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Introduction to Bayesian Reasoning

Below are the notes for the recording.

My girlfriend and I caught COVID a few weeks ago.

We were both sick with a cold, and when the 15-minute rapid antigen at-home test said “positive”, of course our first question was, what’s the false positive rate?

Now according to the internet, 1 in 100 people without COVID get a false positive with the test. So is that the percent change we got a false positive? No, that’s out of all the healthy people, not out of all the people with a positive result.

Bayesian probability is the name for the explicit or implicit math you do when you have to convert a number like that to your situation. The catch is that it only works with a base rate.

  • Suppose in an area, 1/100,000 asymptomatic
    • 1000 false positives
    • 1 COVID case
    • .1 % actually COVID, 99.9% false positive
  • However, my girlfriend and I had the omicron symptoms, similar to cold, and omicron was prevalent at the time
    • Suppose without the test, there was only a 20% chance we had COVID – 1/5
    • 20% COVID, 1% false positive
    • 20:1 — 95% it’s real COVID

So, with symptoms AND a test, we’ve got a 95% chance.

However, I made up the number 20%! This is the catch with Bayesian reasoning — its precision is limited by your base rate.

But, given that we had the symptoms, and omicron rates were super high, the real number can’t be far off. We can be confident we were in a much different base rate category than the asymptomatic person is. So the final result might not be 95%, it might be 92 or 98 – but we can still be pretty confident we had COVID.

You don’t need numbers, you can also do this logic implicitly (with less precision).

And indeed when we told a doctor about the test he said, yeah, you should figure you’ve got COVID because you pretty definitely do.