msPriorSpec-class {mombf}

Class "msPriorSpec"

Description

Stores the prior distributions to be used for Bayesian variable selection in normal regression models. This class can be used to specify the prior on non-zero regression coefficients, the model indicator or the nuisance parameters.

Usage

aic()
bic()
bicprior()
ic(penalty)

momprior(taustd=1, tau, tau.adj=10^6, r=1)
imomprior(tau, tau.adj=10^6)
emomprior(tau, tau.adj=10^6)
zellnerprior(taustd=1, tau, tau.adj=10^6)
normalidprior(taustd=1, tau, tau.adj=10^6)

exponentialprior(lambda = 1)

groupmomprior(taustd=1, tau, tau.adj=10^6)
groupimomprior(tau, tau.adj=10^6)
groupemomprior(tau, tau.adj=10^6)
groupzellnerprior(taustd=1, tau, tau.adj=10^6)

modelunifprior()
modelbinomprior(p=0.5)
modelbbprior(alpha.p=1, beta.p=1)
modelcomplexprior(c=1)

igprior(alpha=.01, lambda=.01)

Arguments

penalty

Penalty on model dimension, i.e. for the AIC penalty=2

tau

Prior dispersion parameter for covariates undergoing selection

taustd

Prior dispersion parameter for covariates undergoing selection. It is calibrated so that 'taustd=1' equals the unit information prior.

tau.adj

Prior variance in Normal prior for covariates not undergoing selection

r

MOM prior parameter is 2*r

p

Prior inclusion probability for binomial prior on model space

alpha.p

Beta-binomial prior on model space has parameters alpha.p, beta.p

beta.p

Beta-binomial prior on model space has parameters alpha.p, beta.p

c

Under the Complexity prior the prior probability of having k variables in the model is proportional to 1/p^(ck)

alpha

Inverse gamma prior has parameters alpha/2, lambda/2

lambda

igprior defines an inverse gamma prior with parameters alpha/2, lambda/2. exponentialprior defines an exponential prior with rate parameter lambda

Details

DISCUSSION OF PRIOR ON PARAMETERS

Let beta=(beta_1,...,beta_p) be the regression coefficients for individual variables and delta=(delta_1,...,delta_q) those for grouped variables (e.g. factors or smooth terms in modelSelection).

momprior, emomprior, imomprior, zellnerprior and normalid can be priors on both beta or delta. For further information see the vignette.

groupzellnerprior is the prior density on delta

p_z(\delta; \tau)= \prod_j N(\delta_j; 0, (\tau/p_j)) (X_j'X_j)^{-1}

where X_j are the design matrix columns associated to delta_j and p_j=ncol(X_j) is the number of covariates in the group (for groupmomprior, the term in the denominator is (p_j +2) instead of p_j). A default tau=n=nrow(X_j) mimics the unit information prior and implies that the ratio of variance explained by X_j / residual variance is expected to be 1 a priori. To set the dispersion in terms of unit information prior, taustd is also available.

groupmomprior adds a quadratic MOM penalty

p_m(delta; tau)= p_z(delta; tau * n) prod_j delta_j'X_j'X_jdelta_j ncol(X_j)/(tau * n * p_j / (p_j + 2))

and analogously for eMOM and iMOM. Note that unlike groupzellnerprior, the nrow(X_j) factor is already included in the code. This is done to give user introduced tau values a roughly similar meaning between momprior and groupmomprior.

DISCUSSION OF PRIOR ON MODELS

Under the uniform prior, the prior probability of any model is 1 / number of models.

Under the Binomial, Beta-Binomial and Complexity priors a model with k out of K active variables has prior probability P(Z=k) / (K choose k), where where Z ~ Binomial(K,p), Z ~ BetaBinomial(K,alpha.p,beta.p) or for the Complexity prior P(Z=k) proportional to 1/K^(c*k).

Objects from the Class

Objects can be created by calls of the form new("msPriorSpec", ...), but it is easier to use creator functions.

For priors on regression coefficients use momprior, imomprior or emomprior. For prior on model space modelunifprior, modelbinomprior modelbbprior, or modelcomplexprior. For prior on residual variance use igprior.

Slots

priorType:

Object of class "character". "coefficients" indicates that the prior is for the non-zero regression coefficients. "modelIndicator" that it is for the model indicator, and "nuisancePars" that it is for the nuisance parameteres. Several prior distributions are available for each choice of priorType, and these can be speicified in the slot priorDist.

priorDistr:

Object of class "character". If priorType=="coefficients", priorDistr can be equal to "pMOM", "piMOM", "peMOM", "zellner", "normalid", "groupMOM" or "groupzellner" (product moment, product inverse moment, product exponential moment, Zellner prior, normal prior with \Sigma=\mathbf{I}, respectively). If priorType=="modelIndicator", priorDistr can be equal to "uniform" or "binomial" to specify a uniform prior (all models equaly likely a priori) or a binomial prior, or to "complexity" for the Complexity prior of Castillo et al 2015. For a binomial prior, the prior inclusion probability for any single variable must be specified in slot priorPars['p']. For a beta-binomial prior, the Beta hyper-prior parameters must be in priorPars['alpha.p'] and priorPars['beta.p']. For the Complexity prior, the prior parameter must be in the slot priorPars['c']. If priorType=="nuisancePars", priorDistr must be equal to "invgamma". This corresponds to an inverse gamma distribution for the residual variance, with parameters specified in the slot priorPars.

priorPars:

Object of class "vector", where each element must be named. For priorDistr=='pMOM', there must be an element "r" (MOM power is 2r). For any priorDistr there must be either an element "tau" indicating the prior dispersion or elements "a.tau" and "b.tau" specifying an inverse gamma hyper-prior for "tau". Optionally, there may be an element "tau.adj" indicating the prior dispersion for the adjustment variables (i.e. not undergoing variable selection). If not defined, "tau.adj" is set to 0.001 by default. For priorDistr=='binomial', there must be either an element "p" specifying the prior inclusion probability for any single covariate, or a vector with elements "alpha.p" and "beta.p" specifying a Beta(alpha.p,beta.p) hyper-prior on p. For priorDistr=='invgamma' there must be elements "alpha" and "lambda". The prior for the residual variance is an inverse gamma with parameteres .5*alpha and .5*lambda.

Methods

No methods defined with class "msPriorSpec" in the signature.

Note

When new instances of the class are created a series of check are performed to ensure that a valid prior specification is produced.

Author(s)

David Rossell

References

Johnson VE, Rossell D. Non-Local Prior Densities for Default Bayesian Hypothesis Tests. Journal of the Royal Statistical Society B, 2010, 72, 143-170

Johnson VE, Rossell D. Bayesian model selection in high-dimensional settings. Journal of the American Statistical Association, 107, 498:649-660.

See Also

See also modelSelection for an example of defining an instance of the class and perform Bayesian model selection.

Examples

showClass("msPriorSpec")