Bayesian Prior Update Calculator

Work out bayesian prior update instantly with clear inputs, formula shown and shareable results.

About Bayesian Prior Update Calculator

Bayes' theorem combines a prior with test performance: posterior = sensitivity x prior divided by the total probability of a positive result. The counter-intuitive consequence is that with a 1 percent prior, a 95 percent sensitive and 90 percent specific test still yields a posterior under 10 percent, because false positives from the large negative population swamp the true positives.

Formula

Bayes' theorem

P(D|+) = sens x prior / (sens x prior + (1 - spec) x (1 - prior)); LR+ = sens / (1 - spec)

Worked example

With the default values below, Bayesian Prior Update Calculator returns Posterior probability after a positive result of 8.756%. Change any input above and every figure here is recalculated in your browser.

Inputs used

Prior probabilityaccepts 0–99.99 %: 1 %
Test sensitivity (true positive rate)accepts 0.01–100 %: 95 %
Test specificity (true negative rate)accepts 0.01–100 %: 90 %

Results

Posterior probability after a positive result: 8.756%
Positive likelihood ratio: 9.500
Probability of a positive result: 0.10850
Posterior odds: 0.0960

Step by step

1Values used
2Bayes' theorem
3Posterior probability after a positive result= 8.756
4Positive likelihood ratio= 9.500
5Probability of a positive result= 0.10850
6Posterior odds= 0.0960

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 5 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Why is the posterior so low with a good test?

Base rates dominate. At a 1 percent prior, out of 10,000 cases there are 95 true positives but around 990 false positives, so most positive results are wrong.

How is this used outside medicine?

Identically for alerting and fraud. A rare-event detector with high accuracy still produces mostly false alarms, which is why precision at the operating point matters more than accuracy.