The goal of this paper is to generalize classical d-separation and classical Belief Propagation (BP) to the quantum realm. Classical d-separation is an essential ingredient of most of Judea Pearl's work. It is crucial to all 3 rungs of what Pearl calls the 3 rungs of Causation. So having a quantum version of d-separation and BP probably implies that most of Pearl's Bayesian networks work, including his theory of causality, can be translated in a straightforward manner to the quantum realm.

The recent success of Bayesian methods in neuroscience and artificial intelligence gives rise to the hypothesis that the brain is a Bayesian machine. Since logic and learning are both practices of the human brain, it leads to another hypothesis that there is a Bayesian interpretation underlying both logical reasoning and machine learning. In this paper, we introduce a generative model of logical consequence relations. It formalises the process of how the truth value of a sentence is probabilistically generated from the probability distribution over states of the world. We show that the generative model characterises a classical consequence relation, paraconsistent consequence relation and nonmonotonic consequence relation. In particular, the generative model gives a new consequence relation that outperforms them in reasoning with inconsistent knowledge. We also show that the generative model gives a new classification algorithm that outperforms several representative algorithms in predictive accuracy and complexity on the Kaggle Titanic dataset.

Probabilistic solvers for ordinary differential equations (ODEs) assign a posterior measure to the solution of an initial value problem. The joint covariance of this distribution provides an estimate of the (global) approximation error. The contraction rate of this error estimate as a function of the solver's step size identifies it as a well-calibrated worst-case error. But its explicit numerical value for a certain step size, which depends on certain parameters of this class of solvers, is not automatically a good estimate of the explicit error. Addressing this issue, we introduce, discuss, and assess several probabilistically motivated ways to calibrate the uncertainty estimate. Numerical experiments demonstrate that these calibration methods interact efficiently with adaptive step-size selection, resulting in descriptive, and efficiently computable posteriors. We demonstrate the efficiency of the methodology by benchmarking against the classic, widely used Dormand-Prince 4/5 Runge-Kutta method.

While energy-based models (EBMs) exhibit a number of desirable properties, training and sampling on high-dimensional datasets remains challenging. Inspired by recent progress on diffusion probabilistic models, we present a diffusion recovery likelihood method to tractably learn and sample from a sequence of EBMs trained on increasingly noisy versions of a dataset. Each EBM is trained by maximizing the recovery likelihood: the conditional probability of the data at a certain noise level given their noisy versions at a higher noise level. The recovery likelihood objective is more tractable than the marginal likelihood objective, since it only requires MCMC sampling from a relatively concentrated conditional distribution. Moreover, we show that this estimation method is theoretically consistent: it learns the correct conditional and marginal distributions at each noise level, given sufficient data. After training, synthesized images can be generated efficiently by a sampling process that initializes from a spherical Gaussian distribution and progressively samples the conditional distributions at decreasingly lower noise levels. Our method generates high fidelity samples on various image datasets. On unconditional CIFAR-10 our method achieves FID 9.60 and inception score 8.58, superior to the majority of GANs. Moreover, we demonstrate that unlike previous work on EBMs, our long-run MCMC samples from the conditional distributions do not diverge and still represent realistic images, allowing us to accurately estimate the normalized density of data even for high-dimensional datasets.

We consider the problem of online learning in the presence of sudden distribution shifts as frequently encountered in applications such as autonomous navigation. Distribution shifts require constant performance monitoring and re-training. They may also be hard to detect and can lead to a slow but steady degradation in model performance. To address this problem we propose a new Bayesian meta-algorithm that can both (i) make inferences about subtle distribution shifts based on minimal sequential observations and (ii) accordingly adapt a model in an online fashion. The approach uses beam search over multiple change point hypotheses to perform inference on a hierarchical sequential latent variable modeling framework. Our proposed approach is model-agnostic, applicable to both supervised and unsupervised learning, and yields significant improvements over state-of-the-art Bayesian online learning approaches.

