poisson.zellner.prior {BoomSpikeSlab}  R Documentation 
Zellner Prior for Poisson Regression
Description
A Zellnerstyle 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 
exposure 
A vector of the same length as 
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

expected.model.size 
A positive number less than 
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. 
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 
Details
A Zellnerstyle 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 \ne 0
if \gamma_i =
1
and \beta_i = 0
if \gamma_i =
0
. The prior is
\beta  \gamma \sim N(b, V)
\gamma \sim B(\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
.
Author(s)
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 NotesMonograph Series, Vol. 38, pp. 65134. Institute of Mathematical Statistics.