Bayes In The Time Of COVID
This example is from very recently and is what inspired me to write this up as a post. Here’s what happened.
Even though I wasn’t aware of any exposures or had any symptoms, I decided to have a COVID test since I wanted to spend New Year’s with my parents in person. I thought it would be wise to get tested just in case, even though I had been going to the doctor and going to work in person. Surprise, the test was positive (otherwise, why am I telling you this story?). As a result, I canceled my plans for the New Year and holed up in my bedroom for 10 days. Although I had been sharing a home and a bed with my fiancé before to learning the test result, we were concerned for a while that we might both fall ill.
Thankfully, that didn’t occur; he tested negative twice (maybe a little early the first time, but in the theoretical peak detection time the second), and neither of us ever had any symptoms. That got me to wondering: Was my test a false positive, or did I actually have COVID but didn’t pass it on to him? It’s time to apply Bayes’ Theorem.
Feel free to just look at the graphic and skip this section if you’re comfortable with your Bayes skills by this point. But I’ll walk you through the procedure here, step by step.
I define my variables first. Apostrophes are read as “not-X” and are used to denote values that are less than one. So far, it has been simple.
Then, I state my priors and assumptions:
- • Prior to the test, I had a 10% chance of having COVID, or, to put it another way, I was 90% confident that I did not. This is merely my hazy estimation.
- P(T | C’) = 0.5%: I looked up the false positive rate and this is what I found. A few studies have been done on this, and the majority of them agree that 0.5%, or one in 200, is approximately correct.
- P(T’ | C) = 0: I’m making the assumption that there aren’t any instances in which a COVID test comes back negative for me even if I’m ill.
- P(F | C’) = 0: I’m guessing that I am the only way my fiance could have acquired COVID. Given that hubby works from home and we had our groceries delivered, I was the only one who was going anywhere.
- P(F | C) = 95%: I also estimated the transmission rate, which is as follows. Given that we spent the most of the four consecutive days of peak contagiousness in the same room, there is a very high probability that my fiancé would contract COVID if I did. However, I don’t use a confidence level of 100% or even 99% since I believe that 95% is about as high as one should realistically gamble on anything that isn’t a proven truth. (Cromwell’s Rule is what this is called.)
- Take note: This also applies to my fiancé since I’m assuming no false negatives (P(T’ | C) = 0). This implies that P(fiancé tests negative) and P(fiancé does not have COVID) are similar.
Note also: In the United States, the sample is multiplied and each test is run 40 times as part of the standard process, and if any one test returns positive, the entire test is marked positive. The false negative rate for a single PCR-type COVID test is actually 20%. This indicates that the false negative rate is 0.240=10280.240=1028, which is essentially zero (i.e., 0.00000000000000000000000000001).
I’m prepared to make the typical two-variable table at this point. I increased my level of confidence in my COVID from 90% to 96% without having any knowledge about my fiancé’s health. The extremely low false-positive rate is to blame for this.
The next phase is the most intriguing: we add a third dimension to the table and “spread” its probability over it by using the conditional assumptions. The situations I’m assuming have a chance of zero are those for which the probabilities for many of them do not spread at all. However, one of them does: in the case of (C,T,F), the 10% breaks into a 9.5% block and a 0.5% block in the case of (C,T,F’). Applying the 95% transmission rate yields that result.
Finally, I can do the calculation:
P(C|T,F′)=0.50.5+0.45=52.6%P(C|T,F′)=0.50.5+0.45=52.6%
Wow, that’s really close. I’m now 52.6% certain, if that’s a word, that I had COVID asymptomatically. It is up to the reader to expand the table into 4D using the probability that a patient will exhibit symptoms given an illness (P(S | C) = 40%, P(S | C’) = 0).
https://www.significancemagazine.com/science/660-bayes-theorem-and-covid-19-testing