Density operators representing ensembles of pure states of sample wave functions are used in place probability densities. We show that such representation allows to formulate the statistical Bayesian learning problem in different coordinate systems on the sample space. We further show that such representation allows to learn projections of density operators using a kernel trick. In particular, the study highlights that decomposing wave functions rather than probability densities, as it is done in kernel embedding, allows to preserve the nature of probability operators. Results are illustrated with a simple example using discrete orthogonal wavelet transform of density operators.

Table of contents: What is Conformal Prediction? What is Conformal Prediction used for? Why should I use Conformal Prediction? Why shouldn’t I use Conformal Prediction? How can Conformal Prediction be used in Finance? How can Conformal Prediction be used in Algorithmic Trading? What are some Conformal Prediction alternatives? Understanding Conformal Prediction What is MAPIE? How […]

Recent developments using End-to-End Deep Learning models have been shown to have near or better performance than state of the art Recurrent Neural Networks (RNNs) on Automatic Speech Recognition tasks. These models tend to be lighter weight and require less training time than traditional RNN-based approaches. However, these models take frequentist approach to weight training. In theory, network weights are drawn from a latent, intractable probability distribution. We introduce BayesSpeech for end-to-end Automatic Speech Recognition. BayesSpeech is a Bayesian Transformer Network where these intractable posteriors are learned through variational inference and the local reparameterization trick without recurrence. We show how the introduction of variance in the weights leads to faster training time and near state-of-the-art performance on LibriSpeech-960.

Estimating the effect of a treatment on an outcome is a fundamental problem in many fields such as medicine [33, 34, 61], public policy [20] and more [2, 37]. For example, doctors might be interested in how a treatment, such as a drug, affects the recovery of patients [18], economists may be interested in how a job training program affects employment prospectives [35], and advertisers may want to model the average effect an advertisement has on sales [36]. However, individuals may react differently to the treatment of interest, and knowing only the average treatment effect in the population is insufficient. For example, a drug may have adverse effects on some individuals but not others [61], or a person's education and background may affect how much they benefit from job training [35, 50]. Measuring the extent to which different individuals react differently to treatment is known as heterogeneous treatment effect (HTE) estimation. Traditionally, HTE estimation has been done through subgroup analysis [9, 19]. However, this can lead to cherry-picking since the practitioner is the one who identifies subgroups for estimating effects. Recently, there has been more focus on data-driven estimation of heterogeneous treatment effects by letting the data identify which features are important for treatment effect estimation using machine learning techniques [28, 39, 61, 69]. A straightforward approach is to create interaction terms between all covariates and use them in a regression [6].

Mesoscale structures are an integral part of the abstraction and analysis of complex systems. They reveal a node's function in the network, and facilitate our understanding of the network dynamics. For example, they can represent communities in social or citation networks, roles in corporate interactions, or core-periphery structures in transportation networks. We usually detect mesoscale structures under the assumption of independence of interactions. Still, in many cases, the interactions invalidate this assumption by occurring in a specific order. Such patterns emerge in pathway data; to capture them, we have to model the dependencies between interactions using higher-order network models. However, the detection of mesoscale structures in higher-order networks is still under-researched. In this work, we derive a Bayesian approach that simultaneously models the optimal partitioning of nodes in groups and the optimal higher-order network dynamics between the groups. In synthetic data we demonstrate that our method can recover both standard proximity-based communities and role-based groupings of nodes. In synthetic and real world data we show that it can compete with baseline techniques, while additionally providing interpretable abstractions of network dynamics.

This thesis focuses on data that has complex spatio-temporal structure and on probabilistic graphical models that learn the structure in an interpretable and scalable manner. We target two research areas of interest: Gaussian graphical models for tensor-variate data and summarization of complex time-varying texts using topic models. This work advances the state-of-the-art in several directions. First, it introduces a new class of tensor-variate Gaussian graphical models via the Sylvester tensor equation. Second, it develops an optimization technique based on a fast-converging proximal alternating linearized minimization method, which scales tensor-variate Gaussian graphical model estimations to modern big-data settings. Third, it connects Kronecker-structured (inverse) covariance models with spatio-temporal partial differential equations (PDEs) and introduces a new framework for ensemble Kalman filtering that is capable of tracking chaotic physical systems. Fourth, it proposes a modular and interpretable framework for unsupervised and weakly-supervised probabilistic topic modeling of time-varying data that combines generative statistical models with computational geometric methods. Throughout, practical applications of the methodology are considered using real datasets. This includes brain-connectivity analysis using EEG data, space weather forecasting using solar imaging data, longitudinal analysis of public opinions using Twitter data, and mining of mental health related issues using TalkLife data. We show in each case that the graphical modeling framework introduced here leads to improved interpretability, accuracy, and scalability.

