Selecting Variables in High Dimensional Data

Given a dataset with n samples and p features where p is large (possibly p >> n):

1. Check the n/p ratio and correlation structure among covariates.
2. Standardize features; center the response.
3. Fit a penalized model as a baseline:

   R
   library(glmnet)
   cv_fit <- cv.glmnet(X, y, alpha = 1)   # LASSO
   coef(cv_fit, s = "lambda.min")

4. If sparsity assumptions are weak or you need uncertainty quantification, fit a Bayesian shrinkage model (e.g., spike-and-slab or horseshoe) instead of/alongside LASSO.
5. Validate selected variables via stability selection or cross-validated prediction error, not just in-sample fit.

Progress:

• Step 1: Characterize the problem (n vs p, signal sparsity, correlation among predictors, response type: continuous/binary/survival)
• Step 2: Choose a modeling framework (frequentist penalization vs Bayesian shrinkage) based on goals (point estimate vs full posterior/uncertainty)
• Step 3: Select and fit an appropriate method
• Step 4: Tune hyperparameters (penalty λ, prior scale) via cross-validation or empirical Bayes
• Step 5: Assess stability and control false discoveries
• Step 6: Validate on held-out data and interpret selected model

Step 1: Characterize the problem

• If p > n substantially: assume sparsity is necessary for identifiability.
• Check pairwise correlations: high collinearity favors elastic net or group-structured penalties over plain LASSO.
• Note response type to pick the right base model (linear, logistic, Cox).

Step 2: Choose a framework

Frequentist (penalization-based) — fast, scalable, good default for point estimates:

• LASSO (L1): sparse solutions, but biased estimates and unstable selection under correlation.
• Ridge (L2): no sparsity, use only for prediction with dense signals.
• Elastic Net: L1+L2, handles grouped/correlated predictors.
• SCAD / MCP: non-convex penalties reducing bias of LASSO, harder to optimize but better oracle properties.
• Group LASSO / Sparse Group LASSO: when predictors have natural grouping (e.g., dummy variables, pathways).

Bayesian (prior-based shrinkage) — better uncertainty quantification, natural handling of multiplicity:

• Spike-and-slab priors: explicit variable inclusion/exclusion, gives posterior inclusion probabilities.
• Continuous shrinkage priors (horseshoe, Bayesian LASSO, normal-gamma): easier computation (no combinatorial search), good adaptive shrinkage.
• Empirical Bayes: estimate prior hyperparameters (e.g., sparsity level) from data — preferred over fully subjective priors in high dimensions.
• Objective/reference priors: when no strong prior information exists.

Step 3: Fit the method

Use established software (glmnet, ncvreg for SCAD/MCP, BoomSpikeSlab/horseshoe packages) rather than custom optimization unless justified.

Step 4: Tune hyperparameters

• Frequentist: k-fold CV on prediction loss; consider CV on multiple λ's along the path, not just lambda.min (use lambda.1se for parsimony).
• Bayesian: empirical Bayes marginal likelihood maximization, or hierarchical priors with hyperpriors on sparsity/shrinkage parameters.

Step 5: Assess stability & control false discoveries

• Stability selection: resample/subsample and check selection frequency of each variable; keep variables selected above a threshold.
• Knockoffs: construct knockoff variables to control FDR rigorously in high dimensions.
• Posterior inclusion probability thresholds (Bayesian): e.g., select variables with PIP > 0.5 (median probability model).

Step 6: Validate

• Hold out a test set or use nested CV to get honest prediction error.
• Check selected model against domain knowledge (e.g., known relevant genes/pathways).
• Report uncertainty: confidence/credible intervals for selected effects using post-selection inference or Bayesian posteriors.

Example 1: Input: Gene expression dataset, n=150 samples, p=20,000 genes, binary disease outcome. Output: Use sparse logistic regression with LASSO or elastic net (correlated gene pathways favor elastic net); tune λ via 10-fold CV on deviance; validate selected genes via stability selection across 100 bootstrap resamples; report final model with genes selected in >80% of resamples, cross-validated AUC, and known pathway annotations for interpretation.

Example 2: Input: Genomic dataset with strong prior belief that only a small fraction of SNPs are causal, and researcher wants posterior probability of association for each SNP. Output: Fit a Bayesian spike-and-slab linear model with empirical-Bayes-estimated sparsity level; use MCMC or variational inference for posterior inclusion probabilities; report SNPs with PIP > 0.5 (median probability model) along with 95% credible intervals for effect sizes.