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High dimensional categorical data are routinely collected in biomedical andsocial sciences. It is of great importance to build interpretable parsimoniousmodels that perform dimension reduction and uncover meaningful latentstructures from such discrete data. Identifiability is a fundamentalrequirement for valid modeling and inference in such scenarios, yet ischallenging to address when there are complex latent structures. In thisarticle, we propose a class of identifiable multilayer (potentially deep)discrete latent structure models for discrete data, termed Bayesian pyramids.We establish the identifiability of Bayesian pyramids by developing noveltransparent conditions on the pyramid-shaped deep latent directed graph. Theproposed identifiability conditions can ensure Bayesian posterior consistencyunder suitable priors. As an illustration, we consider the two-latent-layermodel and propose a Bayesian shrinkage estimation approach. Simulation resultsfor this model corroborate the identifiability and estimability of modelparameters. Applications of the methodology to DNA nucleotide sequence datauncover useful discrete latent features that are highly predictive of sequencetypes. The proposed framework provides a recipe for interpretable unsupervisedlearning of discrete data, and can be a useful alternative to popular machinelearning methods.

Bayesian Pyramids: Identifiable Multilayer Discrete Latent Structure Models for Discrete Data