What does this Bayesian inference calculator do?

This Bayesian inference calculator combines a Beta prior with binomial data (successes and trials) to produce a Beta posterior distribution for the underlying probability parameter. It reports the posterior alpha and beta, posterior mean, posterior mode when defined, and an approximate central credible interval based on the normal approximation to the Beta distribution.

Which assumptions does the binomial Beta model make?

The binomial Beta model assumes independent and identically distributed Bernoulli trials, a fixed but unknown probability of success, and a conjugate Beta prior. Under these assumptions the posterior distribution for the success probability remains in the Beta family, which allows for closed-form updates and transparent interpretation in terms of "prior" and "pseudo counts".

What is the difference between prior, likelihood, and posterior in this tool?

The prior is your initial belief about the success probability before seeing the data, represented as a Beta(alpha, beta) distribution. The likelihood is the binomial model that connects the probability parameter to the observed number of successes in a given number of trials. The posterior is the updated Beta distribution that combines prior and data and is proportional to prior times likelihood.

How is the credible interval computed and how should I interpret it?

The calculator reports an approximate central credible interval by using the normal approximation to the Beta posterior, with bounds equal to the posterior mean plus or minus a z-multiplier times the posterior standard deviation. A 95 percent credible interval means that, under the chosen model and prior, there is 95 percent posterior probability that the true success probability lies within that interval.

Bayesian Inference Calculator (Binomial)

Combine a Beta prior with binomial data (successes and trials) to obtain the posterior Beta distribution, posterior mean, and an approximate credible interval for an unknown probability.

Educational and exploratory use only. For regulatory, clinical, or safety-critical work, always verify calculations with a statistical package and follow your organisation’s validation procedures.

1. Choose a prior (Beta distribution)

The prior describes your beliefs about the success probability \( \theta \) before seeing the data, using a Beta distribution \( \mathrm{Beta}(\alpha,\beta) \).

Prior type

Uniform is non-informative on \( \theta \); Jeffreys is invariant and often used for binomial data.

2. Enter binomial data

Model: \( X \mid \theta \sim \mathrm{Binomial}(n,\theta) \) where \( n \) is the number of trials and \( X \) the number of observed successes.

Number of trials \( n \)

Successes \( x \)

Must satisfy \( 0 \le x \le n \).

Credible level (%)

Used for an approximate central credible interval.

Outputs are rounded for display; internal calculations use full precision.

Bayesian inference for a binomial proportion

In this calculator we model a sequence of independent Bernoulli trials with unknown success probability \( \theta \). If we observe \( x \) successes out of \( n \) trials, the likelihood under a binomial model is

\[ p(x \mid \theta, n) = \binom{n}{x} \theta^x (1 - \theta)^{n-x}, \quad x = 0, 1, \dots, n. \]

We place a Beta prior \( \theta \sim \mathrm{Beta}(\alpha_{\text{prior}}, \beta_{\text{prior}}) \). The Beta distribution is conjugate to the binomial, which means the posterior is also Beta:

\[ \theta \mid x, n \sim \mathrm{Beta}(\alpha_{\text{post}}, \beta_{\text{post}}), \] \[ \alpha_{\text{post}} = \alpha_{\text{prior}} + x,\qquad \beta_{\text{post}} = \beta_{\text{prior}} + n - x. \]

The parameters \( \alpha_{\text{prior}} - 1 \) and \( \beta_{\text{prior}} - 1 \) can be interpreted as pseudo counts of prior successes and failures. After observing the data, the posterior simply adds the new successes and failures to these prior pseudo counts.

Posterior mean, variance, and predictive probability

For a Beta\( (\alpha, \beta) \) distribution, the mean, variance and mode (when defined) are:

\[ \mathbb{E}[\theta] = \frac{\alpha}{\alpha + \beta}, \quad \mathrm{Var}(\theta) = \frac{\alpha \beta}{(\alpha + \beta)^2(\alpha + \beta + 1)}, \] \[ \mathrm{mode}(\theta) = \begin{cases} \dfrac{\alpha - 1}{\alpha + \beta - 2}, & \alpha > 1, \beta > 1, \\ \text{boundary (0 or 1)}, & \text{otherwise.} \end{cases} \]

