Bayesian Nonparametric Spectral Estimation

Spectral estimation (SE) aims to identify how the energy of a signal (e.g., a time series) is distributed across different frequencies. This can become particularly challenging when only partial and noisy observations of the signal are available, where current methods fail to handle uncertainty appropriately. In this context, we propose a joint probabilistic model for signals, observations and spectra, where SE is addressed as an exact inference problem. Assuming a Gaussian process prior over the signal, we apply Bayes' rule to find the analytic posterior distribution of the spectrum given a set of observations. Besides its expressiveness and natural account of spectral uncertainty, the proposed model also provides a functional-form representation of the power spectral density, which can be optimised efficiently. Comparison with previous approaches, in particular against Lomb-Scargle, is addressed theoretically and also experimentally in three different scenarios. Code and demo available at https://github.com/GAMES-UChile/BayesianSpectralEstimation.Comment: 11 pages. In Advances in Neural Information Processing Systems, 201

Gradient Distribution Priors for Biomedical Image Processing

Ill-posed inverse problems are commonplace in biomedical image processing. Their solution typically requires imposing prior knowledge about the latent ground truth. While this regularizes the problem to an extent where it can be solved, it also biases the result toward the expected. With inappropriate priors harming more than they use, it remains unclear what prior to use for a given practical problem. Priors are hence mostly chosen in an {\em ad hoc} or empirical fashion. We argue here that the gradient distribution of natural-scene images may provide a versatile and well-founded prior for biomedical images. We provide motivation for this choice from different points of view, and we fully validate the resulting prior for use on biomedical images by showing its stability and correlation with image quality. We then provide a set of simple parametric models for the resulting prior, leading to straightforward (quasi-)convex optimization problems for which we provide efficient solver algorithms. We illustrate the use of the present models and solvers in a variety of common image-processing tasks, including contrast enhancement, noise level estimation, denoising, blind deconvolution, zooming/up-sampling, and dehazing. In all cases we show that the present method leads to results that are comparable to or better than the state of the art; always using the same, simple prior. We conclude by discussing the limitations and possible interpretations of the prior.Comment: submitted to journa

Variational Inference over Non-differentiable Cardiac Simulators using Bayesian Optimization

Performing inference over simulators is generally intractable as their runtime means we cannot compute a marginal likelihood. We develop a likelihood-free inference method to infer parameters for a cardiac simulator, which replicates electrical flow through the heart to the body surface. We improve the fit of a state-of-the-art simulator to an electrocardiogram (ECG) recorded from a real patient.Comment: Workshops on Deep Learning for Physical Sciences and Machine Learning 4 Health, NIPS 201

Universal Approximation of Edge Density in Large Graphs

In this paper, we present a novel way to summarize the structure of large graphs, based on non-parametric estimation of edge density in directed multigraphs. Following coclustering approach, we use a clustering of the vertices, with a piecewise constant estimation of the density of the edges across the clusters, and address the problem of automatically and reliably inferring the number of clusters, which is the granularity of the coclustering. We use a model selection technique with data-dependent prior and obtain an exact evaluation criterion for the posterior probability of edge density estimation models. We demonstrate, both theoretically and empirically, that our data-dependent modeling technique is consistent, resilient to noise, valid non asymptotically and asymptotically behaves as an universal approximator of the true edge density in directed multigraphs. We evaluate our method using artificial graphs and present its practical interest on real world graphs. The method is both robust and scalable. It is able to extract insightful patterns in the unsupervised learning setting and to provide state of the art accuracy when used as a preparation step for supervised learning

A Bayesian spatial temporal mixtures approach to kinetic parametric images in dynamic Positron Emission Tomography

We present a fully Bayesian statistical approach to the problem of compartmental modelling in the context of Positron Emission Tomography. We cluster homogeneous region of interest and perform kinetic parameter estimation simultaneously. A mixture modelling approach is adopted, incorporating both spatial and temporal information based on reconstructed dynamic PET image. Our modelling approach is flexible, and provides uncertainty estimates for the estimated kinetic parameters. Crucially, the proposed method allows us to determine the unknown number of clusters, which has a great impact on resulting estimated kinetic parameters. We demonstrate our method on simulated dynamic Myocardial PET data, and show that our method is superior to standard curve-fitting approach.Comment: 30 page

A spatio-spectral hybridization for edge preservation and noisy image restoration via local parametric mixtures and Lagrangian relaxation

This paper investigates a fully unsupervised statistical method for edge preserving image restoration and compression using a spatial decomposition scheme. Smoothed maximum likelihood is used for local estimation of edge pixels from mixture parametric models of local templates. For the complementary smooth part the traditional L2-variational problem is solved in the Fourier domain with Thin Plate Spline (TPS) regularization. It is well known that naive Fourier compression of the whole image fails to restore a piece-wise smooth noisy image satisfactorily due to Gibbs phenomenon. Images are interpreted as relative frequency histograms of samples from bi-variate densities where the sample sizes might be unknown. The set of discontinuities is assumed to be completely unsupervised Lebesgue-null, compact subset of the plane in the continuous formulation of the problem. Proposed spatial decomposition uses a widely used topological concept, partition of unity. The decision on edge pixel neighborhoods are made based on the multiple testing procedure of Holms. Statistical summary of the final output is decomposed into two layers of information extraction, one for the subset of edge pixels and the other for the smooth region. Robustness is also demonstrated by applying the technique on noisy degradation of clean images.Comment: 29 Pages, 13 figure

