Beta-Binomial Update

Bayesian beta-binomial conjugate posterior update for estimating probabilities from data with prior beliefs.

Installation

npm install @tpmjs/tools-beta-binomial-update

Usage

import { betaBinomialUpdateTool } from '@tpmjs/tools-beta-binomial-update';

// Example: Estimate conversion rate with prior belief
// Prior: Beta(2, 2) = uniform-ish prior slightly favoring 0.5
// Data: 15 conversions out of 100 trials
const result = await betaBinomialUpdateTool.execute({
  priorAlpha: 2,
  priorBeta: 2,
  successes: 15,
  trials: 100,
  credibleLevel: 0.95, // 95% credible interval
});

console.log(result);
// {
//   posteriorAlpha: 17,      // 2 + 15
//   posteriorBeta: 87,       // 2 + (100 - 15)
//   posteriorMean: 0.163,    // Best estimate
//   posteriorMode: 0.157,    // Most likely value
//   posteriorVariance: 0.001,
//   credibleInterval: {
//     lower: 0.098,
//     upper: 0.239,
//     level: 0.95
//   },
//   statistics: {
//     effectiveSampleSize: 4,
//     priorMean: 0.5,
//     dataLikelihood: 0.15
//   }
// }

API

Input

• priorAlpha (required): Prior successes + 1 (e.g., 1 for uninformative, 2 for weak prior)
• priorBeta (required): Prior failures + 1
• successes (required): Number of successes observed
• trials (required): Total number of trials
• credibleLevel (optional): Credible interval level (default: 0.95)

Output

• posteriorAlpha: Updated alpha parameter
• posteriorBeta: Updated beta parameter
• posteriorMean: Expected value of probability
• posteriorMode: Most likely probability value
• posteriorVariance: Uncertainty in estimate
• credibleInterval: Bayesian confidence interval
• statistics: Prior mean, likelihood, effective sample size

Algorithm

Uses conjugate Beta-Binomial model:

Prior: θ ~ Beta(α, β) Likelihood: X ~ Binomial(n, θ) Posterior: θ|X ~ Beta(α + k, β + (n - k))

Where:

• k = successes
• n = trials
• θ = unknown probability

The Beta distribution is conjugate to the Binomial, making the update simple and exact.

Common Priors

• Uninformative: Beta(1, 1) = Uniform[0, 1]
• Jeffreys: Beta(0.5, 0.5) = Uninformative invariant prior
• Weak: Beta(2, 2) = Slight preference for θ = 0.5
• Strong: Beta(20, 20) = Strong belief in θ = 0.5

Use Cases

• A/B test analysis (conversion rates)
• Click-through rate estimation
• Medical test sensitivity/specificity
• Quality control (defect rates)
• Sports analytics (win probabilities)

Credible Interval

The credible interval is the Bayesian analog of a confidence interval. A 95% credible interval means "there is a 95% probability that θ lies in this interval given the data."

License

MIT