Bayesian Estimation: Covid Test Confidence
Why a positive test for a rare disease doesn't necessarily mean you are sick.
The Base Rate Fallacy
When Covid-19 rapid antigen tests were first introduced, many people struggled to understand their efficacy. If a test is 95% accurate, and you get a positive result, what is the probability you actually have Covid?
Surprisingly, it’s often much lower than 95%. This is due to the base rate fallacy and can be explained using Bayes’ Theorem.
Bayesian Estimation Model
Test Confidence Simulation (Placeholder)
Prior Confidence: 50%
Posterior Probability (Given positive test): 75.0%
Bayes’ Theorem in Practice
Bayes’ Theorem relates current probability to prior probability. It is stated mathematically as:
P(A|B) = [P(B|A) * P(A)] / P(B)
Where:
- P(A|B) is the probability of having the disease given a positive test (Posterior).
- P(B|A) is the probability of a positive test given you have the disease (True Positive Rate).
- P(A) is the overall prevalence of the disease in the population (Prior / Base Rate).
- P(B) is the overall probability of a positive test.
If the disease is very rare (low base rate), the number of false positives can easily outnumber the true positives, leading to a surprisingly low confidence in a single positive result.