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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