In the continual effort to improve product quality and decrease operations costs, computational modeling is increasingly being deployed to determine feasibility of product designs or configurations. Surrogate modeling of these computer experiments via local models, which induce sparsity by only considering short range interactions, can tackle huge analyses of complicated input-output relationships. However, narrowing focus to local scale means that global trends must be re-learned over and over again. In this article, we propose a framework for incorporating information from a global sensitivity analysis into the surrogate model as an input rotation and rescaling preprocessing step. We discuss the relationship between several sensitivity analysis methods based on kernel regression before describing how they give rise to a transformation of the input variables. Specifically, we perform an input warping such that the "warped simulator" is equally sensitive to all input directions, freeing local models to focus on local dynamics. Numerical experiments on observational data and benchmark test functions, including a high-dimensional computer simulator from the automotive industry, provide empirical validation.

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.

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 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.

Recent advances in probabilistic modelling have led to a large number of simulation-based inference algorithms which do not require numerical evaluation of likelihoods. However, a public benchmark with appropriate performance metrics for such 'likelihood-free' algorithms has been lacking. This has made it difficult to compare algorithms and identify their strengths and weaknesses. We set out to fill this gap: We provide a benchmark with inference tasks and suitable performance metrics, with an initial selection of algorithms including recent approaches employing neural networks and classical Approximate Bayesian Computation methods. We found that the choice of performance metric is critical, that even state-of-the-art algorithms have substantial room for improvement, and that sequential estimation improves sample efficiency. Neural network-based approaches generally exhibit better performance, but there is no uniformly best algorithm. We provide practical advice and highlight the potential of the benchmark to diagnose problems and improve algorithms. The results can be explored interactively on a companion website. All code is open source, making it possible to contribute further benchmark tasks and inference algorithms.

The analysis is borrowed from the regression setting and aims to decompose the prediction error of a given classifier into the terms of B&V to evaluate their effects on the performance. Therefore, it can help answer questions such as "How can we compare the accuracy of two different types of classifiers?", "What is it that makes stronger classifiers perform well? Is it the reduction in the bias they bring about, or in variance, or both?". Other than being theoretically interesting, the answers to these questions are also meant to provide better classifier design strategies which bring about improved prediction performance. After the initial decomposition of the prediction error into the standard B&V terms in the regression setting by [1], different studies have attempted to carry over this analysis into the classification setting while preserving the meanings of the terms and the additive property of the decomposition.

Due to the importance of uncertainty quantification (UQ), Bayesian approach to inverse problems has recently gained popularity in applied mathematics, physics, and engineering. However, traditional Bayesian inference methods based on Markov Chain Monte Carlo (MCMC) tend to be computationally intensive and inefficient for such high dimensional problems. To address this issue, several methods based on surrogate models have been proposed to speed up the inference process. More specifically, the calibration-emulation-sampling (CES) scheme has been proven to be successful in large dimensional UQ problems. In this work, we propose a novel CES approach for Bayesian inference based on deep neural network (DNN) models for the emulation phase. The resulting algorithm is not only computationally more efficient, but also less sensitive to the training set. Further, by using an Autoencoder (AE) for dimension reduction, we have been able to speed up our Bayesian inference method up to three orders of magnitude. Overall, our method, henceforth called \emph{Dimension-Reduced Emulative Autoencoder Monte Carlo (DREAM)} algorithm, is able to scale Bayesian UQ up to thousands of dimensions in physics-constrained inverse problems. Using two low-dimensional (linear and nonlinear) inverse problems we illustrate the validity this approach. Next, we apply our method to two high-dimensional numerical examples (elliptic and advection-diffussion) to demonstrate its computational advantage over existing algorithms.