Neural marked temporal point processes have been a valuable addition to the existing toolbox of statistical parametric models for continuous-time event data. These models are useful for sequences where each event is associated with a single item (a single type of event or a "mark") -- but such models are not suited for the practical situation where each event is associated with a set of items. In this work, we develop a general framework for modeling set-valued data in continuous-time, compatible with any intensity-based recurrent neural point process model. In addition, we develop inference methods that can use such models to answer probabilistic queries such as "the probability of item $A$ being observed before item $B$," conditioned on sequence history. Computing exact answers for such queries is generally intractable for neural models due to both the continuous-time nature of the problem setting and the combinatorially-large space of potential outcomes for each event. To address this, we develop a class of importance sampling methods for querying with set-based sequences and demonstrate orders-of-magnitude improvements in efficiency over direct sampling via systematic experiments with four real-world datasets. We also illustrate how to use this framework to perform model selection using likelihoods that do not involve one-step-ahead prediction.

Bayesian networks (BNs) are a foundational model in machine learning and causal inference. Their graphical structure can handle high-dimensional problems, divide them into a sparse collection of smaller ones, underlies Judea Pearl's causality, and determines their explainability and interpretability. Despite their popularity, there are almost no resources in the literature on how to compute Shannon's entropy and the Kullback-Leibler (KL) divergence for BNs under their most common distributional assumptions. In this paper, we provide computationally efficient algorithms for both by leveraging BNs' graphical structure, and we illustrate them with a complete set of numerical examples. In the process, we show it is possible to reduce the computational complexity of KL from cubic to quadratic for Gaussian BNs.

Gradient-enhanced Kriging (GE-Kriging) is a well-established surrogate modelling technique for approximating expensive computational models. However, it tends to get impractical for high-dimensional problems due to the size of the inherent correlation matrix and the associated high-dimensional hyper-parameter tuning problem. To address these issues, a new method, called sliced GE-Kriging (SGE-Kriging), is developed in this paper for reducing both the size of the correlation matrix and the number of hyper-parameters. We first split the training sample set into multiple slices, and invoke Bayes' theorem to approximate the full likelihood function via a sliced likelihood function, in which multiple small correlation matrices are utilized to describe the correlation of the sample set rather than one large one. Then, we replace the original high-dimensional hyper-parameter tuning problem with a low-dimensional counterpart by learning the relationship between the hyper-parameters and the derivative-based global sensitivity indices. The performance of SGE-Kriging is finally validated by means of numerical experiments with several benchmarks and a high-dimensional aerodynamic modeling problem. The results show that the SGE-Kriging model features an accuracy and robustness that is comparable to the standard one but comes at much less training costs. The benefits are most evident for high-dimensional problems with tens of variables.

To further reduce the computational effort in evaluating the approximate posterior, stochastic methods such as We introduce ensembles of stochastic neural networks to Monte Carlo dropout [47] and DropConnect [51] inference approximate the Bayesian posterior, combining stochastic have also been used extensively [13, 14, 40]. They benefit methods such as dropout with deep ensembles. The stochastic from computationally cheaper inference by virtue of sampling ensembles are formulated as families of distributions and stochastically from a single model. Formulated as a trained to approximate the Bayesian posterior with variational variational approximation to the posterior, dropout samples inference. We implement stochastic ensembles based from a family of parameter distributions where parameters on Monte Carlo dropout, DropConnect and a novel nonparametric can be randomly set to zero. Although this particular family version of dropout and evaluate them on a toy of distributions might seem unnatural [11], it turns out problem and CIFAR image classification. For both tasks, that the stochastic property can help to find more robust regions we test the quality of the posteriors directly against Hamiltonian of the parameter space, a fact well-known from a long Monte Carlo simulations. Our results show that history of using dropout as a regularization method.

We present a unified framework for deriving PAC-Bayesian generalization bounds. Unlike most previous literature on this topic, our bounds are anytime-valid (i.e., time-uniform), meaning that they hold at all stopping times, not only for a fixed sample size. Our approach combines four tools in the following order: (a) nonnegative supermartingales or reverse submartingales, (b) the method of mixtures, (c) the Donsker-Varadhan formula (or other convex duality principles), and (d) Ville's inequality. Our main result is a PAC-Bayes theorem which holds for a wide class of discrete stochastic processes. We show how this result implies time-uniform versions of well-known classical PAC-Bayes bounds, such as those of Seeger, McAllester, Maurer, and Catoni, in addition to many recent bounds. We also present several novel bounds. Our framework also enables us to relax traditional assumptions; in particular, we consider nonstationary loss functions and non-i.i.d. data. In sum, we unify the derivation of past bounds and ease the search for future bounds: one may simply check if our supermartingale or submartingale conditions are met and, if so, be guaranteed a (time-uniform) PAC-Bayes bound.

