sven {bravo}

Selection of variables with embedded screening using Bayesian methods (SVEN) in Gaussian linear models (ultra-high, high or low dimensional).

Description

SVEN is an approach to selecting variables with embedded screening using a Bayesian hierarchical model. It is also a variable selection method in the spirit of the stochastic shotgun search algorithm. However, by embedding a unique model based screening and using fast Cholesky updates, SVEN produces a highly scalable algorithm to explore gigantic model spaces and rapidly identify the regions of high posterior probabilities. It outputs the log (unnormalized) posterior probability of a set of best (highest probability) models. For more details, see Li et al. (2023, https://doi.org/10.1080/10618600.2022.2074428)

Usage

sven(
  X,
  y,
  w = NULL,
  lam = NULL,
  Ntemp = 10,
  Tmax = NULL,
  Miter = 50,
  wam.threshold = 0.5,
  log.eps = -16,
  L = 20,
  verbose = FALSE
)

Arguments

X

The n\times p covariate matrix or list of two matrices without intercept. The following classes are supported: matrix and dgCMatrix. Every care is taken not to make copies of these (typically) giant matrices. No need to center or scale these matrices manually. Scaling is performed implicitly and regression coefficient are returned on the original scale. Typically, in a combined GWAS-TWAS type analysis, X[[1]] should be a sparse matrix and X[[2]] should be a dense matrix.

y

The response vector of length n. No need to center or scale.

w

The prior inclusion probability of each variable. Default: NULL, whence it is set as \sqrt{n}/p if X is a matrix. Or (\sqrt{n}/p_1,\sqrt{n}/p_2) if $X$ is a list of two matrices with p_1 and p_2 columns.

lam

The slab precision parameter. Default: NULL, whence it is set as n/p^2 for as suggested by the theory of Li et al. (2023). Similarly, it's a vector of length two with values \sqrt{n}/P_1^2 and \sqrt{n}/p_2^2 when X is a list.

Ntemp

The number of temperatures. Default: 10.

Tmax

The maximum temperature. Default: \log\log p+\log p.

Miter

The number of iterations per temperature. Default: 50.

wam.threshold

The threshold probability to select the covariates for WAM. A covariate will be included in WAM if its corresponding marginal inclusion probability is greater than the threshold. Default: 0.5.

log.eps

The tolerance to choose the number of top models. See detail. Default: -16.

L

The minimum number of neighboring models screened. Default: 20.

verbose

If FALSE, the function prints the current temperature SVEN is at; the default is TRUE.

Details

SVEN is developed based on a hierarchical Gaussian linear model with priors placed on the regression coefficients as well as on the model space as follows:

y | X, \beta_0,\beta,\gamma,\sigma^2,w,\lambda \sim N(\beta_01 + X_\gamma\beta_\gamma,\sigma^2I_n)

\beta_i|\beta_0,\gamma,\sigma^2,w,\lambda \stackrel{indep.}{\sim} N(0, \gamma_i\sigma^2/\lambda),~i=1,\ldots,p,

(\beta_0,\sigma^2)|\gamma,w,p \sim p(\beta_0,\sigma^2) \propto 1/\sigma^2

\gamma_i|w,\lambda \stackrel{iid}{\sim} Bernoulli(w)

where X_\gamma is the n \times |\gamma| submatrix of X consisting of those columns of X for which \gamma_i=1 and similarly, \beta_\gamma is the |\gamma| subvector of \beta corresponding to \gamma. Degenerate spike priors on inactive variables and Gaussian slab priors on active covariates makes the posterior probability (up to a normalizing constant) of a model P(\gamma|y) available in explicit form (Li et al., 2020).

The variable selection starts from an empty model and updates the model according to the posterior probability of its neighboring models for some pre-specified number of iterations. In each iteration, the models with small probabilities are screened out in order to quickly identify the regions of high posterior probabilities. A temperature schedule is used to facilitate exploration of models separated by valleys in the posterior probability function, thus mitigate posterior multimodality associated with variable selection models. The default maximum temperature is guided by the asymptotic posterior model selection consistency results in Li et al. (2020).

SVEN provides the maximum a posteriori (MAP) model as well as the weighted average model (WAM). WAM is obtained in the following way: (1) keep the best (highest probability) K distinct models \gamma^{(1)},\ldots,\gamma^{(K)} with

\log P\left(\gamma^{(1)}|y\right) \ge \cdots \ge \log P\left(\gamma^{(K)}|y\right)

where K is chosen so that \log \left\{P\left(\gamma^{(K)}|y\right)/P\left(\gamma^{(1)}|y\right)\right\} > \code{log.eps}; (2) assign the weights

w_i = P(\gamma^{(i)}|y)/\sum_{k=1}^K P(\gamma^{(k)}|y)

to the model \gamma^{(i)}; (3) define the approximate marginal inclusion probabilities for the jth variable as

\hat\pi_j = \sum_{k=1}^K w_k I(\gamma^{(k)}_j = 1).

Then, the WAM is defined as the model containing variables j with \hat\pi_j > \code{wam.threshold}. SVEN also provides all the top K models which are stored in an p \times K sparse matrix, along with their corresponding log (unnormalized) posterior probabilities.

When X is a list with two matrices, say, W and Z, the above method is extended to ncol(W)+ncol(Z) dimensional regression. However, the hyperparameters lam and w are chosen separately for the two matrices, the default values being nrow(W)/ncol(W)^2 and nrow(Z)/ncol(Z)^2 for lam and sqrt(nrow(W))/ncol(W) and sqrt(nrow(Z))/ncol(Z) for w.

The marginal inclusion probabities can be extracted by using the function mip.

Value

A list with components

model.map

A vector of indices corresponding to the selected variables in the MAP model.

model.wam

A vector of indices corresponding to the selected variables in the WAM.

model.top

A sparse matrix storing the top models.

beta.map

The ridge estimator of regression coefficients in the MAP model.

beta.wam

The ridge estimator of regression coefficients in the WAM.

mip.map

The marginal inclusion probabilities of the variables in the MAP model.

mip.wam

The marginal inclusion probabilities of the variables in the WAM.

pprob.map

The log (unnormalized) posterior probability corresponding to the MAP model.

pprob.top

A vector of the log (unnormalized) posterior probabilities corresponding to the top models.

stats

Additional statistics.

Author(s)

Dongjin Li, Debarshi Chakraborty, and Somak Dutta
Maintainer: Dongjin Li <[e-mail]>

References

Li, D., Dutta, S., and Roy, V. (2023). Model based screening embedded Bayesian variable selection for ultra-high dimensional settings. Journal of Computational and Graphical Statistics, 32(1), 61-73.

See Also

[mip.sven()] for marginal inclusion probabilities, [predict.sven()](via [predict()]) for prediction for .

Examples

n <- 50; p <- 100; nonzero <- 3
trueidx <- 1:3
truebeta <- c(4,5,6)
X <- matrix(rnorm(n*p), n, p) # n x p covariate matrix
y <- 0.5 + X[,trueidx] %*% truebeta + rnorm(n)
res <- sven(X=X, y=y)
res$model.map # the MAP model


Z <- matrix(rnorm(n*p), n, p) # another covariate matrix
y2 = 0.5 + X[,trueidx] %*% truebeta  + Z[,1:2] %*% c(-2,-2) + rnorm(n)
res2 <- sven(X=list(X,Z), y=y2)