Meta-learning is a framework in which machine learning models train over a set of datasets in order to produce predictions on new datasets at test time. Probabilistic meta-learning has received an abundance of attention from the research community in recent years, but a problem shared by many existing probabilistic meta-models is that they require a very large number of datasets in order to produce high-quality predictions with well-calibrated uncertainty estimates. In many applications, however, such quantities of data are simply not available. In this dissertation we present a significantly more data-efficient approach to probabilistic meta-learning through per-datapoint amortisation of inference in Bayesian neural networks, introducing the Amortised Pseudo-Observation Variational Inference Bayesian Neural Network (APOVI-BNN). First, we show that the approximate posteriors obtained under our amortised scheme are of similar or better quality to those obtained through traditional variational inference, despite the fact that the amortised inference is performed in a single forward pass. We then discuss how the APOVI-BNN may be viewed as a new member of the neural process family, motivating the use of neural process training objectives for potentially better predictive performance on complex problems as a result. Finally, we assess the predictive performance of the APOVI-BNN against other probabilistic meta-models in both a one-dimensional regression problem and in a significantly more complex image completion setting. In both cases, when the amount of training data is limited, our model is the best in its class.

Score-based approaches in the structure learning task are thriving because of their scalability. Continuous relaxation has been the key reason for this advancement. Despite achieving promising outcomes, most of these methods are still struggling to ensure that the graphs generated from the latent space are acyclic by minimizing a defined score. There has also been another trend of permutation-based approaches, which concern the search for the topological ordering of the variables in the directed acyclic graph in order to limit the search space of the graph. In this study, we propose an alternative approach for strictly constraining the acyclicty of the graphs with an integration of the knowledge from the topological orderings. Our approach can reduce inference complexity while ensuring the structures of the generated graphs to be acyclic. Our empirical experiments with simulated and real-world data show that our approach can outperform related Bayesian score-based approaches.

In Bayesian statistics, posterior contraction rates (PCRs) quantify the speed at which the posterior distribution concentrates on arbitrarily small neighborhoods of a true model, in a suitable way, as the sample size goes to infinity. In this paper, we develop a new approach to PCRs, with respect to strong norm distances on parameter spaces of functions. Critical to our approach is the combination of a local Lipschitz-continuity for the posterior distribution with a dynamic formulation of the Wasserstein distance, which allows to set forth an interesting connection between PCRs and some classical problems arising in mathematical analysis, probability and statistics, e.g., Laplace methods for approximating integrals, Sanov's large deviation principles in the Wasserstein distance, rates of convergence of mean Glivenko-Cantelli theorems, and estimates of weighted Poincar\'e-Wirtinger constants. We first present a theorem on PCRs for a model in the regular infinite-dimensional exponential family, which exploits sufficient statistics of the model, and then extend such a theorem to a general dominated model. These results rely on the development of novel techniques to evaluate Laplace integrals and weighted Poincar\'e-Wirtinger constants in infinite-dimension, which are of independent interest. The proposed approach is applied to the regular parametric model, the multinomial model, the finite-dimensional and the infinite-dimensional logistic-Gaussian model and the infinite-dimensional linear regression. In general, our approach leads to optimal PCRs in finite-dimensional models, whereas for infinite-dimensional models it is shown explicitly how the prior distribution affect PCRs.

The efficient representation of random fields on geometrically complex domains is crucial for Bayesian modelling in engineering and machine learning. Today's prevalent random field representations are either intended for unbounded domains or are too restrictive in terms of possible field properties. Because of these limitations, techniques leveraging the historically established link between stochastic PDEs (SPDEs) and random fields have been gaining interest. The SPDE representation is especially appealing for engineering applications which already have a finite element discretisation for solving the physical conservation equations. In contrast to the dense covariance matrix of a random field, its inverse, the precision matrix, is usually sparse and equal to the stiffness matrix of an elliptic SPDE. We use the SPDE representation to develop a scalable framework for large-scale statistical finite element analysis and Gaussian process (GP) regression on complex geometries. The statistical finite element method (statFEM) introduced by Girolami et al. (2022) is a novel approach for synthesising measurement data and finite element models. In both statFEM and GP regression, we use the SPDE formulation to obtain the relevant prior probability densities with a sparse precision matrix. The properties of the priors are governed by the parameters and possibly fractional order of the SPDE so that we can model on bounded domains and manifolds anisotropic, non-stationary random fields with arbitrary smoothness. The observation models for statFEM and GP regression are such that the posterior probability densities are Gaussians with a closed-form mean and precision. The respective mean vector and precision matrix and can be evaluated using only sparse matrix operations. We demonstrate the versatility of the proposed framework and its convergence properties with Poisson and thin-shell examples.

