Scientists often want to learn about cause and effect from hierarchical data, collected from subunits nested inside units. Consider students in schools, cells in patients, or cities in states. In such settings, unit-level variables (e.g. each school's budget) may affect subunit-level variables (e.g. the test scores of each student in each school) and vice versa. To address causal questions with hierarchical data, we propose hierarchical causal models, which extend structural causal models and causal graphical models by adding inner plates. We develop a general graphical identification technique for hierarchical causal models that extends do-calculus. We find many situations in which hierarchical data can enable causal identification even when it would be impossible with non-hierarchical data, that is, if we had only unit-level summaries of subunit-level variables (e.g. the school's average test score, rather than each student's score). We develop estimation techniques for hierarchical causal models, using methods including hierarchical Bayesian models. We illustrate our results in simulation and via a reanalysis of the classic "eight schools" study.

We present a new Subset Simulation approach using Hamiltonian neural network-based Monte Carlo sampling for reliability analysis. The proposed strategy combines the superior sampling of the Hamiltonian Monte Carlo method with computationally efficient gradient evaluations using Hamiltonian neural networks. This combination is especially advantageous because the neural network architecture conserves the Hamiltonian, which defines the acceptance criteria of the Hamiltonian Monte Carlo sampler. Hence, this strategy achieves high acceptance rates at low computational cost. Our approach estimates small failure probabilities using Subset Simulations. However, in low-probability sample regions, the gradient evaluation is particularly challenging. The remarkable accuracy of the proposed strategy is demonstrated on different reliability problems, and its efficiency is compared to the traditional Hamiltonian Monte Carlo method. We note that this approach can reach its limitations for gradient estimations in low-probability regions of complex and high-dimensional distributions. Thus, we propose techniques to improve gradient prediction in these particular situations and enable accurate estimations of the probability of failure. The highlight of this study is the reliability analysis of a system whose parameter distributions must be inferred with Bayesian inference problems. In such a case, the Hamiltonian Monte Carlo method requires a full model evaluation for each gradient evaluation and, therefore, comes at a very high cost. However, using Hamiltonian neural networks in this framework replaces the expensive model evaluation, resulting in tremendous improvements in computational efficiency.

Gaussian processes provide a flexible, non-parametric framework for the approximation of functions in high-dimensional spaces. The covariance kernel is the main engine of Gaussian processes, incorporating correlations that underpin the predictive distribution. For applications with spatiotemporal datasets, suitable kernels should model joint spatial and temporal dependence. Separable space-time covariance kernels offer simplicity and computational efficiency. However, non-separable kernels include space-time interactions that better capture observed correlations. Most non-separable kernels that admit explicit expressions are based on mathematical considerations (admissibility conditions) rather than first-principles derivations. We present a hybrid spectral approach for generating covariance kernels which is based on physical arguments. We use this approach to derive a new class of physically motivated, non-separable covariance kernels which have their roots in the stochastic, linear, damped, harmonic oscillator (LDHO). The new kernels incorporate functions with both monotonic and oscillatory decay of space-time correlations. The LDHO covariance kernels involve space-time interactions which are introduced by dispersion relations that modulate the oscillator coefficients. We derive explicit relations for the spatiotemporal covariance kernels in the three oscillator regimes (underdamping, critical damping, overdamping) and investigate their properties. We further illustrate the hybrid spectral method by deriving covariance kernels that are based on the Ornstein-Uhlenbeck model.

Recent Vision-Language Pretrained (VLP) models have become the backbone for many downstream tasks, but they are utilized as frozen model without learning. Prompt learning is a method to improve the pre-trained VLP model by adding a learnable context vector to the inputs of the text encoder. In a few-shot learning scenario of the downstream task, MLE training can lead the context vector to over-fit dominant image features in the training data. This overfitting can potentially harm the generalization ability, especially in the presence of a distribution shift between the training and test dataset. This paper presents a Bayesian-based framework of prompt learning, which could alleviate the overfitting issues on few-shot learning application and increase the adaptability of prompts on unseen instances. Specifically, modeling data-dependent prior enhances the adaptability of text features for both seen and unseen image features without the trade-off of performance between them. Based on the Bayesian framework, we utilize the Wasserstein Gradient Flow in the estimation of our target posterior distribution, which enables our prompt to be flexible in capturing the complex modes of image features. We demonstrate the effectiveness of our method on benchmark datasets for several experiments by showing statistically significant improvements on performance compared to existing methods. The code is available at https://github.com/youngjae-cho/APP.

