Prior probability models are a central component of many image processing problems, but density estimation is notoriously difficult for high-dimensional signals such as photographic images. Deep neural networks have provided state-of-the-art solutions for problems such as denoising, which implicitly rely on a prior probability model of natural images. Here, we develop a robust and general methodology for making use of this implicit prior. We rely on a little-known statistical result due to Miyasawa (1961), who showed that the least-squares solution for removing additive Gaussian noise can be written directly in terms of the gradient of the log of the noisy signal density. We use this fact to develop a stochastic coarse-to-fine gradient ascent procedure for drawing high-probability samples from the implicit prior embedded within a CNN trained to perform blind (i.e., unknown noise level) least-squares denoising. A generalization of this algorithm to constrained sampling provides a method for using the implicit prior to solve any linear inverse problem, with no additional training. We demonstrate this general form of transfer learning in multiple applications, using the same algorithm to produce high-quality solutions for deblurring, super-resolution, inpainting, and compressive sensing.

This article deals with the sequential design of experiments for (deterministic or stochastic) multi-fidelity numerical simulators, that is, simulators that offer control over the accuracy of simulation of the physical phenomenon or system under study. Very often, accurate simulations correspond to high computational efforts whereas coarse simulations can be obtained at a smaller cost. In this setting, simulation results obtained at several levels of fidelity can be combined in order to estimate quantities of interest (the optimal value of the output, the probability that the output exceeds a given threshold...) in an efficient manner. To do so, we propose a new Bayesian sequential strategy called Maximal Rate of Stepwise Uncertainty Reduction (MR-SUR), that selects additional simulations to be performed by maximizing the ratio between the expected reduction of uncertainty and the cost of simulation. This generic strategy unifies several existing methods, and provides a principled approach to develop new ones. We assess its performance on several examples, including a computationally intensive problem of fire safety analysis where the quantity of interest is the probability of exceeding a tenability threshold during a building fire.

In this note we explore a fully unsupervised deep-learning framework for simulating non-linear structural equation models from observational training data. The main contribution of this note is an architecture for applying moment-matching loss functions to the edges of a causal Bayesian graph, resulting in a generative conditional-moment-matching graph-neural-network. This framework thus enables automated sampling of latent space conditional probability distributions for various graphical interventions, and is capable of generating out-of-sample interventional probabilities that are often faithful to the ground truth distributions well beyond the range contained in the training set. These methods could in principle be used in conjunction with any existing autoencoder that produces a latent space representation containing causal graph structures.

The SARS-CoV-2 (Covid-19) pandemic has caused significant strain on public health institutions around the world. Contact tracing is an essential tool to change the course of the Covid-19 pandemic. Manual contact tracing of Covid-19 cases has significant challenges that limit the ability of public health authorities to minimize community infections. Personalized peer-to-peer contact tracing through the use of mobile apps has the potential to shift the paradigm. Some countries have deployed centralized tracking systems, but more privacy-protecting decentralized systems offer much of the same benefit without concentrating data in the hands of a state authority or for-profit corporations. Machine learning methods can circumvent some of the limitations of standard digital tracing by incorporating many clues and their uncertainty into a more graded and precise estimation of infection risk. The estimated risk can provide early risk awareness, personalized recommendations and relevant information to the user. Finally, non-identifying risk data can inform epidemiological models trained jointly with the machine learning predictor. These models can provide statistical evidence for the importance of factors involved in disease transmission. They can also be used to monitor, evaluate and optimize health policy and (de)confinement scenarios according to medical and economic productivity indicators. However, such a strategy based on mobile apps and machine learning should proactively mitigate potential ethical and privacy risks, which could have substantial impacts on society (not only impacts on health but also impacts such as stigmatization and abuse of personal data). Here, we present an overview of the rationale, design, ethical considerations and privacy strategy of `COVI,' a Covid-19 public peer-to-peer contact tracing and risk awareness mobile application developed in Canada.

