The field of probabilistic numerics (PN), loosely speaking, attempts to provide a statistical treatment of the errors and/or approximations that are made en route to the output of a deterministic numerical method, e.g. the approximation of an integral by quadrature, or the discretised solution of an ordinary or partial differential equation. This decade has seen a surge of activity in this field. In comparison with historical developments that can be traced back over more than a hundred years, the most recent developments are particularly interesting because they have been characterised by simultaneous input from multiple scientific disciplines: mathematics, statistics, machine learning, and computer science. The field has, therefore, advanced on a broad front, with contributions ranging from the building of overarching generaltheory to practical implementations in specific problems of interest. Over the same period of time, and because of increased interaction among researchers coming from different communities, the extent to which these developments were -- or were not -- presaged by twentieth-century researchers has also come to be better appreciated. Thus, the time appears to be ripe for an update of the 2014 Tübingen Manifesto on probabilistic numerics[Hennig, 2014, Osborne, 2014d,c,b,a] and the position paper[Hennig et al., 2015] to take account of the developments between 2014 and 2019, an improved awareness of the history of this field, and a clearer sense of its future directions. In this article, we aim to summarise some of the history of probabilistic perspectives on numerics (Section 2), to place more recent developments into context (Section 3), and to articulate a vision for future research in, and use of, probabilistic numerics (Section 4).

In machine learning, if the training data is an unbiased sample of an underlying distribution, then the learned classification function will make accurate predictions for new samples. However, if the training data is not an unbiased sample, then there will be differences between how the training data is distributed and how the test data is distributed. Standard classifiers cannot cope with changes in data distributions between training and test phases, and will not perform well. Domain adaptation and transfer learning are sub-fields within machine learning that are concerned with accounting for these types of changes. Here, we present an introduction to these fields, guided by the question: when and how can a classifier generalize from a source to a target domain? We will start with a brief introduction into risk minimization, and how transfer learning and domain adaptation expand upon this framework. Following that, we discuss three special cases of data set shift, namely prior, covariate and concept shift. For more complex domain shifts, there are a wide variety of approaches. These are categorized into: importance-weighting, subspace mapping, domain-invariant spaces, feature augmentation, minimax estimators and robust algorithms. A number of points will arise, which we will discuss in the last section. We conclude with the remark that many open questions will have to be addressed before transfer learners and domain-adaptive classifiers become practical.

Model discrimination identifies a mathematical model that usefully explains and predicts a given system's behaviour. Researchers will often have several models, i.e.\ hypotheses, about an underlying system mechanism, but insufficient experimental data to discriminate between the models, i.e.\ discard inaccurate models. Given rival mathematical models and an initial experimental data set, optimal design of experiments suggests maximally informative experimental observations that maximise a design criterion weighted by prediction uncertainty. The model uncertainty requires gradients, which may not be readily available for black-box models. This paper (i) proposes a new design criterion using the Jensen-R\'enyi divergence, and (ii) develops a novel method replacing black-box models with Gaussian process surrogates. Using the surrogates, we marginalise out the model parameters with approximate inference. Results show these contributions working well for both classical and new test instances. We also (iii) introduce and discuss GPdoemd, the open-source implementation of the Gaussian process surrogate method.

Generative dynamic texture models (GDTMs) are widely used for dynamic texture (DT) segmentation in the video sequences. GDTMs represent DTs as a set of linear dynamical systems (LDSs). A major limitation of these models concerns the automatic selection of a proper number of DTs. Dirichlet process mixture (DPM) models which have appeared recently as the cornerstone of the non-parametric Bayesian statistics, is an optimistic candidate toward resolving this issue. Under this motivation to resolve the aforementioned drawback, we propose a novel non-parametric fully Bayesian approach for DT segmentation, formulated on the basis of a joint DPM and GDTM construction. This interaction causes the algorithm to overcome the problem of automatic segmentation properly. We derive the Variational Bayesian Expectation-Maximization (VBEM) inference for the proposed model. Moreover, in the E-step of inference, we apply Rauch-Tung-Striebel smoother (RTSS) algorithm on Variational Bayesian LDSs. Ultimately, experiments on different video sequences are performed. Experiment results indicate that the proposed algorithm outperforms the previous methods in efficiency and accuracy noticeably.

Abstract--Deep neural networks (DNNs) are increasingly being adopted for sensing and control functions in a variety of safety and mission-critical systems such as self-driving cars, autonomous air vehicles, medical diagnostics and industrial robotics. Failures of such systems can lead to loss of life or property, which necessitates stringent verification and validation for providing high assurance. Though formal verification approaches are being investigated, testing remains the primary technique for assessing the dependability of such systems. Due to the nature of the tasks handled by DNNs, the cost of obtaining test oracle data--the expected output, a.k.a. Thus, prioritizing input data for testing DNNs in meaningful ways to reduce the cost of labeling can go a long way in increasing testing efficacy. This paper proposes using gauges of the DNN's sentiment derived from the computation performed by the model, as a means to identify inputs that are likely to reveal weaknesses. We empirically assessed the efficacy of three such sentiment measures for prioritization--confidence, uncertainty and surprise--and compare their effectiveness in terms of their fault-revealing capability and retraining effectiveness. The results indicate that sentiment measures can effectively flag inputs that expose unacceptable DNN behavior . For MNIST models, the average percentage of inputs correctly flagged ranged from 88% to 94.8%.

