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.

We present a sound and complete algorithm for recovering causal graphs from observed, non-interventional data, in the possible presence of latent confounders and selection bias. We rely on the causal Markov and faithfulness assumptions and recover the equivalence class of the underlying causal graph by performing a series of conditional independence (CI) tests between observed variables. We propose a single step that is applied iteratively, such that the independence and causal relations entailed from the resulting graph, after any iteration, is correct and becomes more informative with successive iteration. Essentially, we tie the size of the CI condition set to its distance from the tested nodes on the resulting graph. Each iteration refines the skeleton and orientation by performing CI tests having condition sets that are larger than in the preceding iteration. In an iteration, condition sets of CI tests are constructed from nodes that are within a specified search distance, and the sizes of these condition sets is equal to this search distance. The algorithm then iteratively increases the search distance along with the condition set sizes. Thus, each iteration refines a graph, that was recovered by previous iterations having smaller condition sets---having a higher statistical power. We demonstrate that our algorithm requires significantly fewer CI tests and smaller condition sets compared to the FCI algorithm. This is evident for both recovering the true underlying graph using a perfect CI oracle, and accurately estimating the graph using limited observed data.

A key task for speech recognition systems is to reduce the mismatch between the training and evaluation data that is often attributable to speaker differences. To this end, speaker adaptation techniques play a vital role to reduce the mismatch. Model-based speaker adaptation approaches often require sufficient amounts of target speaker data to ensure robustness. When the amount of speaker level data is limited, speaker adaptation is prone to overfitting and poor generalization. To address the issue, this paper proposes a full Bayesian learning based DNN speaker adaptation framework to model speaker-dependent (SD) parameter uncertainty given limited speaker specific adaptation data. This framework is investigated in three forms of model based DNN adaptation techniques: Bayesian learning of hidden unit contributions (BLHUC), Bayesian parameterized activation functions (BPAct), and Bayesian hidden unit bias vectors (BHUB). In all three Bayesian adaptation methods, deterministic SD parameters are replaced by latent variable posterior distributions to be learned for each speaker, whose parameters are efficiently estimated using a variational inference based approach. Experiments conducted on 300-hour speed perturbed Switchboard corpus trained LF-MMI factored TDNN/CNN-TDNN systems featuring i-vector speaker adaptation suggest the proposed Bayesian adaptation approaches consistently outperform the adapted systems using deterministic parameters on the NIST Hub5'00 and RT03 evaluation sets in both unsupervised test time speaker adaptation and speaker adaptive training. The efficacy of the proposed Bayesian adaptation techniques is further demonstrated in a comparison against the state-of-the-art performance obtained on the same task using the most recent hybrid and end-to-end systems reported in the literature.