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Module 4: Conjugate Models

Purpose

Conjugate models give closed-form Bayesian updates. They are the best way to understand posterior learning before using MCMC.

Learning Objectives

After this module, learners should be able to:

  • Explain what conjugacy means.
  • Update beta-binomial, gamma-Poisson, and normal-normal models.
  • Interpret prior strength as pseudo-data.
  • Use conjugate models for quick uncertainty analysis.

What Is Conjugacy?

A prior is conjugate to a likelihood when the posterior belongs to the same distribution family as the prior.

This makes updating simple:

prior parameters + data summaries = posterior parameters

Beta-Binomial Model

Use this model for a probability.

theta ~ Beta(alpha, beta)
y ~ Binomial(n, theta)
theta | y ~ Beta(alpha + y, beta + n - y)

Interpretation:

  • alpha - 1 can be thought of as prior successes.
  • beta - 1 can be thought of as prior failures.

Example:

Prior: Beta(2, 2)
Data: 30 successes out of 40
Posterior: Beta(32, 12)

Gamma-Poisson Model

Use this model for count rates.

lambda ~ Gamma(alpha, beta)
y_i ~ Poisson(lambda)
lambda | y ~ Gamma(alpha + sum(y), beta + n)

Here beta is a rate parameter.

Example:

Prior crash rate: Gamma(3, 1)
Monthly counts: 2, 4, 3, 5, 4
Posterior: Gamma(21, 6)

Normal-Normal Model

Use this model for a mean when observation variance is known or treated as known.

mu ~ Normal(mu0, tau0^2)
y_i ~ Normal(mu, sigma^2)

Posterior precision is:

1 / tau_n^2 = 1 / tau0^2 + n / sigma^2

Posterior mean is a precision-weighted average of the prior mean and sample mean:

mu_n = tau_n^2 * (mu0 / tau0^2 + n * ybar / sigma^2)

Why Conjugate Models Matter

Conjugate models teach:

  • How priors behave like information.
  • How larger data reduce prior influence.
  • How posterior uncertainty changes with sample size.
  • Why Bayesian inference is a full distribution, not one estimate.

Practice

  1. Start with Beta(1, 1) and observe 14 successes in 20 trials. Find the posterior.
  2. Start with Beta(10, 10) and observe the same data. Compare the posterior.
  3. For crash counts 3, 4, 1, 2, start with Gamma(2, 1). Find the posterior.
  4. Explain why conjugate models may be insufficient for complex research questions.

Lab

Complete Lab 2: Normal-normal updating.

Assignment

Start Assignment 2: Conjugate Modeling.

Next Module

Continue to Module 5: Monte Carlo Computation.