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Bayesian Hierarchical Model Ying Nian Wu UCLA Department of Statistics IPAM Summer School July 12, 2007. Plan Bayesian inference Learning the prior Examples Josh’s example. Inference of normal mean. independently. unknown parameter. given constant. Example:. one’s height.
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Bayesian Hierarchical Model Ying Nian Wu UCLA Department of Statistics IPAM Summer School July 12, 2007
Plan • Bayesian inference • Learning the prior • Examples • Josh’s example
Inference of normal mean independently unknown parameter given constant Example: one’s height repeated measurements known precision
Prior distribution known hyper-parameters The larger , the more uncertain about , prior becomes non-informative
Bayesian inference Prior: Data: independently Posterior: Compromise between prior and data
Bayesian inference Prior: Data: Posterior:
Illustration Prior: Data:
Inference of normal mean Prior: Data: independently Sufficient statistic:
Combining prior and data large small
Combining prior and data small large
Learning the prior Prior: Data: independently Prior distribution cannot be learned from single realization of
Learning the prior Prior: Data: Prior distribution can be learned from multiple experiences
Hierarchical model Prior: Data: …… ……
Hierarchical model …… ……
Collapsing projecting
Prior: Data: Sufficient statistics
Collapsing Integrating out
Empirical Bayes Borrowing strength from other observations
Full Bayesian e.g., constant Hyper prior: …… ……
Stein’s estimator Example: measure each person’s height
Stein’s estimator Empirical Bayes interpretation
Beta-Binomial example Data: e.g., flip a coin, is probability of head flips is number of heads out of Pre-election poll
Data: Prior: Posterior:
Hierarchical model Examples: a number of coins probs of head a number of MLB players probs of hit pre-election poll in different states
Dirichlet-Multinomial Roll a die:
Data Prior Posterior