Scientists from South Ural State University, in collaboration with foreign colleagues, have proposed a new model for the classification of MRI images based on a deep-belief network that will help to detect malignant brain tumors faster and more accurately. The research study was published in the Journal of Big Data, indexed in the scientometric Scopus database. Glioblastoma (GBM) is a stage 4 malignant brain tumor in which a large proportion of tumor cells are reproducing at any given moment. Such tumors are life-threatening and can lead to partial or complete mental and physical disability. The study was carried out by an international group of scientists from Indian universities and South Ural State University.

This version has preliminary implementations of automatic differentiation and of arithmetic on lists. These are both useful for gradient-based optimization, such as maximum likelihood estimation and neural network training, as well as gradient-based MCMC methods. List arithmetic is helpful when dealing with models that have several groups of parameters, which are most conveniently represented using a list of vectors or matrices, rather than a single vector. You can read the documentation on these facilities here and here. Some example programs are in this repository.

Relevance Vector Machine (RVM) is a supervised learning algorithm extended from Support Vector Machine (SVM) based on the Bayesian sparsity model. Compared with the regression problem, RVM classification is difficult to be conducted because there is no closed-form solution for the weight parameter posterior. Original RVM classification algorithm used Newton's method in optimization to obtain the mode of weight parameter posterior then approximated it by a Gaussian distribution in Laplace's method. It would work but just applied the frequency methods in a Bayesian framework. This paper proposes a Generic Bayesian approach for the RVM classification. We conjecture that our algorithm achieves convergent estimates of the quantities of interest compared with the nonconvergent estimates of the original RVM classification algorithm. Furthermore, a Fully Bayesian approach with the hierarchical hyperprior structure for RVM classification is proposed, which improves the classification performance, especially in the imbalanced data problem. By the numeric studies, our proposed algorithms obtain high classification accuracy rates. The Fully Bayesian hierarchical hyperprior method outperforms the Generic one for the imbalanced data classification.

Prob-solvable loops are probabilistic programs with polynomial assignments over random variables and parametrised distributions, for which the full automation of moment-based invariant generation is decidable. In this paper we extend Prob-solvable loops with new features essential for encoding Bayesian networks (BNs). We show that various BNs, such as discrete, Gaussian, conditional linear Gaussian and dynamic BNs, can be naturally encoded as Prob-solvable loops. Thanks to these encodings, we can automatically solve several BN related problems, including exact inference, sensitivity analysis, filtering and computing the expected number of rejecting samples in sampling-based procedures. We evaluate our work on a number of BN benchmarks, using automated invariant generation within Prob-solvable loop analysis.

In spatial statistics, a common objective is to predict the values of a spatial process at unobserved locations by exploiting spatial dependence. In geostatistics, Kriging provides the best linear unbiased predictor using covariance functions and is often associated with Gaussian processes. However, when considering non-linear prediction for non-Gaussian and categorical data, the Kriging prediction is not necessarily optimal, and the associated variance is often overly optimistic. We propose to use deep neural networks (DNNs) for spatial prediction. Although DNNs are widely used for general classification and prediction, they have not been studied thoroughly for data with spatial dependence. In this work, we propose a novel neural network structure for spatial prediction by adding an embedding layer of spatial coordinates with basis functions. We show in theory that the proposed DeepKriging method has multiple advantages over Kriging and classical DNNs only with spatial coordinates as features. We also provide density prediction for uncertainty quantification without any distributional assumption and apply the method to PM$_{2.5}$ concentrations across the continental United States.

This paper will focus on the process of 'fusing' several observations or models of uncertainty into a single resultant model. Many existing approaches to fusion use subjective quantities such as 'strengths of belief' and process these quantities with heuristic algorithms. This paper argues in favor of quantities that can be objectively measured, as opposed to the subjective 'strength of belief' values. This paper will focus on probability distributions, and more importantly, structures that denote sets of probability distributions known as 'credal sets'. The novel aspect of this paper will be a taxonomy of models of fusion that use specific types of credal sets, namely probability interval distributions and Dempster-Shafer models. An objective requirement for information fusion algorithms is provided, and is satisfied by all models of fusion presented in this paper. Dempster's rule of combination is shown to not satisfy this requirement. This paper will also assess the computational challenges involved for the proposed fusion approaches.