The problem of jointly analysing functional connectomics and behavioral data is extremely challenging owing to the complex interactions between the two domains. In addition, clinical rs-fMRI studies often have to contend with limited samples, especially in the case of rare disorders. This data-starved regimen can severely restrict the reliability of classical machine learning or deep learning designed to predict behavior from connectivity data. In this work, we approach this problem from the lens of representation learning and bayesian modeling. To model the distributional characteristics of the domains, we first examine the ability of approaches such as Bayesian Linear Regression, Stochastic Search Variable Selection after performing a classical covariance decomposition. Finally, we present a fully bayesian formulation for joint representation learning and prediction.

The goal of sequential event prediction is to estimate the next event based on a sequence of historical events, with applications to sequential recommendation, user behavior analysis and clinical treatment. In practice, the next-event prediction models are trained with sequential data collected at one time and need to generalize to newly arrived sequences in remote future, which requires models to handle temporal distribution shift from training to testing. In this paper, we first take a data-generating perspective to reveal a negative result that existing approaches with maximum likelihood estimation would fail for distribution shift due to the latent context confounder, i.e., the common cause for the historical events and the next event. Then we devise a new learning objective based on backdoor adjustment and further harness variational inference to make it tractable for sequence learning problems. On top of that, we propose a framework with hierarchical branching structures for learning context-specific representations. Comprehensive experiments on diverse tasks (e.g., sequential recommendation) demonstrate the effectiveness, applicability and scalability of our method with various off-the-shelf models as backbones.

Recent advances in coreset methods have shown that a selection of representative datapoints can replace massive volumes of data for Bayesian inference, preserving the relevant statistical information and significantly accelerating subsequent downstream tasks. Existing variational coreset constructions rely on either selecting subsets of the observed datapoints, or jointly performing approximate inference and optimizing pseudodata in the observed space akin to inducing points methods in Gaussian Processes. So far, both approaches are limited by complexities in evaluating their objectives for general purpose models, and require generating samples from a typically intractable posterior over the coreset throughout inference and testing. In this work, we present a black-box variational inference framework for coresets that overcomes these constraints and enables principled application of variational coresets to intractable models, such as Bayesian neural networks. We apply our techniques to supervised learning problems, and compare them with existing approaches in the literature for data summarization and inference.

Creating autonomous robots that can actively explore the environment, acquire knowledge and learn skills continuously is the ultimate achievement envisioned in cognitive and developmental robotics. Their learning processes should be based on interactions with their physical and social world in the manner of human learning and cognitive development. Based on this context, in this paper, we focus on the two concepts of world models and predictive coding. Recently, world models have attracted renewed attention as a topic of considerable interest in artificial intelligence. Cognitive systems learn world models to better predict future sensory observations and optimize their policies, i.e., controllers. Alternatively, in neuroscience, predictive coding proposes that the brain continuously predicts its inputs and adapts to model its own dynamics and control behavior in its environment. Both ideas may be considered as underpinning the cognitive development of robots and humans capable of continual or lifelong learning. Although many studies have been conducted on predictive coding in cognitive robotics and neurorobotics, the relationship between world model-based approaches in AI and predictive coding in robotics has rarely been discussed. Therefore, in this paper, we clarify the definitions, relationships, and status of current research on these topics, as well as missing pieces of world models and predictive coding in conjunction with crucially related concepts such as the free-energy principle and active inference in the context of cognitive and developmental robotics. Furthermore, we outline the frontiers and challenges involved in world models and predictive coding toward the further integration of AI and robotics, as well as the creation of robots with real cognitive and developmental capabilities in the future.