Variable Selection Priors Module

[NathanielF]

Variable selection priors were shown to be useful for causal inference here: #537

and I can think of more use cases especially for synthetic control methods. I think the functionality is general enough to warrant a module for the different classes of variable selection priors: spike and slab, horseshoe, etc...

I've added a staggered commit on PR 537 here: NathanielF#1

Just to make the idea a bit more concrete. But open to feedback and implementation suggestions.

[drbenvincent]

Brainstorming...

ITS with many covariates learns a counterfactual for one treated series by fitting a time-series state (level/trend/seasonality) plus a regression on a (potentially) large set of predictors, then comparing observed vs predicted post-intervention. With variable-selection priors (e.g., regularized horseshoe or spike-and-slab), the model keeps a small predictive subset and shrinks the rest. This looks similar to SC because both: (1) train only on the pre-period, (2) generate a post-period counterfactual, and (3) tend to be sparse so a few predictors dominate. If you restrict ITS covariates to donor outcomes and apply strong shrinkage, it approaches regression-based SC.

Key differences

• Coefficients: ITS uses unconstrained coefficients (can be positive or negative, need not sum to one). SC uses nonnegative weights that sum to one.
• Structure: ITS models level/trend/seasonality explicitly; SC relies on the donor mix to match dynamics.
• Predictors: ITS can mix donor outcomes with non-outcome covariates (weather, marketing, holidays). SC typically uses donor outcomes, optionally balancing covariates but not as outcome regressors.
• Extrapolation: ITS allows extrapolation via signed coefficients; SC restricts to a convex combination of donors, which limits extrapolation.
• When high-dimensional: Variable-selection priors make ITS practical when p≈n or p>n (p = number of candidate predictors; n = number of pre-period observations).