Uncertainty representation and quantification are paramount in machine learning and constitute an important prerequisite for safety-critical applications. In this paper, we propose novel measures for the quantification of aleatoric and epistemic uncertainty based on proper scoring rules, which are loss functions with the meaningful property that they incentivize the learner to predict ground-truth (conditional) probabilities. We assume two common representations of (epistemic) uncertainty, namely, in terms of a credal set, i.e. a set of probability distributions, or a second-order distribution, i.e., a distribution over probability distributions. Our framework establishes a natural bridge between these representations. We provide a formal justification of our approach and introduce new measures of epistemic and aleatoric uncertainty as concrete instantiations.

Observable operator models (OOMs) offer a powerful framework for modelling stochastic processes, surpassing the traditional hidden Markov models (HMMs) in generality and efficiency. However, using OOMs to model infinite-dimensional processes poses significant theoretical challenges. This article explores a rigorous approach to developing an approximation theory for OOMs of infinite-dimensional processes. Building upon foundational work outlined in an unpublished tutorial [Jae98], an inner product structure on the space of future distributions is rigorously established and the continuity of observable operators with respect to the associated 2-norm is proven. The original theorem proven in this thesis describes a fundamental obstacle in making an infinite-dimensional space of future distributions into a Hilbert space. The presented findings lay the groundwork for future research in approximating observable operators of infinite-dimensional processes, while a remedy to the encountered obstacle is suggested.

We propose a self-supervised learning approach for solving the following constrained optimization task in log-linear models or Markov networks. Let $f$ and $g$ be two log-linear models defined over the sets $\mathbf{X}$ and $\mathbf{Y}$ of random variables respectively. Given an assignment $\mathbf{x}$ to all variables in $\mathbf{X}$ (evidence) and a real number $q$, the constrained most-probable explanation (CMPE) task seeks to find an assignment $\mathbf{y}$ to all variables in $\mathbf{Y}$ such that $f(\mathbf{x}, \mathbf{y})$ is maximized and $g(\mathbf{x}, \mathbf{y})\leq q$. In our proposed self-supervised approach, given assignments $\mathbf{x}$ to $\mathbf{X}$ (data), we train a deep neural network that learns to output near-optimal solutions to the CMPE problem without requiring access to any pre-computed solutions. The key idea in our approach is to use first principles and approximate inference methods for CMPE to derive novel loss functions that seek to push infeasible solutions towards feasible ones and feasible solutions towards optimal ones. We analyze the properties of our proposed method and experimentally demonstrate its efficacy on several benchmark problems.

Forecasting tasks using large datasets gathering thousands of heterogeneous time series is a crucial statistical problem in numerous sectors. The main challenge is to model a rich variety of time series, leverage any available external signals and provide sharp predictions with statistical guarantees. In this work, we propose a new forecasting model that combines discrete state space hidden Markov models with recent neural network architectures and training procedures inspired by vector quantized variational autoencoders. We introduce a variational discrete posterior distribution of the latent states given the observations and a two-stage training procedure to alternatively train the parameters of the latent states and of the emission distributions. By learning a collection of emission laws and temporarily activating them depending on the hidden process dynamics, the proposed method allows to explore large datasets and leverage available external signals. We assess the performance of the proposed method using several datasets and show that it outperforms other state-of-the-art solutions.

This paper considers a distributed adaptive optimization problem, where all agents only have access to their local cost functions with a common unknown parameter, whereas they mean to collaboratively estimate the true parameter and find the optimal solution over a connected network. A general mathematical framework for such a problem has not been studied yet. We aim to provide valuable insights for addressing parameter uncertainty in distributed optimization problems and simultaneously find the optimal solution. Thus, we propose a novel Prediction while Optimization scheme, which utilizes distributed fractional Bayesian learning through weighted averaging on the log-beliefs to update the beliefs of unknown parameters, and distributed gradient descent for renewing the estimation of the optimal solution. Then under suitable assumptions, we prove that all agents' beliefs and decision variables converge almost surely to the true parameter and the optimal solution under the true parameter, respectively. We further establish a sublinear convergence rate for the belief sequence. Finally, numerical experiments are implemented to corroborate the theoretical analysis.