For situations that may benefit from information sharing among datasets, e.g., population-based SHM of similar structures, the hierarchical Bayesian approach provides a useful modelling structure. Hierarchical Bayesian models learn statistical distributions at the population (or parent) and the domain levels simultaneously, to bolster statistical strength among the parameters. As a result, variance is reduced among the parameter estimates, particularly when data are limited. In this paper, a combined probabilistic FRF model is developed for a small population of nominally-identical helicopter blades, using a hierarchical Bayesian structure, to support information transfer in the context of sparse data. The modelling approach is also demonstrated in a traditional SHM context, for a single helicopter blade exposed to varying temperatures, to show how the inclusion of physics-based knowledge can improve generalisation beyond the training data, in the context of scarce data. These models address critical challenges in SHM, by accommodating benign variations that present as differences in the underlying dynamics, while also considering (and utilising), the similarities among the domains.

Causal inference is a crucial goal of science, enabling researchers to arrive at meaningful conclusions regarding the predictions of hypothetical interventions using observational data. Path models, Structural Equation Models (SEMs), and, more generally, Directed Acyclic Graphs (DAGs), provide a means to unambiguously specify assumptions regarding the causal structure underlying a phenomenon. Unlike DAGs, which make very few assumptions about the functional and parametric form, SEM assumes linearity. This can result in functional misspecification which prevents researchers from undertaking reliable effect size estimation. In contrast, we propose Super Learner Equation Modeling, a path modeling technique integrating machine learning Super Learner ensembles. We empirically demonstrate its ability to provide consistent and unbiased estimates of causal effects, its competitive performance for linear models when compared with SEM, and highlight its superiority over SEM when dealing with non-linear relationships. We provide open-source code, and a tutorial notebook with example usage, accentuating the easy-to-use nature of the method.

This paper presents a series of new results for domain adaptation in the multi-view learning setting. The incorporati on of multiple views in the domain adaptation was paid little attention in t he previous studies. In this way, we propose an analysis of generaliz ation bounds with Pac-Bayesian theory to consolidate the two paradigms, which are currently treated separately. Firstly, building on previo us work by Ger-main et al. [7,8], we adapt the distance between distributio n proposed by Germain et al. for domain adaptation with the concept of mu lti-view learning. Thus, we introduce a novel distance that is ta ilored for the multi-view domain adaptation setting. Then, we give Pac -Bayesian bounds for estimating the introduced divergence. Finally, we compare the different new bounds with the previous studies.

Recently, Gaussian processes have been used to model the vector field of continuous dynamical systems, referred to as GPODEs, which are characterized by a probabilistic ODE equation. Bayesian inference for these models has been extensively studied and applied in tasks such as time series prediction. However, the use of standard GPs with basic kernels like squared exponential kernels has been common in GPODE research, limiting the model's ability to represent complex scenarios. To address this limitation, we introduce normalizing flows to reparameterize the ODE vector field, resulting in a data-driven prior distribution, thereby increasing flexibility and expressive power. We develop a data-driven variational learning algorithm that utilizes analytically tractable probability density functions of normalizing flows, enabling simultaneous learning and inference of unknown continuous dynamics. Additionally, we also apply normalizing flows to the posterior inference of GP ODEs to resolve the issue of strong mean-field assumptions in posterior inference. By applying normalizing flows in both these ways, our model improves accuracy and uncertainty estimates for Bayesian Gaussian Process ODEs. We validate the effectiveness of our approach on simulated dynamical systems and real-world human motion data, including time series prediction and missing data recovery tasks. Experimental results show that our proposed method effectively captures model uncertainty while improving accuracy.

Quantification (variously called learning to quantify or class prevalence estimation) is the area of supervised machine learning concerned with estimating the percentages of instances from a population (hereafter, a bag of examples) belonging to each of the classes of interest [González et al., 2017, Esuli et al., 2023]. Quantification finds applications in many disciplines, like the social sciences, epidemiology, or market research, in which the interest lies at the aggregate level, i.e., in which inferring characteristics of the single individual (e.g., via classification, or via regression) is of little concern since knowing group-level information is all we need. Despite the fact that binary quantification (i.e., the setting in which the classes of interest are positive vs. negative) has been, by far, the most studied scenario in the quantification literature [Card and Smith, 2018, Forman, 2008, Bella et al., 2010, Esuli and Sebastiani, 2015, Hassan et al., 2020, Moreo and Sebastiani, 2021], the truth is that many of the applications of quantification naturally arise in the multiclass regime, i.e., in cases in which there are more than two mutually exclusive classes. Examples of multiclass settings are ubiquitous, and may include the allocation of human resources to different departments in a company [Forman, 2005], the analysis of different phytoplankton species that could exist in a water sample [González et al., 2019], or the analysis of the various causes of death studied in verbal autopsies [King and Lu, 2008], to name a few. A more concrete example could consist of providing answers to questions like: "What is the percentage of tweets conveying positive, neutral, and negative opinions concerning a specific hashtag?"