text{infected}|\text{positive})\), and the probability of being uninfected given a positive diagnosis, \(P(\text{uninfected}|\text{positive})\), are given by Equation 3.
\[ \begin{align} P(\text{infected}|\text{positive}) \propto P(\text{positive}|\text{infected})P(\text{infected}) &= 0.99 \times 0.01 = 0.0099 \\ P(\text{uninfected}|\text{positive}) \propto P(\text{positive}|\text{uninfected})P(\text{uninfected}) &= (1 - 0.99) \times (1 - 0.01) = 0.0099 \end{align} \tag{3}\]
From Equation 3, we can see that \(P(\text{infected}|\text{positive})\) and \(P(\text{uninfected}|\text{positive})\) are equal, meaning the positive predictive value is 50%. In other words, among the people tested who receive a positive diagnosis, only 1 in 2 is actually infected.
In this way, using Bayes’ theorem reveals an important relationship: even if a test’s accuracy (sensitivity and specificity) is high, the positive predictive value will be low if the pre-test probability is low.
Summary
Using the PCR testing that became widespread during the COVID-19 pandemic as an example, we explained the importance of Bayes’ theorem.
If you are interested in more realistic calculation examples, we also recommend reading this column published by the Statistics Bureau (in Japanese).