  Distributed Lag Models (DLMs) and similar regression approaches such as MIDAS
have been used for many decades in econometrics, and more recently in the study
of air quality and its impact on human health. They are useful not only for
quantifying accumulating and delayed effects, but also for estimating the lags
that are most susceptible to these effects. Among other things, they have been
used to infer the period of exposure to poor air quality which might negatively
impact child birth weight. The increased attention DLMs have received in recent
years is reflective of their potential to help us understand a great many
issues, particularly in the investigation of how the environment affects human
health. In this paper we describe how to expand the utility of these models for
Bayesian inference by leveraging latent-variables. In particular we explain how
to perform binary regression to better handle imbalanced data, how to
incorporate negative binomial regression, and how to estimate the probability
of predictor inclusion. Extra parameters introduced through the DLM framework
may require calibration for the MCMC algorithm, but this will not be the case
in DLM-based analyses often seen in pollution exposure literature. In these
cases, the parameters are inferred through a fully automatic Gibbs sampling
procedure.