Current machine learning methods provide unprecedented accuracy across a range of domains, from computer vision to natural language processing. However, in many important high-stakes applications, such as medical diagnosis or autonomous driving, rare mistakes can be extremely costly, and thus effective deployment of learned models requires not only high accuracy, but also a way to measure the certainty in a model's predictions. Reliable uncertainty quantification is especially important when faced with out-of-distribution inputs, as model accuracy tends to degrade heavily on inputs that differ significantly from those seen during training. In this blog post, we will discuss how we can get reliable uncertainty estimation with a strategy that does not simply rely on a learned model to extrapolate to out-of-distribution inputs, but instead asks: "given my training data, which labels would make sense for this input?". To illustrate how this can allow for more reasonable predictions on out-of-distribution data, consider the following example where we attempt to classify automobiles, where all the class 1 training examples are sedans and class 2 examples are large buses.

Learning about many things can provide numerous benefits to a reinforcement learning system. For example, learning many auxiliary value functions, in addition to optimizing the environmental reward, appears to improve both exploration and representation learning. The question we tackle in this paper is how to sculpt the stream of experience—how to adapt the learning system’s behavior—to optimize the learning of a collection of value functions. A simple answer is to compute an intrinsic reward based on the statistics of each auxiliary learner, and use reinforcement learning to maximize that intrinsic reward. Unfortunately, implementing this simple idea has proven difficult, and thus has been the focus of decades of study. It remains unclear which of the many possible measures of learning would work well in a parallel learning setting where environmental reward is extremely sparse or absent. In this paper, we investigate and compare different intrinsic reward mechanisms in a new bandit-like parallel-learning testbed. We discuss the interaction between reward and prediction learners and highlight the importance of introspective prediction learners: those that increase their rate of learning when progress is possible, and decrease when it is not. We provide a comprehensive empirical comparison of 14 different rewards, including well-known ideas from reinforcement learning and active learning. Our results highlight a simple but seemingly powerful principle: intrinsic rewards based on the amount of learning can generate useful behavior, if each individual learner is introspective.

Inference and forecast problems of the nonlinear dynamical system have arisen in a variety of contexts. Reservoir computing and deep sequential models, on the one hand, have demonstrated efficient, robust, and superior performance in modeling simple and chaotic dynamical systems. However, their innate deterministic feature has partially detracted their robustness to noisy system, and their inability to offer uncertainty measurement has also been an insufficiency of the framework. On the other hand, the traditional state-space model framework is robust to noise. It also carries measured uncertainty, forming a just-right complement to the reservoir computing and deep sequential model framework. We propose the unscented reservoir smoother, a model that unifies both deep sequential and state-space models to achieve both frameworks' superiorities. Evaluated in the option pricing setting on top of noisy datasets, URS strikes highly competitive forecasting accuracy, especially those of longer-term, and uncertainty measurement. Further extensions and implications on URS are also discussed to generalize a full integration of both frameworks.

Deep Gaussian processes (DGPs) are increasingly popular as predictive models in machine learning (ML) for their non-stationary flexibility and ability to cope with abrupt regime changes in training data. Here we explore DGPs as surrogates for computer simulation experiments whose response surfaces exhibit similar characteristics. In particular, we transport a DGP's automatic warping of the input space and full uncertainty quantification (UQ), via a novel elliptical slice sampling (ESS) Bayesian posterior inferential scheme, through to active learning (AL) strategies that distribute runs non-uniformly in the input space -- something an ordinary (stationary) GP could not do. Building up the design sequentially in this way allows smaller training sets, limiting both expensive evaluation of the simulator code and mitigating cubic costs of DGP inference. When training data sizes are kept small through careful acquisition, and with parsimonious layout of latent layers, the framework can be both effective and computationally tractable. Our methods are illustrated on simulation data and two real computer experiments of varying input dimensionality. We provide an open source implementation in the "deepgp" package on CRAN.

This work focuses on the development of a new family of decision-making algorithms for adaptation and learning, which are specifically tailored to decision problems and are constructed by building up on first principles from decision theory. A key observation is that estimation and decision problems are structurally different and, therefore, algorithms that have proven successful for the former need not perform well when adjusted for decision problems. We propose a new scheme, referred to as BLLR (barrier log-likelihood ratio algorithm) and demonstrate its applicability to real-data from the COVID-19 pandemic in Italy. The results illustrate the ability of the design tool to track the different phases of the outbreak.