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

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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

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nlpMarginal(sel, y, x, family="normal", priorCoef=momprior(tau=0.348),
priorVar=igprior(alpha=0.01,lambda=0.01), priorSkew=momprior(tau=0.348),
method='auto', hess='asymp', optimMethod='CDA', 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

y

Vector with observed responses

x

Design matrix with covariates. Only the columns specified in sel are included in the model, the rest are disregarded

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.

priorVar

Inverse gamma prior on scale parameter, created by igprior(). For Normal variance=scale, for Laplace variance=2*scale.

priorSkew

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. method=='auto' uses closed-form expressions whenever possible and Laplace approximations otherwise. method=='Laplace' for Laplace approx. method=='MC' for Monte Carlo

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' or method=='MC'. optimMethod=='LMA' uses modified Newton-Raphson algorithm, 'CDA' coordinate descent algorithm

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

Residual variance, assumed to be known by pimomMarginalK and pmomMarginalK

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. mode2g for prior elicitation.

Examples

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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')