Do You Think Like A Bayesian?

You’re watching American TV, and an overproduced commercial informs you there’s a rare genetic condition preventing you from strolling through life surrounded by ridiculously attractive people frolicking through a lush park. While gleeful couples smile and breathe in vibrant, CGI-enhanced air, the ad insists you might be missing out on life because of this obscure ailment you never knew you had. Clearly, the only solution is to ask your doctor about Griftapril™.

The doctor is going to get paid either way, so he agrees to run a test for a rare condition that only one in every ten thousand people have. The test is highly accurate — 99% sensitive and 99% specific — which you interpret as: “If I test positive, I’m basically guaranteed to have it.”

And guess what? You tested positive.

Luckily for you, though, your doctor read this exceptionally clear blog post about Bayesian reasoning and tells you the chance of having the disease is not 99% but actually less than 1%.

And guess what? He’s right!

How can that be?

Deriving Bayes’ Rule

Bayes’ rule comes from a simple fact about probabilities. For two events, A (e.g., having the disease) and B (e.g., testing positive), the probability of both events happening can be written in two ways: