  In dynamic topic modeling, the proportional contribution of a topic to a
document depends on the temporal dynamics of that topic's overall prevalence in
the corpus. We extend the Dynamic Topic Model of Blei and Lafferty (2006) by
explicitly modeling document level topic proportions with covariates and
dynamic structure that includes polynomial trends and periodicity. A Markov
Chain Monte Carlo (MCMC) algorithm that utilizes Polya-Gamma data augmentation
is developed for posterior inference. Conditional independencies in the model
and sampling are made explicit, and our MCMC algorithm is parallelized where
possible to allow for inference in large corpora. To address computational
bottlenecks associated with Polya-Gamma sampling, we appeal to the Central
Limit Theorem to develop a Gaussian approximation to the Polya-Gamma random
variable. This approximation is fast and reliable for parameter values relevant
in the text mining domain. Our model and inference algorithm are validated with
multiple simulation examples, and we consider the application of modeling
trends in PubMed abstracts. We demonstrate that sharing information across
documents is critical for accurately estimating document-specific topic
proportions. We also show that explicitly modeling polynomial and periodic
behavior improves our ability to predict topic prevalence at future time
points.
