Gaussian Processes (GPs) has experienced tremendous success in geoscience in general and for bio-geophysical parameter retrieval in the last years. GPs constitute a solid Bayesian framework to formulate many function approximation problems consistently. This paper reviews the main theoretical GP developments in the field. We review new algorithms that respect the signal and noise characteristics, that provide feature rankings automatically, and that allow applicability of associated uncertainty intervals to transport GP models in space and time. All these developments are illustrated in the field of geoscience and remote sensing at a local and global scales through a set of illustrative examples.

We combine two popular optimization approaches to derive learning algorithms for generative models: variational optimization and evolutionary algorithms. The combination is realized for generative models with discrete latents by using truncated posteriors as the family of variational distributions. The variational parameters of truncated posteriors are sets of latent states. By interpreting these states as genomes of individuals and by using the variational lower bound to define a fitness, we can apply evolutionary algorithms to realize the variational loop. The used variational distributions are very flexible and we show that evolutionary algorithms can effectively and efficiently optimize the variational bound. Furthermore, the variational loop is generally applicable ("black box") with no analytical derivations required. To show general applicability, we apply the approach to three generative models (we use noisy-OR Bayes Nets, Binary Sparse Coding, and Spike-and-Slab Sparse Coding). To demonstrate effectiveness and efficiency of the novel variational approach, we use the standard competitive benchmarks of image denoising and inpainting. The benchmarks allow quantitative comparisons to a wide range of methods including probabilistic approaches, deep deterministic and generative networks, and non-local image processing methods. In the category of "zero-shot" learning (when only the corrupted image is used for training), we observed the evolutionary variational algorithm to significantly improve the state-of-the-art in many benchmark settings. For one well-known inpainting benchmark, we also observed state-of-the-art performance across all categories of algorithms although we only train on the corrupted image. In general, our investigations highlight the importance of research on optimization methods for generative models to achieve performance improvements.

Epidemiology models are central in understanding and controlling large scale pandemics. Several epidemiology models require simulation-based inference such as Approximate Bayesian Computation (ABC) to fit their parameters to observations. ABC inference is highly amenable to efficient hardware acceleration. In this work, we develop parallel ABC inference of a stochastic epidemiology model for COVID-19. The statistical inference framework is implemented and compared on Intel Xeon CPU, NVIDIA Tesla V100 GPU and the Graphcore Mk1 IPU, and the results are discussed in the context of their computational architectures. Results show that GPUs are 4x and IPUs are 30x faster than Xeon CPUs. Extensive performance analysis indicates that the difference between IPU and GPU can be attributed to higher communication bandwidth, closeness of memory to compute, and higher compute power in the IPU. The proposed framework scales across 16 IPUs, with scaling overhead not exceeding 8% for the experiments performed. We present an example of our framework in practice, performing inference on the epidemiology model across three countries, and giving a brief overview of the results.

When the dimension of data is comparable to or larger than the number of available data samples, Principal Components Analysis (PCA) is known to exhibit problematic phenomena of high-dimensional noise. In this work, we propose an Empirical Bayes PCA method that reduces this noise by estimating a structural prior for the joint distributions of the principal components. This EB-PCA method is based upon the classical Kiefer-Wolfowitz nonparametric MLE for empirical Bayes estimation, distributional results derived from random matrix theory for the sample PCs, and iterative refinement using an Approximate Message Passing (AMP) algorithm. In theoretical "spiked" models, EB-PCA achieves Bayes-optimal estimation accuracy in the same settings as the oracle Bayes AMP procedure that knows the true priors. Empirically, EB-PCA can substantially improve over PCA when there is strong prior structure, both in simulation and on several quantitative benchmarks constructed using data from the 1000 Genomes Project and the International HapMap Project. A final illustration is presented for an analysis of gene expression data obtained by single-cell RNA-seq.

Users can be supported to adopt healthy behaviors, such as regular physical activity, via relevant and timely suggestions on their mobile devices. Recently, reinforcement learning algorithms have been found to be effective for learning the optimal context under which to provide suggestions. However, these algorithms are not necessarily designed for the constraints posed by mobile health (mHealth) settings, that they be efficient, domain-informed and computationally affordable. We propose an algorithm for providing physical activity suggestions in mHealth settings. Using domain-science, we formulate a contextual bandit algorithm which makes use of a linear mixed effects model. We then introduce a procedure to efficiently perform hyper-parameter updating, using far less computational resources than competing approaches. Not only is our approach computationally efficient, it is also easily implemented with closed form matrix algebraic updates and we show improvements over state of the art approaches both in speed and accuracy of up to 99% and 56% respectively.

