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.
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)
Marginal density of the observed data under the specified prior.
Vector with indexes of columns in x to be included in the model.
Ignored if y is a formula
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
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
If y is a formula then data should be a data
frame containing the variables in the model
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
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
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)
Residual distribution. Possible values are 'normal','twopiecenormal','laplace', 'twopiecelaplace'
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.
Prior on grouped coefficients (e.g. categorical
predictors with >2 categories, splines). Created by
groupmomprior, groupemomprior,
groupimomprior or groupzellnerprior
Inverse gamma prior on scale parameter, created by
igprior(). For Normal variance=scale, for Laplace variance=2*scale.
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 to approximate the integral. See
help(modelSelection).
Only used for method=='ALA'. Over-dispersion adjustment for models with fixed dispersion parameter such as logistic and Poisson regression
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).
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
Maximum number of iterations when method=='Laplace'
Initial regression parameter values when finding the posterior mode to approximate the integrated likelihood. See help(modelSelection)
Number of Monte Carlo samples to use (ignored unless
method=='MC')
If logscale==TRUE the log marginal density is returned.
Optionally, specify the matrix X'X. Useful when the function must be called a large number of times.
Optionally, specify the vector y'X. Useful when the function must be called a large number of times.
Disperson parameter. See help(modelSelection)
Prior for phi is inverse gamma alpha/2,
lambda/2
Prior for phi is inverse gamma alpha/2,
lambda/2
Prior dispersion parameter for MOM and iMOM priors (see details)
Prior power parameter for MOM prior is 2*r
David Rossell
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.
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.
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
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')
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