Let’s revisit Sophie and the equation of life (a.k.a. Bayes’ theorem). We know that the chance of breast cancer became about 12%, starting from a state of no symptoms and a positive test result. That too from a test that has 95% sensitivity and 90% specificity. And the secret behind this mysterious result was the low prevalence or prior probability of the disease.
P(D|TP) = P(TP|D) x P(D) /[P(TP|D) x P(D) + P(TP|!D) x P(!D)]
Here, TP represents test positive, D denotes disease and !D is no disease.
Rare disease
How do you describe a rare disease? As a simple approximation, let’s define it as an illness with a chance of 1% or lower to occur. We’ll now apply the Bayes’ rule for a few cases of P(D) (0.01, 0.005, 0.001, etc) and see how the probability updates when the test comes positive.
| P(D) Prior | P(D|TP) Posterior | Posterior / Prior (ratio) |
| 0.01 | 0.088 | 8.8 |
| 0.005 | 0.046 | 9.2 |
| 0.001 | 0.0094 | 9.4 |
Can you estimate the Bayes factor for the above case?
Bayes FactorD-!D = P(TP|D) / P(TP|!D)
P(TP|D) = sensitivity = 0.95
P(TP|!D) = 1 – specificity = 1 – 0.9 = 0.1
BFD-!D = 0.95 / 0.1 = 9.5
As a heuristic, for rare diseases, the updated chance of having the disease post a positive diagnostics equals Bayes factor x prevalence.
