# Selection of variables with embedded screening using Bayesian methods (SVEN) in Gaussian linear models (ultra-high, high or low dimensional). 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) ```r 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. ## Returns 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. ## 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 $j$th 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`. ## Examples ```r 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) ``` ## 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 . ## Author(s) Dongjin Li, Debarshi Chakraborty, and Somak Dutta Maintainer: Dongjin Li

sven function