This study introduces a novel methodology for modelling patient emotions from online patient experience narratives. We employed metadata network topic modelling to analyse patient-reported experiences from Care Opinion, revealing key emotional themes linked to patient-caregiver interactions and clinical outcomes. We develop a probabilistic, context-specific emotion recommender system capable of predicting both multilabel emotions and binary sentiments using a naive Bayes classifier using contextually meaningful topics as predictors. The superior performance of our predicted emotions under this model compared to baseline models was assessed using the information retrieval metrics nDCG and Q-measure, and our predicted sentiments achieved an F1 score of 0.921, significantly outperforming standard sentiment lexicons. This method offers a transparent, cost-effective way to understand patient feedback, enhancing traditional collection methods and informing individualised patient care. Our findings are accessible via an R package and interactive dashboard, providing valuable tools for healthcare researchers and practitioners.

In this work, we propose an original method for aggregating multiple clustering coming from different sources of information. Each partition is encoded by a co-membership matrix between observations. Our approach uses a mixture of multilayer Stochastic Block Models (SBM) to group co-membership matrices with similar information into components and to partition observations into different clusters, taking into account their specificities within the components. The identifiability of the model parameters is established and a variational Bayesian EM algorithm is proposed for the estimation of these parameters. The Bayesian framework allows for selecting an optimal number of clusters and components. The proposed approach is compared using synthetic data with consensus clustering and tensor-based algorithms for community detection in large-scale complex networks. Finally, the method is utilized to analyze global food trading networks, leading to structures of interest.

We consider the problem of sampling from an unknown distribution for which only a sufficiently large number of training samples are available. Such settings have recently drawn considerable interest in the context of generative modelling. In this paper, we propose a generative model combining diffusion maps and Langevin dynamics. Diffusion maps are used to approximate the drift term from the available training samples, which is then implemented in a discrete-time Langevin sampler to generate new samples. By setting the kernel bandwidth to match the time step size used in the unadjusted Langevin algorithm, our method effectively circumvents any stability issues typically associated with time-stepping stiff stochastic differential equations. More precisely, we introduce a novel split-step scheme, ensuring that the generated samples remain within the convex hull of the training samples. Our framework can be naturally extended to generate conditional samples. We demonstrate the performance of our proposed scheme through experiments on synthetic datasets with increasing dimensions and on a stochastic subgrid-scale parametrization conditional sampling problem.

Federated Learning (FL) involves training a model over a dataset distributed among clients, with the constraint that each client's dataset is localized and possibly heterogeneous. In FL, small and noisy datasets are common, highlighting the need for well-calibrated models that represent the uncertainty of predictions. The closest FL techniques to achieving such goals are the Bayesian FL methods which collect parameter samples from local posteriors, and aggregate them to approximate the global posterior. To improve scalability for larger models, one common Bayesian approach is to approximate the global predictive posterior by multiplying local predictive posteriors. In this work, we demonstrate that this method gives systematically overconfident predictions, and we remedy this by proposing $\beta$-Predictive Bayes, a Bayesian FL algorithm that interpolates between a mixture and product of the predictive posteriors, using a tunable parameter $\beta$. This parameter is tuned to improve the global ensemble's calibration, before it is distilled to a single model. Our method is evaluated on a variety of regression and classification datasets to demonstrate its superiority in calibration to other baselines, even as data heterogeneity increases. Code available at https://github.com/hasanmohsin/betaPredBayesFL

The Laplace approximation provides a closed-form model selection objective for neural networks (NN). Online variants, which optimise NN parameters jointly with hyperparameters, like weight decay strength, have seen renewed interest in the Bayesian deep learning community. However, these methods violate Laplace's method's critical assumption that the approximation is performed around a mode of the loss, calling into question their soundness. This work re-derives online Laplace methods, showing them to target a variational bound on a mode-corrected variant of the Laplace evidence which does not make stationarity assumptions. Online Laplace and its mode-corrected counterpart share stationary points where 1. the NN parameters are a maximum a posteriori, satisfying the Laplace method's assumption, and 2. the hyperparameters maximise the Laplace evidence, motivating online methods. We demonstrate that these optima are roughly attained in practise by online algorithms using full-batch gradient descent on UCI regression datasets. The optimised hyperparameters prevent overfitting and outperform validation-based early stopping.

In this manuscript, we present various proposed methods estimate the prevalence of disease, a critical prerequisite for the adequate interpretation of screening tests. To address the limitations of these approaches, which revolve primarily around their a posteriori nature, we introduce a novel method to estimate the pretest probability of disease, a priori, utilizing the Logit function from the logistic regression model. This approach is a modification of McGee's heuristic, originally designed for estimating the posttest probability of disease. In a patient presenting with $n_\theta$ signs or symptoms, the minimal bound of the pretest probability, $\phi$, can be approximated by: $\phi \approx \frac{1}{5}{ln\left[\displaystyle\prod_{\theta=1}^{i}\kappa_\theta\right]}$ where $ln$ is the natural logarithm, and $\kappa_\theta$ is the likelihood ratio associated with the sign or symptom in question.