The posterior predictive probability that the next trial is a success is equal to the posterior mean:

\[ \Pr(\text{next success} \mid x, n) = \mathbb{E}[\theta \mid x, n] = \frac{\alpha_{\text{post}}}{\alpha_{\text{post}} + \beta_{\text{post}}}. \]

Approximate credible interval (normal approximation)

Exact credible intervals for a Beta posterior use Beta quantiles, which require evaluation of the incomplete Beta function. For a quick and transparent calculation in pure JavaScript, this tool uses the normal approximation:

Let \( m = \mathbb{E}[\theta] \) and \( s^2 = \mathrm{Var}(\theta) \) for the posterior \( \mathrm{Beta}(\alpha_{\text{post}}, \beta_{\text{post}}) \).

For a central credible level \( 100(1 - \gamma)\% \) with corresponding standard normal quantile \( z_{1 - \gamma/2} \),

\[ \text{CI}_{1-\gamma} \approx \Big[\, m - z_{1-\gamma/2} s,\; m + z_{1-\gamma/2} s \,\Big]. \]

This approximation is usually reasonable when the posterior is not extremely skewed, for example when both \( \alpha_{\text{post}} \) and \( \beta_{\text{post}} \) are greater than about 3–5. For very small sample sizes or strongly asymmetric posteriors, a full Bayesian analysis with exact Beta quantiles is recommended.

Worked example

Suppose you run \( n = 50 \) trials and observe \( x = 18 \) successes. You start with a uniform prior \( \mathrm{Beta}(1, 1) \), which corresponds to no prior preference for any particular value of \( \theta \) in \( [0, 1] \).

Posterior parameters: \( \alpha_{\text{post}} = 1 + 18 = 19 \), \( \beta_{\text{post}} = 1 + 50 - 18 = 33 \).
Posterior mean: \( 19 / (19 + 33) \approx 0.365 \).
Posterior variance: \( \dfrac{19 \cdot 33}{(52)^2 \cdot 53} \), giving a standard deviation of about 0.065.
Approximate 95% credible interval: mean ± 1.96 × SD ≈ [0.24, 0.49].
Posterior predictive \( \Pr(\text{next success}) \) is also ≈ 0.365.

Enter these values into the calculator to verify the posterior and explore how different priors change the conclusions.

Modeling choices and cautions

Independence and stationarity: The binomial-Beta model assumes that each trial has the same success probability and is independent of previous trials. Time trends or dependence may require more advanced models.
Prior sensitivity: With limited data, different reasonable priors can lead to noticeably different posteriors. Always document how you chose the prior and, where possible, perform sensitivity analysis.
Interpretation: A Bayesian credible interval answers “given the model and prior, what range of parameter values has high posterior probability?”, which is conceptually different from a frequentist confidence interval.
High-stakes applications: For clinical trials, industrial quality control, or regulatory reporting, use vetted statistical libraries or software (for example R, Python, or specialised Bayesian packages) and follow good practice guidelines.

Frequently asked questions

How should I choose between uniform, Jeffreys, and a custom prior?

A uniform prior Beta(1,1) is often used when you want a simple starting point and have little prior information. Jeffreys prior Beta(0.5,0.5) is invariant under reparameterisation and is commonly used in objective Bayesian analyses. A custom Beta prior is appropriate when you can justify prior information based on previous experiments, expert knowledge, or domain constraints.

Can I interpret Beta parameters as pseudo counts?

Yes. For a Beta\( (\alpha, \beta) \) prior you can think of \( \alpha - 1 \) as prior successes and \( \beta - 1 \) as prior failures. After observing \( x \) successes in \( n \) trials, the posterior pseudo counts are \( \alpha - 1 + x \) and \( \beta - 1 + n - x \). This makes it easy to explain the prior and posterior in operational terms.

Why do my Bayesian results differ from a frequentist confidence interval?

Confidence intervals control the long-run frequency of coverage across hypothetical repeated samples, while credible intervals directly quantify posterior uncertainty about the parameter given the observed data and the prior. For moderate to large samples with weak priors the two can be very similar, but in general they answer different questions and need not match.

Is this calculator enough for publication-grade analysis?

It is designed as a teaching and decision-support tool rather than a full statistical environment. It is well suited for quick checks, intuition building, and classroom examples. For complex designs, multiple parameters, or publication-grade work you should rely on specialised Bayesian software, reproducible code, and peer review.