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.

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