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Home/Math Visualization/Bayesian Updating / Conjugate Priors

Bayesian Updating / Conjugate Priors

Beta-binomial Bayesian updating with prior/posterior curves, posterior predictive probability, and credible intervals.

Beta prior

2
2

Binomial data

30
18
1.2

Measured values

Posterior mean0.588
95% credible interval0.421…0.745
Predictive P(success)0.588

Conjugate priors keep the posterior in the same distribution family, making Bayesian updating visible as parameter addition.

Live graphs

About this model

Conjugate priors make Bayesian updating algebraic. In the beta-binomial model, observing k successes in n trials changes Beta(alpha,beta) into Beta(alpha+k,beta+n-k). The simulator plots prior and posterior distributions, the posterior mean, posterior predictive probability, and a 95% credible interval.

Who it's for: Statistics, Bayesian inference, data science, experimental design, epidemiology, and decision science courses.

Key terms

  • Bayesian updating
  • Conjugate prior
  • Beta-binomial model
  • Posterior
  • Credible interval
  • Posterior predictive

How it works

Bayesian updating with beta-binomial conjugate priors, posterior mean, credible intervals, and predictive probability.

Key equations

Beta(α,β) + k successes in n trials → Beta(α+k, β+n−k)
Posterior predictive P(success next) = (α+k)/(α+β+n)

Frequently asked questions

How is a credible interval different from a confidence interval?
A credible interval is a posterior probability statement about the parameter after combining prior and data. A frequentist confidence interval has a long-run coverage interpretation over repeated samples.
Why use conjugate priors?
They keep the posterior in the same distribution family, making updates fast, transparent, and easy to visualize.