Incorporating shape information is essential for the delineation of many organs and anatomical structures in medical images. While previous work has mainly focused on parametric spatial transformations applied on reference template shapes, in this paper, we address the Bayesian inference of parametric shape models for segmenting medical images with the objective to provide interpretable results. The proposed framework defines a likelihood appearance probability and a prior label probability based on a generic shape function through a logistic function. A reference length parameter defined in the sigmoid controls the trade-off between shape and appearance information. The inference of shape parameters is performed within an Expectation-Maximisation approach where a Gauss-Newton optimization stage allows to provide an approximation of the posterior probability of shape parameters. This framework is applied to the segmentation of cochlea structures from clinical CT images constrained by a 10 parameter shape model. It is evaluated on three different datasets, one of which includes more than 200 patient images. The results show performances comparable to supervised methods and better than previously proposed unsupervised ones. It also enables an analysis of parameter distributions and the quantification of segmentation uncertainty including the effect of the shape model.
In a bag-of-words model, the senses of a word with multiple meanings, e.g. "bank" (used either in a river-bank or an institution sense), are represented as probability distributions over context words, and sense prevalence is represented as a probability distribution over senses. Both of these may change with time. Modelling and measuring this kind of sense change is challenging due to the typically high-dimensional parameter space and sparse datasets. A recently published corpus of ancient Greek texts contains expert-annotated sense labels for selected target words. Automatic sense-annotation for the word "kosmos" (meaning decoration, order or world) has been used as a test case in recent work with related generative models and Monte Carlo methods. We adapt an existing generative sense change model to develop a simpler model for the main effects of sense and time, and give MCMC methods for Bayesian inference on all these models that are more efficient than existing methods. We carry out automatic sense-annotation of snippets containing "kosmos" using our model, and measure the time-evolution of its three senses and their prevalence. As far as we are aware, ours is the first analysis of this data, within the class of generative models we consider, that quantifies uncertainty and returns credible sets for evolving sense prevalence in good agreement with those given by expert annotation.
The current pandemic has introduced substantial uncertainty to traditional methods for demand planning. These uncertainties stem from the disease progression, government interventions, economy and consumer behavior. While most of the emerging literature on the pandemic has focused on disease progression, a few have focused on consequent regulations and their impact on individual behavior. The contributions of this paper include a quantitative behavior model of fear of COVID-19, impact of government interventions on consumer behavior, and impact of consumer behavior on consumer choice and hence demand for goods. It brings together multiple models for disease progression, consumer behavior and demand estimation-thus bridging the gap between disease progression and consumer demand. We use panel regression to understand the drivers of demand during the pandemic and Bayesian inference to simplify the regulation landscape that can help build scenarios for resilient demand planning. We illustrate this resilient demand planning model using a specific example of gas retailing. We find that demand is sensitive to fear of COVID-19: as the number of COVID-19 cases increase over the previous week, the demand for gas decreases -- though this dissipates over time. Further, government regulations restrict access to different services, thereby reducing mobility, which in itself reduces demand.
Joint registration of a stack of 2D histological sections to recover 3D structure (``3D histology reconstruction'') finds application in areas such as atlas building and validation of \emph{in vivo} imaging. Straightforward pairwise registration of neighbouring sections yields smooth reconstructions but has well-known problems such as ``banana effect'' (straightening of curved structures) and ``z-shift'' (drift). While these problems can be alleviated with an external, linearly aligned reference (e.g., Magnetic Resonance (MR) images), registration is often inaccurate due to contrast differences and the strong nonlinear distortion of the tissue, including artefacts such as folds and tears. In this paper, we present a probabilistic model of spatial deformation that yields reconstructions for multiple histological stains that that are jointly smooth, robust to outliers, and follow the reference shape. The model relies on a spanning tree of latent transforms connecting all the sections and slices of the reference volume, and assumes that the registration between any pair of images can be see as a noisy version of the composition of (possibly inverted) latent transforms connecting the two images. Bayesian inference is used to compute the most likely latent transforms given a set of pairwise registrations between image pairs within and across modalities. The framework is used for accurate 3D reconstruction of two stains (Nissl and parvalbumin) from the Allen human brain atlas, showing its benefits on real data with severe distortions. Moreover, we also provide the registration of the reconstructed volume to MNI space, bridging the gaps between two of the most widely used atlases in histology and MRI. The 3D reconstructed volumes and atlas registration can be downloaded from https://openneuro.org/datasets/ds003590. The code is freely available at https://github.com/acasamitjana/3dhirest.
