## Bayesian variable selection with shrinking and diffusing priors

### Description

This function performs the Bayesian variable selection procedure with shrinking and diffusing priors via Gibbs sampling. Three different prior options placed on the coefficients are provided: Gaussian, Student's t, Laplace. The posterior estimates of coefficients are returned and the final model is selected either by using the "BIC" criterion or the median probability model.

### Usage

basad(x = NULL, y = NULL, K = -1, df = 5, nburn = 1000, niter = 1000,
alternative = FALSE, verbose = FALSE, nsplit = 20, tau0 = NULL, tau1 = NULL,
prior.dist = "Gauss", select.cri = "median", BIC.maxsize = 20,
standardize = TRUE, intercept = TRUE)


### Arguments

 x The matrix or data frame of covariates. y The response variables. K An initial value for the number of active covariates in the model. This value is related to the prior probability that a covariate is nonzero. If K is not specified greater than 3, this prior probability will be estimated by a Beta prior using Gibbs sampling (see details below). df The degrees of freedom of t prior when prior.dist == "t". nburn The number of iterations for burn-in. niter The number of iterations for estimation. alternative If TRUE, an alternative sampling scheme from Bhattacharya will be used which can accelerate the speed of the algorithm for very large p. However, when using block updating (by setting nsplit to be greater than 1) this alternative sampling will not be invoked. verbose If TRUE, verbose output is sent to the terminal. nsplit Numbers of splits for the block updating scheme. tau0 The scale of the prior distribution for inactive coefficients (see details below). tau1 The scale of the prior distribution for active coefficients (see details below). prior.dist Choice of the base distribution for spike and slab priors. If prior.dist="t", the algorithm will place Student's t prior for regression coefficients. If prior.dist="Laplace", the algorithm will place Laplace prior. Otherwise, it will place the default Gaussian priors. select.cri Model selection criterion. If select.cri="median", the algorithm will use the median probability model to select the active variables. If  select.cri="BIC", the algorithm will use the BIC criterion to select the active variables. BIC.maxsize The maximum number of variables that are chosen to apply BIC criterion based on the ranking of their marginal posterior probabilities. If the dimension is less than the default value of 20, all the variables will be considered to apply BIC. standardize Option for standardization of the covariates. Default is standardize = TRUE. intercept Option to include an intercept in the model. Default is TRUE.

### Details

In the package, the regression coefficients have following hierarchical structure:

β | ( Z = 0, σ^2 ) = N(0, τ_0^2 σ^2 ), β | ( Z = 1, σ^2 ) = N(0, τ_1^2 σ^2 )

where the latent variable Z_i of value 0 or 1 indicates whether ith variable is in the slab and spike part of the prior. The package provides different prior choices for the coefficients: Gaussian, Student's t, Laplace. Through setting the parameter prior.dist, the coefficients will have the corresponding prior densities as follows:

1. The Gaussian priors case:

β | ( Z = k, σ^2 ) = \frac{1}{ √{ 2 π τ^2_k σ^2} } e^{ - \frac{ β^2 }{ 2 τ^2_k σ^2 } }

2. The Student's t prior case:

β | ( Z = k, σ^2 ) = \frac{ Γ ( \frac{ν + 1 }{2} ) }{ Γ ( \frac{ν}{2} ) √{ π ν } τ_k σ } \Bigg( 1 + \frac{1}{ν} \Big( \frac{ β^2 }{ τ^2_k σ^2 } \Big) \Bigg)^{ -\frac{ ν + 1}{ 2 } }

Where ν is the degrees of freedom

3. The Laplace prior case:

β | ( Z = k, σ^2 ) = \frac{1}{2 τ_k^2 σ^2 } \exp \Big( -\frac{ |β| }{ τ_{k}^2 σ^2 } \Big)

The τ_k is the scale for the prior distribution. If users did not set a specific value, the prior scales are specified as follows:

τ_{0}^2 = \frac{1}{n} a_τ , \, τ_{1}^2 = \max\Big( 100 τ_{0}^2, \frac{ τ_{0} p_n }{ (1 - p_n) ρ } \Big),

where ρ is the prior density evaluated at f_p( b_τ \times \log{(p_n + 1)}), f_p is the density function for the corresponding prior distribution. The parameter a and b are a_τ = 1 and b_τ = 2.4 by default.

The prior probability q_n = P(Z_i = 1) that a covariate is nonzero can be specified by value K. The K represents a prior belief of the upper bound of the true covariates in the model. When user specifies a value of K greater than 3, setting q_n = c/p_n, through the calculation(see details in Naveen (2014)):

Φ( (K - c)/ √{c} ) = 1 - α

The prior probability on the models with sizes greater than K will be α, and this α is set to 0.1 in the package.

### Value

An object of class basad with the following components:

 all.var Summary object for all the variables. select.var Summary object for the selected variables. beta.names Variable names for the coefficients. verbose Verbose details (used for printing). posteriorZ A vector of the marginal posterior probabilities for the latent vector Z. model.index A vector containing the indices of selected variables. modelZ A binary vector Z indicating whether the coefficient is true in the selected model. est.B Estimated coefficient values from the posterior distribution through Gibbs sampling. allB A matrix of all sampled coefficient values along the entire chain. Each row represents the sampled values under each iteration. allZ A matrix of all sampled posterior probabilities for the latent variable Z along the entire chain. Each row represents the sampled values under each iteration. x x-matrix used for the algorithm. y y vector used for the algorithm.

### Author(s)

Qingyan Xiang (qyxiang@bu.edu)

Naveen Narisetty (naveen@illinois.edu)

### References

Narisetty, N. N., & He, X. (2014). Bayesian variable selection with shrinking and diffusing priors. The Annals of Statistics, 42(2), 789-817.

Bhattacharya, A., Chakraborty, A., & Mallick, B. K. (2016). Fast sampling with Gaussian scale mixture priors in high-dimensional regression. Biometrika, 4(103), 985-991.

Barbieri, M. M., & Berger, J. O. (2004). Optimal predictive model selection. The Annals of Statistics, 32(3), 870-897.

### Examples



#-----------------------------------------------------------
#Generate Data: The simulated high dimensional data
#-----------------------------------------------------------
n =  100; p = 499; nz = 5

rho1=0.25; rho2=0.25; rho3=0.25  ### correlations
Bc = c(0, seq(0.6, 3, length.out = nz), array(0, p - nz))

covr1 = (1 - rho1) * diag(nz) +  array(rho1, c(nz, nz))
covr3 = (1 - rho3) * diag(p - nz) +  array(rho3, c(p - nz, p - nz))
covr2 = array(rho2, c(nz, p - nz))
covr = rbind(cbind(covr1, covr2), cbind(t(covr2), covr3))

covE = eigen(covr)
covsq = covE$vectors %*% diag(sqrt(covE$values)) %*% t(covE\$vectors)

Xs = matrix(rnorm(n * p), nrow = n); Xn = covsq %*% t(Xs)
X = cbind(array(1, n), t(Xn))
Y = X %*% Bc + rnorm(n); X <- X[, 2:ncol(X)]

#-----------------------------------------------------------
#Example 1: Run the default setting of the Guassian priors
#-----------------------------------------------------------
obj <- basad(x = X, y = Y)
print(obj)

#-----------------------------------------------------------
#Example 2: Use different priors and slection criterion
#-----------------------------------------------------------
obj <- basad(x = X, y = Y, prior.dist = "t", select.cri = "BIC")
print(obj)