Week 05.10.2026 – 11.10.2026

The Monotonicity inequality, introduced by Krylov and Rozovskii (1979), is a tool for establishing the uniqueness of strong solutions to Stochastic PDEs. There is an extensive literature on utilising the Monotonicity inequality to obtain such uniqueness results in the setting of Hilbert spaces, which has been further extended to Stochastic PDEs in the dual of Nuclear spaces. Gawarecki, Mandrekar and Rajeev (2008) showed that such an inequality can also be used to show the existence of strong solutions for Linear Stochastic PDEs in the dual of Nuclear spaces. In this talk, we shall discuss cases where this inequality has been established, specifically in the space of tempered distributions. If time permits, we shall also discuss related results in the setting of non-linear Stochastic PDEs.

Bayesian factor models provide a flexible framework for the analysis of high-dimensional data, combining dimensionality reduction with probabilistic quantification of uncertainty and, importantly, allowing the latent dimension itself to be learned from the data. In recent years, sparse formulations have further expanded their scope. Imposed factor loadings sparsity, indeed, can improve interpretability and identifiability.

Motivated by applications involving multiple related studies, I will first discuss APAFA (Bortolato and Canale, 2026), a Bayesian multi-study factor model designed to flexibly learn which latent components are shared across studies and which are study-specific, while avoiding overly restrictive assumptions on their dependence structure.

Building on this, I will then turn to a complementary perspective in which sparsity is imposed on the factor scores. This leads to a new class of identifiable sparse factor models in which interpretable latent structure can be recovered without relying on conventional constraints on the loadings.

Together, these developments illustrate how different notions of sparsity can be exploited to address some of the central inferential challenges in modern Bayesian factor modelling.

Bortolato, E., & Canale, A. (2026). Adaptive partition factor analysis. Journal of the American Statistical Association, http://doi.org/10.1080/01621459.2026.2621517