Statistical inference of analytically non-tractable posteriors is a difficult problem because of marginalization of correlated variables and stochastic methods such as MCMC and VI are commonly used. We argue that stochastic KL divergence minimization used by MCMC and VI is noisy, and we propose instead EL_2O, expectation optimization of L_2 distance squared between the approximate log posterior q and the un-normalized log posterior of p. When sampling from q the solutions agree with stochastic KL divergence minimization based VI in the large sample limit, however EL_2O method is free of sampling noise, has better optimization properties, and requires only as many sample evaluations as the number of parameters we are optimizing if q covers p. As a consequence, increasing the expressivity of q improves both the quality of results and the convergence rate, allowing EL_2O to approach exact inference. Use of automatic differentiation methods enables us to develop Hessian, gradient and gradient free versions of the method, which can determine M(M+2)/2+1, M+1 and 1 parameter(s) of q with a single sample, respectively. EL_2O provides a reliable estimate of the quality of the approximating posterior, and converges rapidly on full rank gaussian approximation for q and extensions beyond it, such as nonlinear transformations and gaussian mixtures. These can handle general posteriors, while still allowing fast analytic marginalizations. We test it on several examples, including a realistic 13 dimensional galaxy clustering analysis, showing that it is several orders of magnitude faster than MCMC, while giving smooth and accurate non-gaussian posteriors, often requiring a few to a few dozen of iterations only.

This paper presents a non-trivial reconstruction of a previous joint topic-sentiment-preference review model TSPRA with stick-breaking representation under the framework of variational inference (VI) and stochastic variational inference (SVI). TSPRA is a Gibbs Sampling based model that solves topics, word sentiments and user preferences altogether and has been shown to achieve good performance, but for large data set it can only learn from a relatively small sample. We develop the variational models vTSPRA and svTSPRA to improve the time use, and our new approach is capable of processing millions of reviews. We rebuild the generative process, improve the rating regression, solve and present the coordinate-ascent updates of variational parameters, and show the time complexity of each iteration is theoretically linear to the corpus size, and the experiments on Amazon data sets show it converges faster than TSPRA and attains better results given the same amount of time. In addition, we tune svTSPRA into an online algorithm ovTSPRA that can monitor oscillations of sentiment and preference overtime. Some interesting fluctuations are captured and possible explanations are provided. The results give strong visual evidence that user preference is better treated as an independent factor from sentiment.

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.

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%.

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 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.

Actual causation is concerned with the question "what caused what?" Consider a transition between two states within a system of interacting elements, such as an artificial neural network, or a biological brain circuit. Which combination of synapses caused the neuron to fire? Which image features caused the classifier to misinterpret the picture? Even detailed knowledge of the system's causal network, its elements, their states, connectivity, and dynamics does not automatically provide a straightforward answer to the "what caused what?" question. Counterfactual accounts of actual causation based on graphical models, paired with system interventions, have demonstrated initial success in addressing specific problem cases in line with intuitive causal judgments. Here, we start from a set of basic requirements for causation (realization, composition, information, integration, and exclusion) and develop a rigorous, quantitative account of actual causation that is generally applicable to discrete dynamical systems. We present a formal framework to evaluate these causal requirements that is based on system interventions and partitions, and considers all counterfactuals of a state transition. This framework is used to provide a complete causal account of the transition by identifying and quantifying the strength of all actual causes and effects linking the two consecutive system states. Finally, we examine several exemplary cases and paradoxes of causation and show that they can be illuminated by the proposed framework for quantifying actual causation.

Latent class model (LCM), which is a finite mixture of different categorical distributions, is one of the most widely used models in statistics and machine learning fields. Because of its non-continuous nature and the flexibility in shape, researchers in practice areas such as marketing and social sciences also frequently use LCM to gain insights from their data. One likelihood-based method, the Expectation-Maximization (EM) algorithm, is often used to obtain the model estimators. However, the EM algorithm is well-known for its notoriously slow convergence. In this research, we explore alternative likelihood-based methods that can potential remedy the slow convergence of the EM algorithm. More specifically, we regard likelihood-based approach as a constrained nonlinear optimization problem, and apply quasi-Newton type methods to solve them. We examine two different constrained optimization methods to maximize the log likelihood function. We present simulation study results to show that the proposed methods not only converge in less iterations than the EM algorithm but also produce more accurate model estimators.