Numerical lattice quantum chromodynamics (QCD) is an integral part of the modern particle and nuclear theory toolkit [1-9]. In this framework, the discretized path integral is computed using Monte Carlo methods. Computationally, this is very expensive, and grows more so as physical limits of interest are approached [10-12]. Consequently, algorithmic developments are an important driver of progress. For example, resolving topological freezing [12-14]--an issue that arises in sampling gauge field configurations with state-of-the-art Markov chain Monte Carlo (MCMC) algorithms like heatbath [15-19] or Hybrid/Hamiltonian Monte Carlo (HMC) [20-22]--would provide access to finer lattice spacings than presently affordable. To such ends, recent work has explored how emerging machine learning (ML) techniques may be applied to lattice QCD [23, 24]. Of particular interest has been the possibility of accelerating gauge-field sampling [25-34] using normalizing flows [35-37], a class of generative statistical models with tractable density functions. In this framework, a flow is a learned, invertible (diffeomorphic) map between gauge fields. Abstractly, flows may be thought of as bridges between different distributions over gauge fields (or, equivalently, different theories or choices of action parameters).

This paper looks at predictability problems, i.e., wherein an agent must choose its strategy in order to optimize the predictions that an external observer could make. We address these problems while taking into account uncertainties on the environment dynamics and on the observed agent's policy. To that end, we assume that the observer 1. seeks to predict the agent's future action or state at each time step, and 2. models the agent using a stochastic policy computed from a known underlying problem, and we leverage on the framework of observer-aware Markov decision processes (OAMDPs). We propose action and state predictability performance criteria through reward functions built on the observer's belief about the agent policy; show that these induced predictable OAMDPs can be represented by goal-oriented or discounted MDPs; and analyze the properties of the proposed reward functions both theoretically and empirically on two types of grid-world problems.

The application of artificial intelligence (AI) models in fields such as engineering is limited by the known difficulty of quantifying the reliability of an AI's decision. A well-calibrated AI model must correctly report its accuracy on in-distribution (ID) inputs, while also enabling the detection of out-of-distribution (OOD) inputs. A conventional approach to improve calibration is the application of Bayesian ensembling. However, owing to computational limitations and model misspecification, practical ensembling strategies do not necessarily enhance calibration. This paper proposes an extension of variational inference (VI)-based Bayesian learning that integrates calibration regularization for improved ID performance, confidence minimization for OOD detection, and selective calibration to ensure a synergistic use of calibration regularization and confidence minimization. The scheme is constructed successively by first introducing calibration-regularized Bayesian learning (CBNN), then incorporating out-of-distribution confidence minimization (OCM) to yield CBNN-OCM, and finally integrating also selective calibration to produce selective CBNN-OCM (SCBNN-OCM). Selective calibration rejects inputs for which the calibration performance is expected to be insufficient. Numerical results illustrate the trade-offs between ID accuracy, ID calibration, and OOD calibration attained by both frequentist and Bayesian learning methods. Among the main conclusions, SCBNN-OCM is seen to achieve best ID and OOD performance as compared to existing state-of-the-art approaches at the cost of rejecting a sufficiently large number of inputs.

We present a unified framework called deep dependency networks (DDNs) that combines dependency networks and deep learning architectures for multi-label classification, with a particular emphasis on image and video data. The primary advantage of dependency networks is their ease of training, in contrast to other probabilistic graphical models like Markov networks. In particular, when combined with deep learning architectures, they provide an intuitive, easy-to-use loss function for multi-label classification. A drawback of DDNs compared to Markov networks is their lack of advanced inference schemes, necessitating the use of Gibbs sampling. To address this challenge, we propose novel inference schemes based on local search and integer linear programming for computing the most likely assignment to the labels given observations. We evaluate our novel methods on three video datasets (Charades, TACoS, Wetlab) and three image datasets (MS-COCO, PASCAL VOC, NUS-WIDE), comparing their performance with (a) basic neural architectures and (b) neural architectures combined with Markov networks equipped with advanced inference and learning techniques. Our results demonstrate the superiority of our new DDN methods over the two competing approaches.

We introduce a deterministic variational formulation for training Bayesian last layer neural networks. This yields a sampling-free, single-pass model and loss that effectively improves uncertainty estimation. Our variational Bayesian last layer (VBLL) can be trained and evaluated with only quadratic complexity in last layer width, and is thus (nearly) computationally free to add to standard architectures. We experimentally investigate VBLLs, and show that they improve predictive accuracy, calibration, and out of distribution detection over baselines across both regression and classification. Finally, we investigate combining VBLL layers with variational Bayesian feature learning, yielding a lower variance collapsed variational inference method for Bayesian neural networks. Well-calibrated uncertainty quantification is essential for reliable decision-making with machine learning systems.