poisson.zellner.prior {BoomSpikeSlab} R Documentation

## Zellner Prior for Poisson Regression

### Description

A Zellner-style spike and slab prior for Poisson regression models. See 'Details' for a definition.

### Usage

```PoissonZellnerPrior(
predictors,
counts = NULL,
exposure = NULL,
prior.event.rate = NULL,
expected.model.size = 1,
prior.information.weight = .01,
diagonal.shrinkage = .5,
optional.coefficient.estimate = NULL,
max.flips = -1,
prior.inclusion.probabilities = NULL)
```

### Arguments

 `predictors` The design matrix for the regression problem. No missing data is allowed. `counts` The vector of responses, This is only used to obtain the empirical overall event rate, so it can be left `NULL` if prior.event.rate is specified. `exposure` A vector of the same length as `counts`, giving the "exposure time" for each observation. This can also be `NULL`, signifying that `exposure = 1.0` for each observation. `prior.event.rate` An a priori guess at the overall event rate. Used in two places: to set the prior mean of the intercept (if `optional.coefficient.estimate` is `NULL`) and to weight the information matrix in the "slab" portion of the prior. `expected.model.size` A positive number less than `ncol(x)`, representing a guess at the number of significant predictor variables. Used to obtain the 'spike' portion of the spike and slab prior. `prior.information.weight` A positive scalar. Number of observations worth of weight that should be given to the prior estimate of beta. `diagonal.shrinkage` The conditionally Gaussian prior for beta (the "slab") starts with a precision matrix equal to the information in a single observation. However, this matrix might not be full rank. The matrix can be made full rank by averaging with its diagonal. `diagonal.shrinkage` is the weight given to the diaonal in this average. Setting this to zero gives Zellner's g-prior. `optional.coefficient.estimate` If desired, an estimate of the regression coefficients can be supplied. In most cases this will be a difficult parameter to specify. If omitted then a prior mean of zero will be used for all coordinates except the intercept, which will be set to mean(y). `max.flips` The maximum number of variable inclusion indicators the sampler will attempt to sample each iteration. If negative then all indicators will be sampled. `prior.inclusion.probabilities` A vector giving the prior probability of inclusion for each variable. If `NULL` then a default set of probabilities is obtained by setting each element equal to `min(1, expected.model.size / ncol(x))`.

### Details

A Zellner-style spike and slab prior for Poisson regression. Denote the vector of coefficients by beta, and the vector of inclusion indicators by gamma. These are linked by the relationship beta[i] != 0 if gamma[i] = 1 and beta[i] == 0 if gamma[i] = 0. The prior is

beta | gamma ~ N(b, V),

gamma ~ Bernoulli(pi)

where pi is the vector of `prior.inclusion.probabilities`, and b is the `optional.coefficient.estimate`. Conditional on gamma, the prior information matrix is

V^{-1} = kappa * ((1 - alpha) * x^Twx / n + alpha * diag(x^Twx/n))

The matrix x^Twx is, for suitable choice of the weight vector w, the total Fisher information available in the data. Dividing by n gives the average Fisher information in a single observation, multiplying by kappa then results in kappa units of "average" information. This matrix is averaged with its diagonal to ensure positive definiteness.

In the formula above, kappa is `prior.information.weight`, alpha is `diagonal.shrinkage`, and w is a diagonal matrix with all elements set to ```prior.success.probability * (1 - prior.success.probability)```. The vector b and the matrix V^{-1} are both implicitly subscripted by gamma, meaning that elements, rows, or columsn corresponding to gamma = 0 should be omitted.

### Value

Returns an object of class `PoissonZellnerPrior`, which is a list with data elements encoding the selected prior values. It inherits from `PoissonPrior` and from `SpikeSlabGlmPrior`, which implies that it contains an element `prior.success.probability`.

This object is intended for use with `poisson.spike`.

Steven L. Scott

### References

Hugh Chipman, Edward I. George, Robert E. McCulloch, M. Clyde, Dean P. Foster, Robert A. Stine (2001), "The Practical Implementation of Bayesian Model Selection" Lecture Notes-Monograph Series, Vol. 38, pp. 65-134. Institute of Mathematical Statistics.

[Package BoomSpikeSlab version 1.2.4 Index]