full_enumeration_pep {PEPBVS}

Bayesian variable selection through exhaustive search

Description

Given a response vector and an input data matrix, performs Bayesian variable selection using full enumeration of the model space. Normal linear models are assumed for the data with the prior distribution on the model parameters (beta coefficients and error variance) being the PEP or the intrinsic. The prior distribution on the model space can be the uniform on the model space or the uniform on the model dimension (special case of the beta-binomial prior). The model space consists of all possible models including an intercept term.

Usage

full_enumeration_pep(
  x,
  y,
  intrinsic = FALSE,
  reference.prior = TRUE,
  beta.binom = TRUE,
  ml_constant.term = FALSE
)

Arguments

x

A matrix of numeric (of size nxp), input data matrix. This matrix contains the values of the p explanatory variables without an intercept column of 1's.

y

A vector of numeric (of length n), response vector.

intrinsic

Boolean, indicating whether the PEP (FALSE) or the intrinsic - which is a special case of it - (TRUE) should be used as prior on the regression parameters. Default value=FALSE.

reference.prior

Boolean, indicating whether the reference prior (TRUE) or the dependence Jeffreys prior (FALSE) is used as baseline. Default value=TRUE.

beta.binom

Boolean, indicating whether the beta-binomial distribution (TRUE) or the uniform distribution (FALSE) should be used as prior on the model space. Default value=TRUE.

ml_constant.term

Boolean, indicating whether the constant (marginal likelihood of the null/intercept-only model) should be included in computing the marginal likelihood of a model (TRUE) or not (FALSE). Default value=FALSE.

Details

The function works when p<=n-2 where p is the number of explanatory variables and n is the sample size.

It is suggested to use this function (i.e. enumeration of the model space) when p is up to 20.

The reference model is the null model (i.e. intercept-only model).

The case of missing data (i.e. presence of NA's either in the input data matrix or the response vector) is not currently supported.

All models considered (i.e. model space) include an intercept term.

If p>1, the input data matrix needs to be of full rank.

The reference prior as baseline corresponds to hyperparameter values d0=0 and d1=0, while the dependence Jeffreys prior corresponds to model-dependent-based values for the hyperparameters d0 and d1, see Fouskakis and Ntzoufras (2022) for more details.

For computing the marginal likelihood of a model, Equation 16 of Fouskakis and Ntzoufras (2022) is used.

When ml_constant.term=FALSE then the log marginal likelihood of a model in the output is shifted by -logC1 (logC1: log marginal likelihood of the null model).

When the prior on the model space is beta-binomial (i.e. beta.binom=TRUE), the following special case is used: uniform prior on model dimension.

Value

full_enumeration_pep returns an object of class pep, i.e. a list with the following elements:

models

A matrix containing information about the models examined. In particular, in row i after representing the model i with variable inclusion indicators, its marginal likelihood (in log scale), the R2, its dimension (including the intercept), the corresponding Bayes factor, posterior odds and its posterior probability are contained. The models are sorted in decreasing order of the posterior probability. For the Bayes factor and the posterior odds, the comparison is done to the model with the largest posterior probability.

inc.probs

A named vector with the posterior inclusion probabilities of the explanatory variables.

x

The input data matrix (of size nxp).

y

The response vector (of length n).

intrinsic

Whether the prior on the model parameters was PEP or intrinsic.

reference.prior

Whether the baseline prior was the reference prior or the dependence Jeffreys prior.

beta.binom

Whether the prior on the model space was beta-binomial or uniform.

References

Fouskakis, D. and Ntzoufras, I. (2022) Power-Expected-Posterior Priors as Mixtures of g-Priors in Normal Linear Models. Bayesian Analysis, 17(4): 1073-1099. doi:10.1214/21-BA1288

See Also

mc3_pep

Examples

data(UScrime_data)
y <- UScrime_data[,"y"]
X <- UScrime_data[,-15]
res <- full_enumeration_pep(X,y)
resu <- full_enumeration_pep(X,y,beta.binom=FALSE)
resi <- full_enumeration_pep(X,y,intrinsic=TRUE)
resiu <- full_enumeration_pep(X,y,intrinsic=TRUE,beta.binom=FALSE)
resj <- full_enumeration_pep(X,y,reference.prior=FALSE)