Bayesian Posterior Calculator
Free Bayesian posterior Calculator for ab testing. Enter values to get step-by-step solutions with formulas and graphs.
Formula
Posterior = Beta(alpha + successes, beta + failures) | P(A|B) = P(B|A)P(A) / P(B)
In the Beta-Binomial model, the posterior distribution is analytically derived by adding observed successes to the prior alpha and failures to the prior beta. For simple Bayes' theorem, the posterior probability is the product of the likelihood and prior divided by the total evidence probability.
Worked Examples
Example 1: A/B Test Conversion Rate
Problem: Prior: Beta(2, 18) (10% baseline conversion). Observed: 45 conversions out of 500 visitors. Calculate the posterior.
Solution: Prior: Beta(alpha=2, beta=18), mean = 2/20 = 10%\nData: s=45 successes, f=455 failures\nPosterior: Beta(2+45, 18+455) = Beta(47, 473)\nPosterior mean = 47/520 = 9.04%\nObserved rate = 45/500 = 9.00%\nPrior mean was 10%, posterior shrunk toward data\n95% CI: approximately [6.7%, 11.7%]
Result: Posterior Beta(47, 473): mean = 9.04%, 95% CI [6.7%, 11.7%]
Example 2: Medical Diagnostic Test
Problem: A disease has 1% prevalence. A test has 95% sensitivity and 5% false positive rate. Patient tests positive. What is the probability of disease?
Solution: P(Disease) = 0.01 (prior)\nP(Positive|Disease) = 0.95 (sensitivity)\nP(Positive|No Disease) = 0.05 (false positive)\nP(Positive) = 0.95 x 0.01 + 0.05 x 0.99 = 0.0095 + 0.0495 = 0.059\nP(Disease|Positive) = (0.95 x 0.01) / 0.059 = 16.1%
Result: Posterior probability of disease = 16.1% (not 95%!)
Frequently Asked Questions
What is Bayesian posterior probability and why is it important?
Bayesian posterior probability represents the updated probability of a hypothesis after observing new evidence, calculated using Bayes' theorem. It combines prior knowledge (what we believed before seeing data) with the likelihood of the observed data to produce a refined estimate. The formula is: P(A|B) = P(B|A) x P(A) / P(B). This framework is foundational in modern statistics, machine learning, medical diagnostics, and A/B testing because it provides a principled way to update beliefs incrementally as new data arrives. Unlike frequentist methods that treat parameters as fixed, Bayesian analysis treats them as probability distributions, enabling richer uncertainty quantification and more intuitive interpretation of results.
What is Bayesian shrinkage and why does it occur?
Bayesian shrinkage is the phenomenon where the posterior estimate is pulled ('shrunk') from the observed data toward the prior mean. This occurs because the Bayesian posterior is a weighted average of the prior belief and the observed evidence. With small sample sizes, the prior dominates and shrinkage is strong — if your prior expects a 10% rate and you observe 3/10 (30%), the posterior might estimate 15% rather than 30%. As sample size increases, the data overwhelms the prior and shrinkage diminishes. Shrinkage is actually beneficial in many scenarios: it reduces overfitting, stabilizes estimates for small samples, and provides natural regularization. In multi-arm A/B tests, Bayesian hierarchical models use shrinkage to 'borrow strength' across variants, producing more reliable estimates for low-traffic segments.
How accurate are the results from Bayesian Posterior Calculator?
All calculations use established mathematical formulas and are performed with high-precision arithmetic. Results are accurate to the precision shown. For critical decisions in finance, medicine, or engineering, always verify results with a qualified professional.
Is Bayesian Posterior Calculator free to use?
Yes, completely free with no sign-up required. All calculators on NovaCalculator are free to use without registration, subscription, or payment.
How do I get the most accurate result?
Enter values as precisely as possible using the correct units for each field. Check that you have selected the right unit (e.g. kilograms vs pounds, meters vs feet) before calculating. Rounding inputs early can reduce output precision.
Can I use the results for professional or academic purposes?
You may use the results for reference and educational purposes. For professional reports, academic papers, or critical decisions, we recommend verifying outputs against peer-reviewed sources or consulting a qualified expert in the relevant field.