Question 1

What does Bayes' theorem calculate?

Accepted Answer

Bayes' theorem updates a prior belief P(A) using new evidence B to produce a posterior belief P(A|B). The formula: P(A|B) = P(B|A)×P(A) / P(B). It answers: 'Given that I observed B, how likely is A?' This is distinct from P(B|A), which is what the data model gives you.

Question 2

Why is the posterior probability so low for rare events, even with accurate tests?

Accepted Answer

This is the base rate fallacy. If a disease affects 1% of people (prior = 0.01) and the test is 99% accurate (sensitivity = 0.99, specificity = 0.99), the false positive rate is 1%. Out of 10,000 people: 100 have the disease (99 test positive), 9,900 are healthy (99 test false positive). Of 198 positive tests, only 99 are true positives — posterior ≈ 50%, not 99%. Low base rates dominate.

Question 3

What is sensitivity and specificity in Bayes' theorem?

Accepted Answer

Sensitivity = P(positive test | disease present) = true positive rate. Specificity = P(negative test | disease absent). False positive rate = 1 − specificity = P(positive test | disease absent). In this calculator: P(B|A) = sensitivity; P(B|¬A) = false positive rate = 1 − specificity. Positive Predictive Value (PPV) = posterior P(A|B) = P(disease | positive test).

Question 4

How do I choose the prior probability?

Accepted Answer

The prior P(A) represents your belief before seeing the evidence. In medical testing, use disease prevalence in the relevant population. In hypothesis testing, use base rates of true effects in your field (estimated 10–50% of tested hypotheses are true in most fields). The prior has a huge impact when it is extreme (very low or very high) — which is why rare diseases with accurate tests still have low PPV.

Question 5

What is sequential Bayes' updating?

Accepted Answer

Bayes' theorem can be applied iteratively: the posterior from one observation becomes the prior for the next. This is the foundation of Bayesian inference. Example: after a positive test (posterior = 50%), if you run a second independent test and it is also positive, use the 50% as your new prior. The result converges to the truth faster than a single test can achieve.

Question 6

What is the difference between frequentist and Bayesian statistics?

Accepted Answer

Frequentist statistics (p-values, confidence intervals) treats parameters as fixed unknowns and data as random. Bayesian statistics (Bayes' theorem, posterior distributions) treats parameters as random variables with probability distributions, updated by data. Bayes allows incorporating prior knowledge, gives direct probability statements about hypotheses, and produces posterior distributions rather than just point estimates.

Term	Symbol	Medical test meaning
Prior probability	P(A)	Disease prevalence in the population
Sensitivity (TPR)	P(B\|A)	P(positive test \| disease present)
False positive rate	P(B\|¬A)	P(positive test \| no disease) = 1 − specificity
Marginal probability	P(B)	Overall P(positive test) across everyone tested
Posterior probability	P(A\|B)	P(disease \| positive test) — the answer we want

LR+	Interpretation	Clinical utility
> 10	Large, clinically important shift	Strong — rules in the condition
5–10	Moderate shift	Good evidence
2–5	Small to moderate shift	Weak evidence — use with other findings
1–2	Minimal shift	Little diagnostic value
= 1	No shift	Positive result is meaningless

Bayes' Theorem Calculator

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Bayes' Theorem Formula

Worked Example: Rare Disease Screening

When to Use Bayes' Theorem

Interpreting Positive Likelihood Ratios

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