Bayesian Inference Calculator (Binomial)
Combine a Beta prior with binomial data (successes and trials) to compute the posterior Beta distribution, posterior mean, and an approximate credible interval for an unknown probability.
1. Choose a prior (Beta distribution)
Select a conjugate prior for the unknown success probability. Custom values are interpreted as pseudo counts.
Uniform ≈ no prior information; Jeffreys is invariant under reparameterizations.
2. Enter binomial data
Model: X ∼ Binomial(n, θ), where n is the number of trials and x the successes.
0 ≤ x ≤ n
Central credible interval for the posterior.
How to Use This Calculator
- Choose a prior type or enter a custom Beta(α, β) that encodes your existing belief about the success probability.
- Provide the number of trials and observed successes inside the binomial model; the tool enforces 0 ≤ x ≤ n.
- Set the credible level for the central interval you want to inspect (default 95%).
- Click Calculate posterior or edit the inputs; the results card shows posterior summary, mode, and credible interval instantly.
Methodology
The calculator implements the conjugate Beta–binomial model. The Beta prior summarizes prior successes and failures, the binomial likelihood updates those counts with the observed data, and the posterior stays within the Beta family for transparent interpretation.
Posterior predictions and credible intervals rely on closed-form expectations and the normal approximation described in the Formulas box below. All intermediate values use full numeric precision before rounding for display.
Bayesian inference for a binomial proportion
We treat each trial as independent with the same unknown success probability θ. Every update adds the observed successes and failures to the prior pseudo counts. The posterior parameters α_post = α_prior + x and β_post = β_prior + n − x are used for subsequent summaries.
Posterior summaries and predictions
The posterior mean, standard deviation, and (when defined) mode are computed from Beta distribution moments. The posterior predictive probability that the next trial succeeds equals the posterior mean because the Beta–binomial predictive expectation matches the posterior expectation.
Credible intervals use the normal approximation: the selected z multiplier is applied to the posterior standard deviation and then clamped to [0, 1] for display. For highly skewed posteriors the approximation is less reliable; in those cases consult exact Beta quantiles.
Worked example
- Use n = 50 trials and x = 18 successes with the default uniform prior Beta(1, 1).
- The posterior parameters become α_post = 19 and β_post = 33.
- The posterior mean is 19 / 52 ≈ 0.365, with a standard deviation around 0.065.
- Approximate 95% credible interval: mean ± 1.96 × SD ≈ [0.24, 0.49].
- The posterior predictive probability that the next trial is a success is also ≈ 0.365.
Enter these values to verify the results and experiment with other priors.
Modeling choices and cautions
- Independence: The binomial-Beta model assumes identical, independent trials with a common θ.
- Prior sensitivity: Different reasonable priors can shift the posterior when data are sparse. Document your choice.
- Interpretation: Credible intervals quantify posterior probability given the model and prior, unlike frequentist confidence intervals.
- High-stakes work: For rigorous studies, validate with trusted statistical software and peer review.
Frequently asked questions
How should I choose between uniform, Jeffreys, and a custom prior?
Uniform Beta(1,1) is a simple default; Jeffreys Beta(0.5,0.5) is invariant under reparameterization and commonly used for objective analyses. Choose a custom prior only when you can justify it with previous data or expert judgment.
Can I interpret Beta parameters as pseudo counts?
Yes. α − 1 acts like prior successes and β − 1 like prior failures. After seeing x successes, the posterior pseudo counts are α − 1 + x and β − 1 + n − x, which makes the updates easy to explain.
Why do my Bayesian results differ from a frequentist confidence interval?
Confidence intervals describe long-run coverage probability, while credible intervals quantify posterior probability for the single observed dataset and chosen prior. They can be similar for large n but are conceptually different.
Is this calculator enough for publication-grade analysis?
It is designed for teaching, intuition, and quick checks. For publication-grade reports, rely on specialised Bayesian packages, close reproducibility, and peer review.
Full original guide (expanded)
More probability & Bayesian tools
About this Bayesian inference tool
This calculator implements the standard conjugate Beta–binomial model taught in undergraduate and graduate Bayesian statistics. It highlights how priors and data combine into posterior summaries that can be cross-checked against textbooks.
Always document your prior choice, model assumptions, and data cleaning. For sensitive decisions, validate with an independent implementation and consult a statistician.