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Many scientific fields collect longitudinal count compositional data. Each observation is a multivariate count vector, where the total counts are arbitrary, and the information lies in the relative frequency of the counts. Multiple authors have proposed Bayesian Multinomial Logistic-Normal Dynamic Linear Models (MLN-DLMs) as a flexible approach to modeling these data. However, adoption of these methods has been limited by computational challenges. This article develops an efficient and accurate approach to posterior state estimation, called $\textit{Fenrir}$. Our approach relies on a novel algorithm for MAP estimation and an accurate approximation to a key posterior marginal of the model. As there are no equivalent methods against which we can compare, we also develop an optimized Stan implementation of MLN-DLMs. Our experiments suggest that Fenrir can be three orders of magnitude more efficient than Stan and can even be incorporated into larger sampling schemes for joint inference of model hyperparameters. Our methods are made available to the community as a user-friendly software library written in C++ with an R interface.

Statistical analysis of microbiome data is challenging. Bayesian multinomial logistic-normal (MLN) models have gained popularity due to their ability to account for the count compositional nature of these data. However, these models are often computationally intractable to infer. Recently, we developed a computationally efficient and accurate approach to inferring MLN models with a Marginally Latent Matrix-T Process (MLTP) form: MLN-MLTPs. Our approach is based on a novel sampler with a marginal Laplace approximation -- called the \textit{Collapse-Uncollapse} (CU) sampler. However, existing work with MLTPs has been limited to linear models or models of a single non-linear process. Moreover, existing methods lack an efficient means of estimating model hyperparameters. This article addresses both deficiencies. We introduce a new class of MLN Additive Gaussian Process models (\textit{MultiAddGPs}) for deconvolution of overlapping linear and non-linear processes. We show that MultiAddGPs are examples of MLN-MLTPs and derive an efficient CU sampler for this model class. Moreover, we derive efficient Maximum Marginal Likelihood estimation for hyperparameters in MLTP models by taking advantage of Laplace approximation in the CU sampler. We demonstrate our approach using simulated and real data studies. Our models produce novel biological insights from a previously published artificial gut study.

Scientific fields such as genomics, ecology, and political science often collect multivariate count data. In these fields, the data are often sufficiently noisy such that inferences regarding the total size of the measured systems have substantial uncertainty. This uncertainty can hinder downstream analyses, such as differential analysis in case-control studies. There have historically been two approaches to this problem: one considers the data as compositional and the other as counts that can be normalized. In this article, we use the framework of partially identified models to rigorously study the types of scientific questions (estimands) that can be answered (estimated) using these data. We prove that satisfying Frequentist inferential criteria is impossible for many estimation problems. In contrast, we find that the criteria for Bayesian inference can be satisfied, yet it requires a particular type of model called a Bayesian partially identified model. We introduce Scale Simulation Random Variables as a flexible and computationally efficient form of Bayesian partially identified models for analyzing these data. We use simulations and data analysis to validate our theory.

Bayesian multinomial logistic-normal (MLN) models are popular for the analysis of sequence count data (e.g., microbiome or gene expression data) due to their ability to model multivariate count data with complex covariance structure. However, existing implementations of MLN models are limited to handling small data sets due to the non-conjugacy of the multinomial and logistic-normal distributions. We introduce MLN models which can be written as marginally latent matrix-t process (LTP) models. Marginally LTP models describe a flexible class of generalized linear regression, non-linear regression, and time series models. We develop inference schemes for Marginally LTP models and, through application to MLN models, demonstrate that our inference schemes are both highly accurate and often 4-5 orders of magnitude faster than MCMC.

Size-constrained clustering (SCC) refers to the dual problem of using observations to determine latent cluster structure while at the same time assigning observations to the unknown clusters subject to an analyst defined constraint on cluster sizes. While several approaches have been proposed, SCC remains a difficult problem due to the combinatorial dependency between observations introduced by the size-constraints. Here we reformulate SCC as a decision problem and introduce a novel loss function to capture various types of size constraints. As opposed to prior work, our approach is uniquely suited to situations in which size constraints reflect and external limitation or desire rather than an internal feature of the data generation process. To demonstrate our approach, we develop a Bayesian mixture model for clustering respondents using both simulated and real categorical survey data. Our motivation for the development of this decision theoretic approach to SCC was to determine optimal team assignments for a Harry Potter themed scavenger hunt based on categorical survey data from participants.