Marginal hyperposterior

Proposition taught in Hierarchical Bayes, part of Bayesian statistics

Marginal hyperposterior

Contours of the rat-tumour hyperposterior in the transformed (z_1,z_2) coordinates, including the Jacobian \alpha\beta. From Hierarchical Bayes.

Type	Proposition
Taught in	Hierarchical Bayes
Part	Bayesian statistics

The binomial coefficient does not depend on (\alpha,\beta), so the hyperposterior kernel is \boxed{ p(\alpha,\beta\mid{\boldsymbol{y}}) \propto p(\alpha,\beta) \prod_{j=1}^J \frac{\Gamma(\alpha+\beta) \Gamma(y_j+\alpha) \Gamma(n_j-y_j+\beta)} {\Gamma(\alpha)\Gamma(\beta) \Gamma(n_j+\alpha+\beta)}}. This expression is available pointwise but is not a familiar normalised density.

Used in

• Taught in Hierarchical Bayes, where the derivation and examples are.
• All 9 questions on Hierarchical Bayes

See also

Hierarchical Bayesian model · Hyperparameter · Normal hierarchical shrinkage · Exchangeability · Empirical Bayes · Study-rate full conditional · Uniformity depends on a domain · Grid Monte Carlo approximation

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