Our method extends recent developments in simulation-based inference (SBI) based on normalizing flows to Bayesian hierarchical models.

Accuracy and generalization capabilities are key objectives when learning dynamical system models. To obtain such models from limited data, current works exploit prior knowledge and assumptions about the system. However, the fusion of diverse prior knowledge, e. g. partially known system equations and smoothness assumptions about unknown model parts, with information contained in the data remains a challenging problem, especially in input-output settings with latent system state. In particular, learning functions that are nested inside known system equations can be a laborious and error-prone expert task. This paper considers inference of latent states and learning of unknown model parts for fusion of data information with different sources of prior knowledge. The main contribution is a general-purpose system identification tool that, for the first time, provides a consistent solution for both, online and offline Bayesian inference and learning while allowing to incorporate explicit and implicit prior system knowledge. We propose a novel interface for combining known dynamics functions with a learning-based approximation of unknown system parts. Based on the proposed model structure, closed-form densities for efficient parameter marginalization are derived. No user-tailored coordinate transformations or model inversions are needed, making the presented framework a general-purpose tool for inference and learning. The broad applicability of the devised framework is illustrated in three distinct case studies, including an experimental data set.

We propose a novel class of prior distributions for sequences of orthogonal functions, which are frequently required in various statistical models such as functional principal component analysis (FPCA). Our approach constructs priors sequentially by imposing adaptive orthogonality constraints through a hierarchical formulation of conditionally normal distributions. The orthogonality is controlled via hyperparameters, allowing for flexible trade-offs between exactness and smoothness, which can be learned from the observed data. We illustrate the properties of the proposed prior and show that it leads to nearly orthogonal posterior estimates. The proposed prior is employed in Bayesian FPCA, providing more interpretable principal functions and efficient low-rank representations. Through simulation studies and analysis of human mobility data in Tokyo, we demonstrate the superior performance of our approach in inducing orthogonality and improving functional component estimation.

This work investigates the generalization behavior of deep neural networks (DNNs), focusing on the phenomenon of "fooling examples," where DNNs confidently classify inputs that appear random or unstructured to humans. To explore this phenomenon, we introduce an analytical framework based on maximum likelihood estimation, without adhering to conventional numerical approaches that rely on gradient-based optimization and explicit labels. Our analysis reveals that DNNs operating in an overparameterized regime exhibit a collapse in the output feature space. While this collapse improves network generalization, adding more layers eventually leads to a state of degeneracy, where the model learns trivial solutions by mapping distinct inputs to the same output, resulting in zero loss. Further investigation demonstrates that this degeneracy can be bypassed using our newly derived "wormhole" solution. The wormhole solution, when applied to arbitrary fooling examples, reconciles meaningful labels with random ones and provides a novel perspective on shortcut learning. These findings offer deeper insights into DNN generalization and highlight directions for future research on learning dynamics in unsupervised settings to bridge the gap between theory and practice.

With the advent of wearable recorders, scientists are increasingly turning to automated methods of analysis of audio and video data in order to measure children's experience, behavior, and outcomes, with a sizable literature employing long-form audio-recordings to study language acquisition. While numerous articles report on the accuracy and reliability of the most popular automated classifiers, less has been written on the downstream effects of classification errors on measurements and statistical inferences (e.g., the estimate of correlations and effect sizes in regressions). This paper proposes a Bayesian approach to study the effects of algorithmic errors on key scientific questions, including the effect of siblings on children's language experience and the association between children's production and their input. In both the most commonly used \gls{lena}, and an open-source alternative (the Voice Type Classifier from the ACLEW system), we find that classification errors can significantly distort estimates. For instance, automated annotations underestimated the negative effect of siblings on adult input by 20--80\%, potentially placing it below statistical significance thresholds. We further show that a Bayesian calibration approach for recovering unbiased estimates of effect sizes can be effective and insightful, but does not provide a fool-proof solution. Both the issue reported and our solution may apply to any classifier involving event detection and classification with non-zero error rates.