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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.

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

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

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.