Gaussian processes (GPs) provide a framework for Bayesian inference that can offer principled uncertainty estimates for a large range of problems. For example, if we consider regression problems with Gaussian likelihoods, a GP model enjoys a posterior in closed form. However, identifying the posterior GP scales cubically with the number of training examples and requires to store all examples in memory. In order to overcome these obstacles, sparse GPs have been proposed that approximate the true posterior GP with pseudo-training examples. Importantly, the number of pseudo-training examples is user-defined and enables control over computational and memory complexity. In the general case, sparse GPs do not enjoy closed-form solutions and one has to resort to approximate inference. In this context, a convenient choice for approximate inference is variational inference (VI), where the problem of Bayesian inference is cast as an optimization problem -- namely, to maximize a lower bound of the log marginal likelihood. This paves the way for a powerful and versatile framework, where pseudo-training examples are treated as optimization arguments of the approximate posterior that are jointly identified together with hyperparameters of the generative model (i.e. prior and likelihood). The framework can naturally handle a wide scope of supervised learning problems, ranging from regression with heteroscedastic and non-Gaussian likelihoods to classification problems with discrete labels, but also multilabel problems. The purpose of this tutorial is to provide access to the basic matter for readers without prior knowledge in both GPs and VI. A proper exposition to the subject enables also access to more recent advances (like importance-weighted VI as well as inderdomain, multioutput and deep GPs) that can serve as an inspiration for new research ideas.

To perform Bayesian inference for stochastic simulator models for which the likelihood is not accessible, Likelihood-Free Inference (LFI) relies on simulations from the model. Standard LFI methods can be split according to how these simulations are used: to build an explicit Surrogate Likelihood, or to accept/reject parameter values according to a measure of distance from the observations (Approximate Bayesian Computation (ABC)). In both cases, simulations are adaptively tailored to the value of the observation. Here, we generate parameter-simulation pairs from the model independently on the observation, and use them to learn a conditional exponential family likelihood approximation; to parametrize it, we use Neural Networks whose weights are tuned with Score Matching. With our likelihood approximation, we can employ MCMC for doubly intractable distributions to draw samples from the posterior for any number of observations without additional model simulations, with performance competitive to comparable approaches. Further, the sufficient statistics of the exponential family can be used as summaries in ABC, outperforming the state-of-the-art method in five different models with known likelihood. Finally, we apply our method to a challenging model from meteorology.

It is argued that 4-valued paraconsistent truth values (called here "p-bits") can serve as a conceptual, mathematical and practical foundation for highly AI-relevant forms of probabilistic logic and probabilistic programming and concept formation. First it is shown that appropriate averaging-across-situations and renormalization of 4-valued p-bits operating in accordance with Constructible Duality (CD) logic yields PLN (Probabilistic Logic Networks) strength-and-confidence truth values. Then variations on the Curry-Howard correspondence are used to map these paraconsistent and probabilistic logics into probabilistic types suitable for use within dependent type based programming languages. Zach Weber's paraconsistent analysis of the sorites paradox is extended to form a paraconsistent / probabilistic / fuzzy analysis of concept boundaries; and a paraconsistent version of concept formation via Formal Concept Analysis is presented, building on a definition of fuzzy property-value degrees in terms of relative entropy on paraconsistent probability distributions. These general points are fleshed out via reference to the realization of probabilistic reasoning and programming and concept formation in the OpenCog AGI framework which is centered on collaborative multi-algorithm updating of a common knowledge metagraph.

Given (small amounts of) time-series' data from a high-dimensional, fine-grained, multiscale dynamical system, we propose a generative framework for learning an effective, lower-dimensional, coarse-grained dynamical model that is predictive of the fine-grained system's long-term evolution but also of its behavior under different initial conditions. We target fine-grained models as they arise in physical applications (e.g. molecular dynamics, agent-based models), the dynamics of which are strongly non-stationary but their transition to equilibrium is governed by unknown slow processes which are largely inaccessible by brute-force simulations. Approaches based on domain knowledge heavily rely on physical insight in identifying temporally slow features and fail to enforce the long-term stability of the learned dynamics. On the other hand, purely statistical frameworks lack interpretability and rely on large amounts of expensive simulation data (long and multiple trajectories) as they cannot infuse domain knowledge. The generative framework proposed achieves the aforementioned desiderata by employing a flexible prior on the complex plane for the latent, slow processes, and an intermediate layer of physics-motivated latent variables that reduces reliance on data and imbues inductive bias. In contrast to existing schemes, it does not require the a priori definition of projection operators from the fine-grained description and addresses simultaneously the tasks of dimensionality reduction and model estimation. We demonstrate its efficacy and accuracy in multiscale physical systems of particle dynamics where probabilistic, long-term predictions of phenomena not contained in the training data are produced.