Determining the Number of Non-Spurious Arcs in a Learned DAG Model: Investigation of a Bayesian and a Frequentist Approach

In many application domains, such as computational biology, the goal of graphical model structure learning is to uncover discrete relationships between entities. For example, in our problem of interest concerning HIV vaccine design, we want to infer which HIV peptides interact with which immune system molecules (HLA molecules). For problems of this nature, we are interested in determining the number of nonspurious arcs in a learned graphical model. We describe both a Bayesian and frequentist approach to this problem. In the Bayesian approach, we use the posterior distribution over model structures to compute the expected number of true arcs in a learned model. In the frequentist approach, we develop a method based on the concept of the False Discovery Rate. On synthetic data sets generated from models similar to the ones learned, we find that both the Bayesian and frequentist approaches yield accurate estimates of the number of non-spurious arcs. In addition, we speculate that the frequentist approach, which is non-parametric, may outperform the parametric Bayesian approach in situations where the models learned are less representative of the data. Finally, we apply the frequentist approach to our problem of HIV vaccine design.Comment: Appears in Proceedings of the Twenty-Third Conference on Uncertainty in Artificial Intelligence (UAI2007

Deep Reinforcement Learning in a Handful of Trials using Probabilistic Dynamics Models

Model-based reinforcement learning (RL) algorithms can attain excellent sample efficiency, but often lag behind the best model-free algorithms in terms of asymptotic performance. This is especially true with high-capacity parametric function approximators, such as deep networks. In this paper, we study how to bridge this gap, by employing uncertainty-aware dynamics models. We propose a new algorithm called probabilistic ensembles with trajectory sampling (PETS) that combines uncertainty-aware deep network dynamics models with sampling-based uncertainty propagation. Our comparison to state-of-the-art model-based and model-free deep RL algorithms shows that our approach matches the asymptotic performance of model-free algorithms on several challenging benchmark tasks, while requiring significantly fewer samples (e.g., 8 and 125 times fewer samples than Soft Actor Critic and Proximal Policy Optimization respectively on the half-cheetah task).Comment: NIPS 2018, video and code available at https://sites.google.com/view/drl-in-a-handful-of-trials

Bayesian active learning for optimization and uncertainty quantification in protein docking

Motivation: Ab initio protein docking represents a major challenge for optimizing a noisy and costly "black box"-like function in a high-dimensional space. Despite progress in this field, there is no docking method available for rigorous uncertainty quantification (UQ) of its solution quality (e.g. interface RMSD or iRMSD). Results: We introduce a novel algorithm, Bayesian Active Learning (BAL), for optimization and UQ of such black-box functions and flexible protein docking. BAL directly models the posterior distribution of the global optimum (or native structures for protein docking) with active sampling and posterior estimation iteratively feeding each other. Furthermore, we use complex normal modes to represent a homogeneous Euclidean conformation space suitable for high-dimension optimization and construct funnel-like energy models for encounter complexes. Over a protein docking benchmark set and a CAPRI set including homology docking, we establish that BAL significantly improve against both starting points by rigid docking and refinements by particle swarm optimization, providing for one third targets a top-3 near-native prediction. BAL also generates tight confidence intervals with half range around 25% of iRMSD and confidence level at 85%. Its estimated probability of a prediction being native or not achieves binary classification AUROC at 0.93 and AUPRC over 0.60 (compared to 0.14 by chance); and also found to help ranking predictions. To the best of our knowledge, this study represents the first uncertainty quantification solution for protein docking, with theoretical rigor and comprehensive assessment. Source codes are available at https://github.com/Shen-Lab/BAL

Forecasting Turbulent Modes with Nonparametric Diffusion Models: Learning from noisy data

In this paper, we apply a recently developed nonparametric modeling approach, the "diffusion forecast", to predict the time-evolution of Fourier modes of turbulent dynamical systems. While the diffusion forecasting method assumes the availability of a noise-free training data set observing the full state space of the dynamics, in real applications we often have only partial observations which are corrupted by noise. To alleviate these practical issues, following the theory of embedology, the diffusion model is built using the delay-embedding coordinates of the data. We show that this delay embedding biases the geometry of the data in a way which extracts the most stable component of the dynamics and reduces the influence of independent additive observation noise. The resulting diffusion forecast model approximates the semigroup solutions of the generator of the underlying dynamics in the limit of large data and when the observation noise vanishes. As in any standard forecasting problem, the forecasting skill depends crucially on the accuracy of the initial conditions. We introduce a novel Bayesian method for filtering the discrete-time noisy observations which works with the diffusion forecast to determine the forecast initial densities. Numerically, we compare this nonparametric approach with standard stochastic parametric models on a wide-range of well-studied turbulent modes, including the Lorenz-96 model in weakly chaotic to fully turbulent regimes and the barotropic modes of a quasi-geostrophic model with baroclinic instabilities. We show that when the only available data is the low-dimensional set of noisy modes that are being modeled, the diffusion forecast is indeed competitive to the perfect model