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Last lecture

Last lecture. Beta-binomial model All types of posterior inference one can make: mean, mode, variance, two types of credible intervals,etc. Prior predictive, posterior predictive Relationship between prior mean and posterior mean, prior variance & post var. Conjugate priors.

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Last lecture

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  1. Last lecture • Beta-binomial model • All types of posterior inference one can make: mean, mode, variance, two types of credible intervals,etc. • Prior predictive, posterior predictive • Relationship between prior mean and posterior mean, prior variance & post var. • Conjugate priors

  2. Sequential Analysis with conjugate prior beta(α,β) data: y1,n1 beta(α+y1,β+n1-y1) data: y2,n2 beta(α+y1+y2, β+n1+n2-y1-y2) ……

  3. Ex. Multinomial Distribution • Example: 1988 CBS pre-election poll: 727 support Republican Candidate, 583 Dem, 137 other Parameters of interest: winning chance for each party. • likelihood=? • What is a conjugate prior for this likelihood?

  4. How to express prior ignorance? • So that the prior plays a minimal role in posterior inference • uniform distribution  likelihood only (MLE) • Jeffrey’s prior

  5. Jeffrey’s rule for noninformative priors • They should be invariant to parameter transformations • For example: • uniform distribution for a proportion: θ • uniform distribution for an odds: θ/(1-θ) • uniform distribution for log odds: log θ/(1-θ)

  6. Jeffrey’s choice of prior • square root of the Fisher information

  7. Jeffrey’s prior for binomial model

  8. Proper vs. Improper • Proper=integrable For example: Jeffrey’s prior is proper beta(1/2, ½) \propto θ-1/2(1-θ)-1/2 θ-1(1-θ)-1 is improper Problem of improper prior: posterior can be improper for certain data. Inference can be problematic…

  9. Reference prior • A type of prior that influences posterior the least • An automated way of constructing prior doesn’t seem to exist.

  10. Normal model

  11. Variance known • y~N(θ, σ2), y=(y1,…,yn) σ2 known • Parameter of interest: mean θ =? • likelihood: • What is a conjugate prior for θ?

  12. Informative Prior for θ

  13. Noninformative Prior for θ

  14. Variance Unknown Mean Known • y~N(θ, σ2), y=(y1,…,yn) θ known • Parameter of interest: σ2 =? • likelihood: • What is a conjugate prior for θ?

  15. Informative Prior for σ2

  16. Noninformative prior for σ2

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