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In prediction problems with more predictors than observations, it cansometimes be helpful to use a joint probability model, $\pi(Y,X)$, rather thana purely conditional model, $\pi(Y \mid X)$, where $Y$ is a scalar responsevariable and $X$ is a vector of predictors. This approach is motivated by thefact that in many situations the marginal predictor distribution $\pi(X)$ canprovide useful information about the parameter values governing the conditionalregression. However, under very mild misspecification, this marginaldistribution can also lead conditional inferences astray. Here, we explorethese ideas in the context of linear factor models, to understand how they playout in a familiar setting. The resulting Bayesian model performs well across awide range of covariance structures, on real and simulated data.

Predictor-dependent shrinkage for linear regression via partial factor modeling