Subsethood, which is to measure the degree of set inclusion relation, is predominant in fuzzy set theory. This paper introduces some basic concepts of spatial granules, coarse-fine relation, and operations like meet, join, quotient meet and quotient join. All the atomic granules can be hierarchized by set-inclusion relation and all the granules can be hierarchized by coarse-fine relation. Viewing an information system from the micro and the macro perspectives, we can get a micro knowledge space and a micro knowledge space, from which a rough set model and a spatial rough granule model are respectively obtained. The classical rough set model is the special case of the rough set model induced from the micro knowledge space, while the spatial rough granule model will be play a pivotal role in the problem-solving of structures. We discuss twelve axioms of monotone increasing subsethood and twelve corresponding axioms of monotone decreasing supsethood, and generalize subsethood and supsethood to conditional granularity and conditional fineness respectively. We develop five conditional granularity measures and five conditional fineness measures and prove that each conditional granularity or fineness measure satisfies its corresponding twelve axioms although its subsethood or supsethood measure only hold one of the two boundary conditions. We further define five conditional granularity entropies and five conditional fineness entropies respectively, and each entropy only satisfies part of the boundary conditions but all the ten monotone conditions.

We consider the problem of optimising an expensive-to-evaluate grey-box objective function, within a finite budget, where known side-information exists in the form of the causal structure between the design variables. Standard black-box optimisation ignores the causal structure, often making it inefficient and expensive. The few existing methods that consider the causal structure are myopic and do not fully accommodate the observation-intervention trade-off that emerges when estimating causal effects. In this paper, we show that the observation-intervention trade-off can be formulated as a non-myopic optimal stopping problem which permits an efficient solution. We give theoretical results detailing the structure of the optimal stopping times and demonstrate the generality of our approach by showing that it can be integrated with existing causal Bayesian optimisation algorithms. Experimental results show that our formulation can enhance existing algorithms on real and synthetic benchmarks.

Multivariate Hawkes processes (MHPs) are versatile probabilistic tools used to model various real-life phenomena: earthquakes, operations on stock markets, neuronal activity, virus propagation and many others. In this paper, we focus on MHPs with exponential decay kernels and estimate connectivity graphs, which represent the Granger causal relations between their components. We approach this inference problem by proposing an optimization criterion and model selection algorithm based on the minimum message length (MML) principle. MML compares Granger causal models using the Occam's razor principle in the following way: even when models have a comparable goodness-of-fit to the observed data, the one generating the most concise explanation of the data is preferred. While most of the state-of-art methods using lasso-type penalization tend to overfitting in scenarios with short time horizons, the proposed MML-based method achieves high F1 scores in these settings. We conduct a numerical study comparing the proposed algorithm to other related classical and state-of-art methods, where we achieve the highest F1 scores in specific sparse graph settings. We illustrate the proposed method also on G7 sovereign bond data and obtain causal connections, which are in agreement with the expert knowledge available in the literature.

Many decision problems cannot be solved exactly and use several estimation algorithms that assign scores to the different available options. The estimation errors can have various correlations, from low (e.g. between two very different approaches) to high (e.g. when using a given algorithm with different hyperparameters). Most aggregation rules would suffer from this diversity of correlations. In this article, we propose different aggregation rules that take correlations into account, and we compare them to naive rules in various experiments based on synthetic data. Our results show that when sufficient information is known about the correlations between errors, a maximum likelihood aggregation should be preferred. Otherwise, typically with limited training data, we recommend a method that we call Embedded Voting (EV).

Causal probabilistic graph-based models have gained widespread utility, enabling the modeling of cause-and-effect relationships across diverse domains. With their rising adoption in new areas, such as automotive system safety and machine learning, the need for an integrated lifecycle framework akin to DevOps and MLOps has emerged. Currently, a process reference for organizations interested in employing causal engineering is missing. To address this gap and foster widespread industrial adoption, we propose CausalOps, a novel lifecycle framework for causal model development and application. By defining key entities, dependencies, and intermediate artifacts generated during causal engineering, we establish a consistent vocabulary and workflow model. This work contextualizes causal model usage across different stages and stakeholders, outlining a holistic view of creating and maintaining them. CausalOps' aim is to drive the adoption of causal methods in practical applications within interested organizations and the causality community.

Learning causal effects from data is a fundamental and well-studied problem across science, especially when the cause-effect relationship is static in nature. However, causal effect is less explored when there are dynamical dependencies, i.e., when dependencies exist between entities across time. Identifying dynamic causal effects from time-series observations is computationally expensive when compared to the static scenario. We demonstrate that the computational complexity of recovering the causation structure for the vector auto-regressive (VAR) model is $O(Tn^3N^2)$, where $n$ is the number of nodes, $T$ is the number of samples, and $N$ is the largest time-lag in the dependency between entities. We report a method, with a reduced complexity of $O(Tn^3 \log N)$, to recover the causation structure to obtain frequency-domain (FD) representations of time-series. Since FFT accumulates all the time dependencies on every frequency, causal inference can be performed efficiently by considering the state variables as random variables at any given frequency. We additionally show that, for systems with interactions that are LTI, do-calculus machinery can be realized in the FD resulting in versions of the classical single-door (with cycles), front and backdoor criteria. We demonstrate, for a large class of problems, graph reconstruction using multivariate Wiener projections results in a significant computational advantage with $O(n)$ complexity over reconstruction algorithms such as the PC algorithm which has $O(n^q)$ complexity, where $q$ is the maximum neighborhood size. This advantage accrues due to some remarkable properties of the phase response of the frequency-dependent Wiener coefficients which is not present in any time-domain approach.