A Zellner-style spike and slab prior for Poisson regression models. See 'Details' for a definition.
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)
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.
The design matrix for the regression problem. No missing data is allowed.
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.
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.
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.
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.
A positive scalar. Number of observations worth of weight that should be given to the prior estimate of beta.
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.
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).
The maximum number of variable inclusion indicators the sampler will attempt to sample each iteration. If negative then all indicators will be sampled.
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)).
Steven L. Scott
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 \(\beta_i \ne 0\) 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.
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.