We consider the optimization problem of minimizing a functional defined over a family of probability distributions, where the objective functional is assumed to possess a variational form. Such a distributional optimization problem arises widely in machine learning and statistics, with Monte-Carlo sampling, variational inference, policy optimization, and generative adversarial network as examples. For this problem, we propose a novel particle-based algorithm, dubbed as variational transport, which approximately performs Wasserstein gradient descent over the manifold of probability distributions via iteratively pushing a set of particles. Specifically, we prove that moving along the geodesic in the direction of functional gradient with respect to the second-order Wasserstein distance is equivalent to applying a pushforward mapping to a probability distribution, which can be approximated accurately by pushing a set of particles. Specifically, in each iteration of variational transport, we first solve the variational problem associated with the objective functional using the particles, whose solution yields the Wasserstein gradient direction. Then we update the current distribution by pushing each particle along the direction specified by such a solution. By characterizing both the statistical error incurred in estimating the Wasserstein gradient and the progress of the optimization algorithm, we prove that when the objective function satisfies a functional version of the Polyak-\L{}ojasiewicz (PL) (Polyak, 1963) and smoothness conditions, variational transport converges linearly to the global minimum of the objective functional up to a certain statistical error, which decays to zero sublinearly as the number of particles goes to infinity.

Evaluating expectations on a pairwise Boltzmann machine (PBM) (or Ising model) is important for various applications, including the statistical machine learning. However, in general the evaluation is computationally difficult because it involves intractable multiple summations or integrations; therefore, it requires an approximation. Monte Carlo integration (MCI) is a well-known approximation method; a more effective MCI-like approximation method was proposed recently, called spatial Monte Carlo integration (SMCI). However, the estimations obtained from SMCI (and MCI) tend to perform poorly in PBMs with low temperature owing to degradation of the sampling quality. Annealed importance sampling (AIS) is a type of importance sampling based on Markov chain Monte Carlo methods, and it can suppress performance degradation in low temperature regions by the force of importance weights. In this study, a new method is proposed to evaluate the expectations on PBMs combining AIS and SMCI. The proposed method performs efficiently in both high- and low-temperature regions, which is theoretically and numerically demonstrated.

This paper introduces a new prior on functions spaces which scales more favourably in the dimension of the function's domain compared to the usual Karhunen-Lo\'eve function space prior, a property we refer to as dimension-robustness. The proposed prior is a Bayesian neural network prior, where each weight and bias has an independent Gaussian prior, but with the key difference that the variances decrease in the width of the network, such that the variances form a summable sequence and the infinite width limit neural network is well defined. We show that our resulting posterior of the unknown function is amenable to sampling using Hilbert space Markov chain Monte Carlo methods. These sampling methods are favoured because they are stable under mesh-refinement, in the sense that the acceptance probability does not shrink to 0 as more parameters are introduced to better approximate the well-defined infinite limit. We show that our priors are competitive and have distinct advantages over other function space priors. Upon defining a suitable likelihood for continuous value functions in a Bayesian approach to reinforcement learning, our new prior is used in numerical examples to illustrate its performance and dimension-robustness.

Crowdsourcing has emerged as a powerful paradigm for efficiently labeling large datasets and performing various learning tasks, by leveraging crowds of human annotators. When additional information is available about the data, semi-supervised crowdsourcing approaches that enhance the aggregation of labels from human annotators are well motivated. This work deals with semi-supervised crowdsourced classification, under two regimes of semi-supervision: a) label constraints, that provide ground-truth labels for a subset of data; and b) potentially easier to obtain instance-level constraints, that indicate relationships between pairs of data. Bayesian algorithms based on variational inference are developed for each regime, and their quantifiably improved performance, compared to unsupervised crowdsourcing, is analytically and empirically validated on several crowdsourcing datasets.

Fuzzy Cognitive Maps (FCMs) is a complex systems modeling technique which, due to its unique advantages, has lately risen in popularity. They are based on graphs that represent the causal relationships among the parameters of the system to be modeled, and they stand out for their interpretability and flexibility. With the late popularity of FCMs, a plethora of research efforts have taken place to develop and optimize the model. One of the most important elements of FCMs is the learning algorithm they use, and their effectiveness is largely determined by it. The learning algorithms learn the node weights of an FCM, with the goal of converging towards the desired behavior. The present study reviews the genetic algorithms used for training FCMs, as well as gives a general overview of the FCM learning algorithms, putting evolutionary computing into the wider context.