Estimate posterior ranges using priors, evidence, and exact beta quantiles. Review uncertainty, interval width, and probability shifts confidently today.
| Scenario | Successes | Trials | Prior | Credibility | Purpose |
|---|---|---|---|---|---|
| Email Conversions | 18 | 30 | Uniform | 95% | Estimate true conversion probability. |
| Machine Pass Rate | 92 | 100 | Jeffreys | 90% | Measure production reliability. |
| Clinical Response | 41 | 60 | Custom Beta(3,2) | 95% | Combine evidence with prior belief. |
| Fraud Detection Flag | 7 | 25 | Skeptical | 99% | Reflect uncertainty with conservative prior. |
This calculator uses a Beta prior with a Binomial likelihood. If the prior is Beta(α, β) and the observed data contain x successes in n trials, the posterior distribution becomes Beta(α + x, β + n − x).
Prior: p ~ Beta(α, β)
Likelihood: X ~ Binomial(n, p)
Posterior: p | X ~ Beta(α + x, β + n − x)
Posterior Mean: (α + x) / (α + β + n)
Equal-Tailed Credible Interval: [ BetaInv(q/2), BetaInv(1 − q/2) ], where q = 1 − credibility level
The lower and upper bounds are posterior quantiles from the inverse regularized incomplete beta function.
A Bayesian credible interval gives a direct probability statement about the parameter after combining prior belief with observed evidence. It is especially helpful for small samples, sparse event counts, and decision problems where prior information matters.
Use it for A/B testing, medical response analysis, manufacturing defect studies, click-through estimates, audit sampling, and reliability tracking.
A Bayesian credible interval is a probability range for an unknown parameter after combining prior belief with observed data. Unlike a frequentist confidence interval, it directly describes the posterior probability of the parameter lying within that range.
It uses a Beta prior and a Binomial likelihood. This combination is conjugate, so the posterior remains Beta. That makes the update mathematically clean and ideal for modeling proportions, rates, and binary outcomes.
Alpha and beta are shape parameters of the Beta prior. Larger values imply stronger prior information. Their ratio influences the prior mean, while their sum controls how concentrated the prior belief is before seeing current data.
Jeffreys prior is often used when you want a weakly informative, symmetry-respecting prior for a binomial proportion. It can behave better than a uniform prior near 0 or 1 and is common in objective Bayesian analysis.
The observed rate uses only current sample data. The posterior mean blends the sample with prior information. When the sample is small, the posterior mean can shrink toward the prior mean more noticeably.
This calculator reports an equal-tailed credible interval. It places equal posterior probability in both tails outside the interval. That makes the method stable, interpretable, and practical for general probability estimation tasks.
Yes. It is well suited for conversion rates, pass rates, defect probabilities, reliability metrics, and any binary outcome. Enter the number of successes and total trials to estimate the posterior range for the true underlying proportion.
More trials provide more information about the underlying probability. As evidence grows, posterior uncertainty decreases, so the credible interval becomes tighter. This reflects greater precision in the estimated parameter value.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.