The drastic increase of data quantity often brings the severe decrease of data quality, such as incorrect label annotations, which poses a great challenge for robustly training Deep Neural Networks (DNNs). Existing learning \mbox{methods} with label noise either employ ad-hoc heuristics or restrict to specific noise assumptions. However, more general situations, such as instance-dependent label noise, have not been fully explored, as scarce studies focus on their label corruption process. By categorizing instances into confusing and unconfusing instances, this paper proposes a simple yet universal probabilistic model, which explicitly relates noisy labels to their instances. The resultant model can be realized by DNNs, where the training procedure is accomplished by employing an alternating optimization algorithm. Experiments on datasets with both synthetic and real-world label noise verify that the proposed method yields significant improvements on robustness over state-of-the-art counterparts.

Counterfactual inference is a useful tool for comparing outcomes of interventions on complex systems. It requires us to represent the system in form of a structural causal model, complete with a causal diagram, probabilistic assumptions on exogenous variables, and functional assignments. Specifying such models can be extremely difficult in practice. The process requires substantial domain expertise, and does not scale easily to large systems, multiple systems, or novel system modifications. At the same time, many application domains, such as molecular biology, are rich in structured causal knowledge that is qualitative in nature. This manuscript proposes a general approach for querying a causal biological knowledge graph, and converting the qualitative result into a quantitative structural causal model that can learn from data to answer the question. We demonstrate the feasibility, accuracy and versatility of this approach using two case studies in systems biology. The first demonstrates the appropriateness of the underlying assumptions and the accuracy of the results. The second demonstrates the versatility of the approach by querying a knowledge base for the molecular determinants of a severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2)-induced cytokine storm, and performing counterfactual inference to estimate the causal effect of medical countermeasures for severely ill patients.

In application areas where data generation is expensive, Gaussian processes are a preferred supervised learning model due to their high data-efficiency. Particularly in model-based control, Gaussian processes allow the derivation of performance guarantees using probabilistic model error bounds. To make these approaches applicable in practice, two open challenges must be solved i) Existing error bounds rely on prior knowledge, which might not be available for many real-world tasks. (ii) The relationship between training data and the posterior variance, which mainly drives the error bound, is not well understood and prevents the asymptotic analysis. This article addresses these issues by presenting a novel uniform error bound using Lipschitz continuity and an analysis of the posterior variance function for a large class of kernels. Additionally, we show how these results can be used to guarantee safe control of an unknown dynamical system and provide numerical illustration examples.

The density estimation is one of the core problems in statistics. Despite this, existing techniques like maximum likelihood estimation are computationally inefficient due to the intractability of the normalizing constant. For this reason an interest to score matching has increased being independent on the normalizing constant. However, such estimator is consistent only for distributions with the full space support. One of the approaches to make it consistent is to add noise to the input data which is called Denoising Score Matching. In this work we derive analytical expression for the Denoising Score matching using the Kernel Exponential Family as a model distribution. The usage of the kernel exponential family is motivated by the richness of this class of densities. To tackle the computational complexity we use Random Fourier Features based approximation of the kernel function. The analytical expression allows to drop additional regularization terms based on the higher-order derivatives as they are already implicitly included. Moreover, the obtained expression explicitly depends on the noise variance, so the validation loss can be straightforwardly used to tune the noise level. Along with benchmark experiments, the model was tested on various synthetic distributions to study the behaviour of the model in different cases. The empirical study shows comparable quality to the competing approaches, while the proposed method being computationally faster. The latter one enables scaling up to complex high-dimensional data.

Trip destination prediction is an area of increasing importance in many applications such as trip planning, autonomous driving and electric vehicles. Even though this problem could be naturally addressed in an online learning paradigm where data is arriving in a sequential fashion, the majority of research has rather considered the offline setting. In this paper, we present a unified framework for trip destination prediction in an online setting, which is suitable for both online training and online prediction. For this purpose, we develop two clustering algorithms and integrate them within two online prediction models for this problem. We investigate the different configurations of clustering algorithms and prediction models on a real-world dataset. By using traditional clustering metrics and accuracy, we demonstrate that both the clustering and the entire framework yield consistent results compared to the offline setting. Finally, we propose a novel regret metric for evaluating the entire online framework in comparison to its offline counterpart. This metric makes it possible to relate the source of erroneous predictions to either the clustering or the prediction model. Using this metric, we show that the proposed methods converge to a probability distribution resembling the true underlying distribution and enjoy a lower regret than all of the baselines.

Traditional communication systems design is dominated by methods that are based on statistical models. These statistical-model-based algorithms, which we refer to henceforth as model-based methods, rely on mathematical models that describe the transmission process, signal propagation, receiver noise, interference, and many other components of the system that affect the end-to-end signal transmission and reception. Such mathematical models use parameters that vary over time as the channel conditions, the environment, network traffic, or network topology change. Therefore, for optimal operation, many of the algorithms used in communication systems rely on the underlying mathematical models as well as the estimation of the model parameters. However, there are cases where this approach fails, in particular when the mathematical models for one or more of the system components are highly complex, hard to estimate, poorly understood, do not well-capture the underlying physics of the system, or do not lend themselves to computationally-efficient algorithms.