Grid Monte Carlo approximation

Definition taught in Hierarchical Bayes, part of Bayesian statistics

Grid Monte Carlo approximation

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

Type	Definition
Taught in	Hierarchical Bayes
Part	Bayesian statistics

Let w_{ab} be the unnormalised hyperposterior value at grid cell (a,b), including the Jacobian \alpha\beta. The discrete approximation is

\widehat p_{ab}=\frac{w_{ab}}{\sum_{r,s}w_{rs}}.

Sampling from the approximate full posterior proceeds as follows.

1. Flatten the grid weights into a one-dimensional list, form cumulative probabilities, draw U\sim\operatorname{Uniform}(0,1), and select the first cell whose cumulative probability exceeds U. Equivalently, draw a grid cell (Z_1,Z_2) with probabilities \widehat p_{ab}. Optionally add independent uniform jitter within the selected cell.

2. Transform (Z_1,Z_2) to (\alpha,\beta) using Equation (10) in Hierarchical Bayes.

3. For each study, draw independently

\theta_j\sim\operatorname{Beta}(y_j+\alpha,n_j-y_j+\beta)

using Equation (7) in Hierarchical Bayes.

4. Repeat to obtain (\alpha^{(l)},\beta^{(l)},{\boldsymbol{\theta}}^{(l)}), l=1,\ldots,L.

These grid-cell draws are independent; this is Monte Carlo from a piecewise-constant approximation, not Markov-chain Monte Carlo. A 100\times100 grid has 10^4 cells and remains practical because only two hyperparameters are integrated numerically. A comparable grid over all J+2 unknowns would be impossible.

Used in

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

See also

Hyperparameter · Hierarchical Bayesian model · Normal hierarchical shrinkage · Exchangeability · Empirical Bayes · Study-rate full conditional · Marginal hyperposterior · Uniformity depends on a domain

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