S3U

Math / Statistics / Graduate / mte01

Bayesian Updating and Model-Conditional Uncertainty

Derive a conjugate posterior and examine what its uncertainty leaves out.

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5 questions0m 0s

Prerequisites

Probability densities, Bayes' theorem, Bernoulli sampling, and integration.

Learn the skill

Multiply the prior density by the likelihood and normalize. For a Beta(a,b) prior and s successes in n independent Bernoulli trials with one fixed probability p, the posterior is Beta(a+s,b+n-s). Inference remains conditional on these assumptions.

Worked example

With a Beta(1,1) prior and two successes in three trials, the posterior is Beta(3,2), whose mean is 3/5. This is a distribution for p, not a guarantee about the next five outcomes.

Model and Assumptions

Use an invented sample of eight successes and two failures. Prior p ~ Beta(2,2). Assume independent, identically distributed Bernoulli trials. A Beta(a,b) mean is a/(a+b).

1. What is the posterior distribution?
2. What is the posterior predictive success probability for one next trial?
3. What does a 95% posterior credible interval describe?
4. How should strongly clustered trials affect the analysis?
5. What is the purpose of comparing plausible priors?

Further inquiry

Derive the posterior kernel and predictive probability. Repeat with Beta(1,1) and Beta(20,20). Explain why the resulting spread is not a full uncertainty assessment if trials are dependent.

Review criteria

  • Track success and failure exponents in the likelihood.
  • Calculate predictive means for all three priors.
  • Separate prior sensitivity from misspecification of the likelihood.