nlpMarginals: Marginal density of the observed data for linear regression...

In mombf: Model Selection with Bayesian Methods and Information Criteria

Marginal density of the observed data for linear regression with Normal, two-piece Normal, Laplace or two-piece Laplace residuals under non-local and Zellner priors

Description

The marginal density of the data, i.e. the likelihood integrated with respect to the given prior distribution on the regression coefficients of the variables included in the model and an inverse gamma prior on the residual variance.

nlpMarginal is the general function, the remaining ones correspond to particular cases and are kept for backwards compatibility with old code, and will be deprecated in the future.

Usage

nlpMarginal(sel, y, x, data, smoothterms, nknots=9, groups=1:ncol(x),
family="normal", priorCoef, priorGroup,
priorVar=igprior(alpha=0.01,lambda=0.01), priorSkew=momprior(tau=0.348),
phi, method='auto', adj.overdisp='intercept', hess='asymp', optimMethod,
optim_maxit, initpar='none', B=10^5, logscale=TRUE, XtX, ytX)

pimomMarginalK(sel, y, x, phi, tau=1, method='Laplace', B=10^5, logscale=TRUE, XtX, ytX)
pimomMarginalU(sel, y, x, alpha=0.001, lambda=0.001, tau=1,
method='Laplace', B=10^5, logscale=TRUE, XtX, ytX)
pmomMarginalK(sel, y, x, phi, tau, r=1, method='auto', B=10^5,
logscale=TRUE, XtX, ytX)
pmomMarginalU(sel, y, x, alpha=0.001, lambda=0.001, tau=1,
r=1, method='auto', B=10^5, logscale=TRUE, XtX, ytX)

Arguments

sel	Vector with indexes of columns in x to be included in the model. Ignored if y is a formula
y	Either a formula with the regression equation or a vector with observed responses. The response can be either continuous or of class Surv (survival outcome). If y is a formula then x, groups and constraints are automatically created
x	Design matrix with linear covariates for which we want to assess if they have a linear effect on the response. Ignored if y is a formula
data	If y is a formula then data should be a data frame containing the variables in the model
smoothterms	Formula for non-linear covariates (cubic splines), modelSelection assesses if the variable has no effect, linear or non-linear effect. smoothterms can also be a design matrix or data.frame containing linear terms, for each column modelSelection creates a spline basis and tests no/linear/non-linear effects
nknots	Number of spline knots. For cubic splines the non-linear basis adds knots-4 coefficients for each linear term, we recommend setting nknots to a small/moderate value
groups	If variables in x such be added/dropped in groups, groups indicates the group that each variable corresponds to (by default each variable goes in a separate group)
family	Residual distribution. Possible values are 'normal','twopiecenormal','laplace', 'twopiecelaplace'
priorCoef	Prior on coefficients, created by momprior, imomprior, emomprior or zellnerprior. Prior dispersion is on coefficients/sqrt(scale) for Normal and two-piece Normal, and on coefficients/sqrt(2*scale) for Laplace and two-piece Laplace.
priorGroup	Prior on grouped coefficients (e.g. categorical predictors with >2 categories, splines). Created by groupmomprior, groupemomprior, groupimomprior or groupzellnerprior
priorVar	Inverse gamma prior on scale parameter, created by igprior(). For Normal variance=scale, for Laplace variance=2*scale.
priorSkew	Either a number fixing tanh(alpha) where alpha is the asymmetry parameter or a prior on residual skewness parameter, assumed to be of the same family as priorCoef. Ignored if family is 'normal' or 'laplace'.
method	Method to approximate the integral. See help(modelSelection).
adj.overdisp	Only used for method=='ALA'. Over-dispersion adjustment for models with fixed dispersion parameter such as logistic and Poisson regression
hess	Method to estimat the hessian in the Laplace approximation to the integrated likelihood under Laplace or asymmetric Laplace errors. When hess=='asymp' the asymptotic hessian is used, hess=='asympDiagAdj' a diagonal adjustment is applied (see Rossell and Rubio for details).
optimMethod	Algorithm to maximize objective function when method=='Laplace'. Leave unspecified or set optimMethod=='auto' for an automatic choice. optimMethod=='LMA' uses modified Newton-Raphson algorithm, 'CDA' coordinate descent algorithm
optim_maxit	Maximum number of iterations when method=='Laplace'
initpar	Initial regression parameter values when finding the posterior mode to approximate the integrated likelihood. See help(modelSelection)
B	Number of Monte Carlo samples to use (ignored unless method=='MC')
logscale	If logscale==TRUE the log marginal density is returned.
XtX	Optionally, specify the matrix X'X. Useful when the function must be called a large number of times.
ytX	Optionally, specify the vector y'X. Useful when the function must be called a large number of times.
phi	Disperson parameter. See help(modelSelection)
alpha	Prior for phi is inverse gamma alpha/2, lambda/2
lambda	Prior for phi is inverse gamma alpha/2, lambda/2
tau	Prior dispersion parameter for MOM and iMOM priors (see details)
r	Prior power parameter for MOM prior is 2*r

Details

The marginal density of the data is equal to the integral of N(y;x[,sel]*theta,phi*I) * pi(theta|phi,tau) * IG(phi;alpha/2,lambda/2) with respect to theta, where pi(theta|phi,tau) is a non-local prior and IG denotes the density of an inverse gamma.

pmomMarginalK and pimomMarginalK assume that the residual variance is known and therefore the inverse-gamma term in the integrand can be ommitted.

The product MOM and iMOM densities can be evaluated using the functions dmom and dimom.

Value

Marginal density of the observed data under the specified prior.

Author(s)

David Rossell

References

Johnson V.E., Rossell D. Non-Local Prior Densities for Default Bayesian Hypothesis Tests. Journal of the Royal Statistical Society B, 2010, 72, 143-170. See http://rosselldavid.googlepages.com for technical reports.

See Also

modelSelection to perform model selection based on product non-local priors. momunknown, imomunknown, momknown, imomknown to compute Bayes factors for additive MOM and iMOM priors

Examples

x <- matrix(rnorm(100*2),ncol=2)
y <- x %*% matrix(c(.5,1),ncol=1) + rnorm(nrow(x))
pmomMarginalK(sel=1, y=y, x=x, phi=1, tau=1, method='Laplace')
pmomMarginalK(sel=1:2, y=y, x=x, phi=1, tau=1, method='Laplace')

mombf documentation built on Sept. 28, 2023, 5:06 p.m.