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About this tool
Calculate the posterior probability from a prior probability and new evidence using Bayes' theorem.
The Bayesian Update Calculator applies Bayes' theorem to turn a starting belief and one piece of evidence into a revised probability, using posterior = P(E|H)·P(H) divided by P(E|H)·P(H) + P(E|¬H)·P(1−H). Enter three percentages — your prior, how likely the evidence is if the hypothesis is true, and how likely it is if the hypothesis is false — and it returns the posterior along with prior odds, the likelihood ratio and posterior odds. It is for anyone who has to reason about a positive test, a warning signal or a noisy indicator and wants the base rate accounted for properly.
Open Bayesian Update Calculator on AltFTool — it loads instantly in your browser.
Type Prior probability (%), P(evidence | hypothesis) (%) and P(evidence | not hypothesis) (%), or press the "10% prior, strong evidence" example.
Change any of the three percentages — each is clamped to the 0 to 100 range — and the posterior recomputes with no submit step; Reset in the Inputs header restores 10, 90 and 5.
The Result panel gives the posterior percentage with rows for Prior, Evidence probability, Prior odds, Likelihood ratio and Posterior odds; Copy or Download saves it as bayesian-update-calculator.txt.
Prior odds, likelihood ratio and posterior odds are shown together, which makes chaining a second piece of evidence straightforward.
Changing only the prior while holding the test quality fixed shows immediately how much the starting rate drives the answer.
The total evidence probability P(E) is reported as its own row, so you can see how many of all positives come from the false-positive branch.
The prior probability that the hypothesis is true, P(evidence | hypothesis) — the true positive rate or sensitivity — and P(evidence | not hypothesis), the false positive rate. All three are entered as percentages between 0 and 100.
Because the false positives are drawn from a much larger group. With a 1 percent prior, 90 percent sensitivity and a 5 percent false positive rate, the posterior is only about 15.4 percent: 0.9 percent of the population are true positives while 4.95 percent are false positives.
It is P(evidence | hypothesis) divided by P(evidence | not hypothesis) — how many times more often the evidence appears when the hypothesis is true. Multiply your prior odds by it to get the posterior odds. A ratio of 18, from 90 percent over 5 percent, means the evidence is 18 times more expected under the hypothesis.
Yes, one step at a time: run the calculation, then feed the posterior back in as the prior for the next piece of evidence. This is only valid when the two pieces of evidence are conditionally independent — two tests that fail in the same way will overstate the result.
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