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