Bayes' theorem and diagnostic tests

Bayes' theorem reverses a conditional probability, combining a base rate with the accuracy of evidence — essential for interpreting screening results.

P(A \given B) = \frac{P(B \given A)\,P(A)}{P(B \given A)\,P(A) + P(B \given A^c)\,P(A^c)}

Bayes' theorem (two hypotheses).

$P(A)$: Prior probability (base rate) of A.
$P(B \given A)$: Likelihood of the evidence B if A is true.
$P(A \given B)$: Posterior probability of A after seeing B.

\text{PPV} = \frac{se\,\pi}{se\,\pi + (1 - sp)(1 - \pi)}, \qquad \text{NPV} = \frac{sp\,(1 - \pi)}{sp\,(1-\pi) + (1 - se)\,\pi}

Predictive values of a diagnostic test.

$se$: Sensitivity, P(test + | condition).
$sp$: Specificity, P(test − | no condition).
$\pi$: Prevalence (base rate) in the tested population.

\text{posterior odds} = \text{prior odds} \times LR, \qquad LR_{+} = \frac{se}{1 - sp}

Odds form with the likelihood ratio.

$LR_{+}$: Positive likelihood ratio: how much a positive result multiplies the odds.

When to use

Interpreting a positive or negative test, alarm, fraud flag or classifier output.
Any "probability of the cause given the effect" question.

How to interpret

The posterior depends on both the accuracy and the base rate. For a rare condition, even an accurate test produces many false positives, so the PPV can be low. Natural frequencies (out of 10 000 people) make the calculation transparent.

Common mistakes

Base-rate fallacy: reading sensitivity (or specificity) as the probability of having the condition after a positive result.
Using a prevalence from a different population (a clinic's patients vs the general public).
Forgetting that PPV and NPV change with prevalence while sensitivity and specificity do not.

Example

Sensitivity 0.90, specificity 0.95, prevalence 2%. Of 10 000 people, 200 have the condition: 180 test positive. Of 9 800 without it, 490 test positive. PPV = 180/670 = 0.269 — about 27% of positives truly have the condition. NPV = 9 310/9 330 = 0.998. $LR_+ = 18$ .