Beta-Binomial Conjugate Prior Bayesian Posterior & 95% Credible Interval Calculator

Update Beta(α₀, β₀) prior with binomial successes/failures for posterior mean, MAP, and 95% CI.

Bayesian Posterior Mean Beta(38, 16)
70.37 %
Maximum A Posteriori (MAP) Mode 71.154 %
Posterior Standard Deviation 6.157 %
95% Credible Half-Width (±1.96 SD) 12.068 ±%
Inputs
⚡ Scenarios:
α₀
Prior Beta shape parameter α₀ (1 = flat uniform prior; 0.5 = Jeffreys prior).
β₀
Prior Beta shape parameter β₀.
successes
Observed binomial successes (s).
failures
Observed binomial failures (f = n − s).
Results & Graph
Bayesian Posterior Mean Beta(38, 16)
70.37 %
Maximum A Posteriori (MAP) Mode
71.154 %
Posterior Standard Deviation
6.157 %
95% Credible Half-Width (±1.96 SD)
12.068 ±%

Bayesian Posterior Mean Beta(38, 16)

Graph 1 • Beta-Binomial Bayesian Update • Domain Architecture & Live Telemetry Maps Prior Alpha0, Prior Beta0, Observed Successes S through Beta-Binomial Bayesian Update Formula.
Bayesian Posterior Mean Beta(3: 70.37 % Beta-Binomial Bayesian Update • Invariants & Metrics Bayesian Posterior Mean Beta 70.37 % Maximum A Posteriori (MAP) M 71.154 % Posterior Standard Deviation 6.157 % 95% Credible Half-Width (±1. 12.068 ±%
Graph 2 • Bayesian Posterior Mean Beta(38, 16) Across Prior Alpha0 Sweep (40% → 160%) Evaluates α_post = α₀ + s; β_post = β₀ + f; PosteriorMean = α_post / (α_post + across 7 operating points (current input highlighted).
0 % 39.06 % 78.12 % 69.70 0.800α₀ 69.92 1.20α₀ 70.15 1.60α₀ 70.37 2α₀ 70.59 2.40α₀ 70.80 2.80α₀ 71.01 3.20α₀
Graph 3 • Joint Sensitivity: Prior Alpha0 × Prior Beta0 Compares Bayesian Posterior Mean Beta(38, 16) when Prior Beta0 is shifted +25% (solid) vs. −25% (dashed).
0 % 39.24 % 78.48 % --- Prior Beta0 @ 75% 69.39 -30% 69.56 -15% 69.72 +0% 69.89 +15% 70.05 +30%
Summary: Updating Beta(2, 2) with 36 successes and 14 failures yields Beta(38, 16) with posterior mean 70.37%, MAP mode 71.15%, and 95% credible band 58.3%–82.44%.
Summary: Updating Beta(2, 2) with 36 successes and 14 failures yields Beta(38, 16) with posterior mean 70.37%, MAP mode 71.15%, and 95% credible band 58.3%–82.44%.
⚡ Send result to: Scientific →Statistics & SD →Probability & P-Value →
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Formula & Mathematical Methodology: Beta-Binomial Bayesian Update Formula

αpost = α₀ + s
βpost = β₀ + f
PosteriorMean = αpost / (αpost + βpost)
MAP = αpost − 1αpost + βpost − 2
α_post = α₀ + s; β_post = β₀ + f; PosteriorMean = α_post / (α_post + β_post); MAP = (α_post − 1) / (α_post + β_post − 2)
  • priorAlpha0: Prior Alpha0 (α₀)
  • priorBeta0: Prior Beta0 (β₀)
  • observedSuccessesS: Observed Successes S (successes)
  • observedFailuresF: Observed Failures F (failures)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: Beta-Binomial Bayesian Update

Update Beta(α₀, β₀) prior with binomial successes/failures for posterior mean, MAP, and 95% CI.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: α_post = α₀ + s; β_post = β₀ + f; PosteriorMean = α_post / (α_post + β_post); MAP = (α_post − 1) / (α_post + β_post − 2).
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Beta-Binomial Bayesian Update via α_post = α₀ + s; β_post = β₀ + f; PosteriorMean = α_post / (α_post + β_post); MAP = (α_post − 1) / (α_post + β_post − 2).

Frequently Asked Questions

How does the Beta-Binomial Conjugate Prior Bayesian Posterior & 95% Credible Interval Calculator compute the solution?

It evaluates α_post = α₀ + s; β_post = β₀ + f; PosteriorMean = α_post / (α_post + β_post); MAP = (α_post − 1) / (α_post + β_post − 2) directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Probability Distributions & Survival Analysis

Beta-Binomial Conjugate Prior Bayesian Posterior & 95% Credible Interval Calculator: inputs, method, and how to read the result

Update Beta(α₀, β₀) prior with binomial successes/failures for posterior mean, MAP, and 95% CI.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Prior Alpha0: Prior Beta shape parameter α₀ (1 = flat uniform prior; 0.5 = Jeffreys prior).
  • Prior Beta0: Prior Beta shape parameter β₀.
  • Observed Successes S: Observed binomial successes (s).
  • Observed Failures F: Observed binomial failures (f = n − s).

What the calculation includes

With the default values shown, Bayesian Posterior Mean Beta(38, 16) is 70.37 %. The calculation follows Beta-Binomial Bayesian Update Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Prior Alpha0 from 2 α₀ to 3 α₀ changes Bayesian Posterior Mean Beta(38, 16) (70.37 %) to Bayesian Posterior Mean Beta(39, 16) (70.909 %). This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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