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Fit linear regression models with a nonparametric Bayesian Lasso prior

Source: R/bnplasso_lm.R
bnplasso.lm.Rd

This function fits linear regression models with a nonparametric Bayesian Lasso prior as in Marin et al. (2025+).

Usage

bnplasso.lm(
  X,
  y,
  intercept = TRUE,
  prior = NULL,
  a = NULL,
  b = NULL,
  alpha = NULL,
  variance.prior.type = NULL,
  max.iters = 6000L,
  burn.in = 1000L,
  thin = 1L,
  polya = TRUE,
  float = FALSE
)

Arguments

X

A matrix of predictors of dimension \(n\)-by-\(p\), where each of the \(n\) rows is an observation vector.

y

Response variable. It should be a numeric vector size \(n\).

intercept

Logical. If TRUE, an intercept term is included in the model; otherwise, the intercept is integrated out. If TRUE, the prior on the intercept would be a non-informative prior of the form \(p(\mu)\propto 1\). Default is TRUE.

prior

A character string denoting which type of shrinkage prior should be employed. The options are either: (1) "bnp.lasso" which implements the nonparametric Bayesian Lasso as in Marin et al. (2025+), (2) "b.lasso" which implements the Bayesian Lasso as in Park and Casella (2008), or (3) "b.adapt.lasso" which implements the Bayesian adaptive Lasso as in Leng et al. (2014). Default is "bnp.lasso".

a

A positive scalar. If prior = "bnp.lasso", it corresponds to the shape parameter in the gamma distribution used as a centering measure in the DP prior. If prior = "b.lasso", it corresponds to the shape parameter in the gamma distribution used as prior in the shrinkage parameter, \(\lambda^{2}\). If prior = "b.adapt.lasso", it corresponds to the shape parameter in the gamma distribution used as prior in the shrinkage parameters, \(\lambda_{j}^{2}\), for \(j\in\{1,\dots,p\}\). If not provided, the function will attempt to determine an appropriate value.

b

A positive scalar. If prior = "bnp.lasso", it corresponds to the rate parameter in the gamma distribution used as a centering measure in the DP prior. If prior = "b.lasso", it corresponds to the rate parameter in the gamma distribution used as prior in the shrinkage parameter, \(\lambda^{2}\). If prior = "b.adapt.lasso", it corresponds to the rate parameter in the gamma distribution used as prior in the shrinkage parameters, \(\lambda_{j}^{2}\), for \(j\in\{1,\dots,p\}\). If not provided, the function will attempt to determine an appropriate value.

alpha

A positive scalar. If prior = "bnp.lasso", it corresponds to the concentration parameter in the DP prior. If prior = "b.lasso" or prior = "b.adapt.lasso", the argument is ignored. If not provided, the function will attempt to determine an appropriate value.

variance.prior.type

A character string denoting whether the prior on the sampling variance should be an independent-type prior or a conjugate-type prior. See Moran et al. (2019) for details. The options are either "independent" or "conjugate". If not provided, the function will attempt to determine an appropriate prior.

max.iters

A positive integer corresponding to the total number of MCMC iterations. Default is 6000.

burn.in

A positive integer corresponding to the number of draws discarded as burn-in. It should be smaller than max.iters. Default is 1000.

thin

A positive integer specifying the period for saving samples. Default is 1.

polya

logical. Whether a generalized Pólya urn sampling scheme or a blocked Gibbs sampling scheme should be employed to update the Dirichlet process mixture in the MCMC algorithm. If TRUE, a generalized Pólya urn sampling scheme would be used. Default is TRUE. Important: This argument is ignored if prior != "bnp.lasso"

float

Logical. If TRUE, some internal routines will use single point precision for improved computational efficiency at the expense of numerical accuracy. If FALSE, double point precision will be used. Default is FALSE.

Value

An object of S3 class, 'lmBayes', containing:

  • post.beta: A matrix of size n.draws-by-n.preds, where each row is a posterior draw of the regression coefficients.

  • post.sigma2: A vector of size n.draws, where each element is a posterior draw of the sampling variance.

  • post.tau2: A matrix of size n.draws-by-n.preds, where each row is a posterior draw of the latent parameters \(\tau_{j}^{2}\).

  • post.lambda2: A matrix of size n.draws-by-n.preds, where each row is a posterior draw of the shrinkage parameters \(\lambda_{j}^{2}\).

  • post.clust_idx: If prior = "bnp.lasso", a matrix of size n.draws-by-n.preds, where each row indicates to which cluster the regression coefficients belong to.

  • post.K: If prior = "bnp.lasso", a vector of size n.draws, where each element indicates the number of clusters in the corresponding MCMC iteration.

  • post.mu: If intercept = "TRUE", a vector of size n.draws, where each entry is a posterior draw of the intercept term.

  • elapsed: The elapsed (wall-clock) time of the MCMC sampler, in seconds.

  • a: If prior = "bnp.lasso", the shape parameter in the gamma distribution used as a centering measure in the DP prior. If prior = "b.lasso", the shape parameter in the gamma distribution used as prior in the shrinkage parameter, \(\lambda^{2}\). If prior = "b.adapt.lasso", the shape parameter in the gamma distribution used as prior in the shrinkage parameters, \(\lambda_{j}^{2}\), for \(j\in\{1,\dots,p\}\).

  • b: If prior = "bnp.lasso", the rate parameter in the gamma distribution used as a centering measure in the DP prior. If prior = "b.lasso", the rate parameter in the gamma distribution used as prior in the shrinkage parameter, \(\lambda^{2}\). If prior = "b.adapt.lasso", the rate parameter in the gamma distribution used as prior in the shrinkage parameters, \(\lambda_{j}^{2}\), for \(j\in\{1,\dots,p\}\).

  • alpha: If prior = "bnp.lasso", the concentration parameter in the Dirichlet process prior.

  • intercept: Whether an intercept was included in the model.

  • variance.prior.type: Whether the variance on the sampling variance was an independent-type prior or a conjugate-type prior.

  • max.iters: The total number of MCMC iterations.

  • burn.in: The number of draws discarded as burn-in.

  • thin: The period for saving draws.

  • n.obs: The sample size.

  • n.preds: The number of predictors.

  • n.draws: The number of posterior draws after burn-in and thinning.

  • X: Matrix of predictors.

  • y: Vector of responses.

  • loglik: Matrix of size n.draws-by-n.obs with the log-likelihood of each observation at each MCMC iteration.

  • post.pred.fitted.values: A matrix of size n.draws-by-n.obs, where each row is a draw from the posterior predictive distribution of the fitted values.

  • post.pred.residuals: A matrix of size n.draws-by-n.obs, where each row is a draw from the posterior predictive distribution of the residuals.

References

C. Leng, MN. Tran, and D. Nott (2014), Bayesian adaptive Lasso. Ann Inst Stat Math, 66:221-244

S. Marin, B. Long,and A. H. Westveld (2025+), Adaptive Shrinkage with a Nonparametric Bayesian Lasso.Journal of Computational and Graphical Statistics. doi:10.1080/10618600.2025.2572327

G. E. Moran, V. Rockova, and E. I. George (2019), Variance Prior Forms for High-Dimensional Bayesian Variable Selection. Bayesian Analysis, 14(4):1091-1119.

T. Park, and G. Casella (2008), The Bayesian Lasso. Journal of the American Statistical Association, 103(482):681-686.

Author

Santiago Marin