We introduce SPFlow, an open-source Python library providing a simple interface to inference, learning and manipulation routines for deep and tractable probabilistic models called Sum-Product Networks (SPNs). The library allows one to quickly create SPNs both from data and through a domain specific language (DSL). It efficiently implements several probabilistic inference routines like computing marginals, conditionals and (approximate) most probable explanations (MPEs) along with sampling as well as utilities for serializing, plotting and structure statistics on an SPN. Moreover, many of the algorithms proposed in the literature to learn the structure and parameters of SPNs are readily available in SPFlow. Furthermore, SPFlow is extremely extensible and customizable, allowing users to promptly distill new inference and learning routines by injecting custom code into a lightweight functional-oriented API framework. This is achieved in SPFlow by keeping an internal Python representation of the graph structure that also enables practical compilation of an SPN into a TensorFlow graph, C, CUDA or FPGA custom code, significantly speeding-up computations.

Due to the phenomenon of "posterior collapse," current latent variable generative models pose a challenging design choice that either weakens the capacity of the decoder or requires augmenting the objective so it does not only maximize the likelihood of the data. In this paper, we propose an alternative that utilizes the most powerful generative models as decoders, whilst optimising the variational lower bound all while ensuring that the latent variables preserve and encode useful information. Our proposed $\delta$-VAEs achieve this by constraining the variational family for the posterior to have a minimum distance to the prior. For sequential latent variable models, our approach resembles the classic representation learning approach of slow feature analysis. We demonstrate the efficacy of our approach at modeling text on LM1B and modeling images: learning representations, improving sample quality, and achieving state of the art log-likelihood on CIFAR-10 and ImageNet $32\times 32$.

Bayesian optimization (BO) based on Gaussian process models is a powerful paradigm to optimize black-box functions that are expensive to evaluate. While several BO algorithms provably converge to the global optimum of the unknown function, they assume that the hyperparameters of the kernel are known in advance. This is not the case in practice and misspecification often causes these algorithms to converge to poor local optima. In this paper, we present the first BO algorithm that is provably no-regret and converges to the optimum without knowledge of the hyperparameters. We slowly adapt the hyperparameters of stationary kernels and thereby expand the associated function class over time, so that the BO algorithm considers more complex function candidates. Based on the theoretical insights, we propose several practical algorithms that achieve the empirical data efficiency of BO with online hyperparameter estimation, but retain theoretical convergence guarantees. We evaluate our method on several benchmark problems.

This paper presents a methodology and numerical algorithms for constructing accelerated gradient flows on the space of probability distributions. In particular, we extend the recent variational formulation of accelerated gradient methods in (wibisono, et. al. 2016) from vector valued variables to probability distributions. The variational problem is modeled as a mean-field optimal control problem. The maximum principle of optimal control theory is used to derive Hamilton's equations for the optimal gradient flow. The Hamilton's equation are shown to achieve the accelerated form of density transport from any initial probability distribution to a target probability distribution. A quantitative estimate on the asymptotic convergence rate is provided based on a Lyapunov function construction, when the objective functional is displacement convex. Two numerical approximations are presented to implement the Hamilton's equations as a system of $N$ interacting particles. The continuous limit of the Nesterov's algorithm is shown to be a special case with $N=1$. The algorithm is illustrated with numerical examples.

This paper presents an approach for constrained Gaussian Process (GP) regression where we assume that a set of linear transformations of the process are bounded. It is motivated by machine learning applications for high-consequence engineering systems, where this kind of information is often made available from phenomenological knowledge, and the resulting constraints may be essential to achieve the level of confidence needed. We consider a GP $f$ over functions on $\mathcal{X} \subset \mathbb{R}^{n}$ taking values in $\mathbb{R}$, where the process $\mathcal{L}f$ is still Gaussian when $\mathcal{L}$ is a linear operator. Our goal is to model $f$ under the constraint that realizations of $\mathcal{L}f$ are confined to a convex set of functions. In particular we require that $a \leq \mathcal{L}f \leq b$ given two functions $a$ and $b$ where $a < b$ pointwise. This formulation provides a consistent way of encoding multiple linear constraints, such as shape-constraints based on e.g. boundedness, monotonicity or convexity as a relevant example. We adopt the approach of using a sufficiently dense set of virtual observation locations where the constraint is required to hold, and derive the exact posterior for a conjugate likelihood. The results needed for stable numerical implementation are derived, together with an efficient sampling scheme for estimating the posterior process which is exact in the limit. A few numerical examples focusing on noiseless observations are given. This is relevant for computer code emulation and is also more computationally demanding than the alternative scenario with i.i.d. Gaussian noise.