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an inferential perspective, most notably the absence of explicit conditioning formula. This paper extends earlier work on linear PDEs to a general class of initial value problems specified by nonlinear PDEs, motivated by problems for which evaluations of the right-hand-side, initial conditions, or boundary conditions of the PDE have a high computational cost. The proposed method can be viewed as exact Bayesian inference under an approximate likelihood, which is based on discretisation of the nonlinear differential operator. Proof-of-concept experimental results demonstrate that meaningful probabilistic uncertainty quantification for the unknown solution of the PDE can be performed, while controlling the number of times the right-hand-side, initial and boundary conditions are evaluated. A suitable prior model for the solution of the PDE is identified using novel theoretical analysis of the sample path properties of Mat\'{e}rn processes, which may be of independent interest.
We study the problem of performing automated experiment design for drug screening through Bayesian inference and optimisation. In particular, we compare and contrast the behaviour of linear-Gaussian models and Gaussian processes, when used in conjunction with upper confidence bound algorithms, Thompson sampling, or bounded horizon tree search. We show that non-myopic sophisticated exploration techniques using sparse tree search have a distinct advantage over methods such as Thompson sampling or upper confidence bounds in this setting. We demonstrate the significant superiority of the approach over existing and synthetic datasets of drug toxicity.
We present an application of invariant polynomials in machine learning. Using the methods developed in previous work, we obtain two types of generators of the Lorentz- and permutation-invariant polynomials in particle momenta; minimal algebra generators and Hironaka decompositions. We discuss and prove some approximation theorems to make use of these invariant generators in machine learning algorithms in general and in neural networks specifically. By implementing these generators in neural networks applied to regression tasks, we test the improvements in performance under a wide range of hyperparameter choices and find a reduction of the loss on training data and a significant reduction of the loss on validation data. For a different approach on quantifying the performance of these neural networks, we treat the problem from a Bayesian inference perspective and employ nested sampling techniques to perform model comparison. Beyond a certain network size, we find that networks utilising Hironaka decompositions perform the best.
The consideration of predictive uncertainty in medical imaging with deep learning is of utmost importance. We apply estimation of both aleatoric and epistemic uncertainty by variational Bayesian inference with Monte Carlo dropout to regression tasks and show that predictive uncertainty is systematically underestimated. We apply $ \sigma $ scaling with a single scalar value; a simple, yet effective calibration method for both types of uncertainty. The performance of our approach is evaluated on a variety of common medical regression data sets using different state-of-the-art convolutional network architectures. In our experiments, $ \sigma $ scaling is able to reliably recalibrate predictive uncertainty. It is easy to implement and maintains the accuracy. Well-calibrated uncertainty in regression allows robust rejection of unreliable predictions or detection of out-of-distribution samples. Our source code is available at https://github.com/mlaves/well-calibrated-regression-uncertainty
Deep learning models, such as convolutional neural networks, have long been applied to image and multi-media tasks, particularly those with structured data. More recently, there has been more attention to unstructured data that can be represented via graphs. These types of data are often found in health and medicine, social networks, and research data repositories. Graph convolutional neural networks have recently gained attention in the field of deep learning that takes advantage of graph-based data representation with automatic feature extraction via convolutions. Given the popularity of these methods in a wide range of applications, robust uncertainty quantification is vital. This remains a challenge for large models and unstructured datasets. Bayesian inference provides a principled approach to uncertainty quantification of model parameters for deep learning models. Although Bayesian inference has been used extensively elsewhere, its application to deep learning remains limited due to the computational requirements of the Markov Chain Monte Carlo (MCMC) methods. Recent advances in parallel computing and advanced proposal schemes in MCMC sampling methods has opened the path for Bayesian deep learning. In this paper, we present Bayesian graph convolutional neural networks that employ tempered MCMC sampling with Langevin-gradient proposal distribution implemented via parallel computing. Our results show that the proposed method can provide accuracy similar to advanced optimisers while providing uncertainty quantification for key benchmark problems.
Generalised Bayesian inference updates prior beliefs using a loss function, rather than a likelihood, and can therefore be used to confer robustness against possible mis-specification of the likelihood. Here we consider generalised Bayesian inference with a Stein discrepancy as a loss function, motivated by applications in which the likelihood contains an intractable normalisation constant. In this context, the Stein discrepancy circumvents evaluation of the normalisation constant and produces generalised posteriors that are either closed form or accessible using standard Markov chain Monte Carlo. On a theoretical level, we show consistency, asymptotic normality, and bias-robustness of the generalised posterior, highlighting how these properties are impacted by the choice of Stein discrepancy. Then, we provide numerical experiments on a range of intractable distributions, including applications to kernel-based exponential family models and non-Gaussian graphical models.
Collaborative bandit learning, i.e., bandit algorithms that utilize collaborative filtering techniques to improve sample efficiency in online interactive recommendation, has attracted much research attention as it enjoys the best of both worlds. However, all existing collaborative bandit learning solutions impose a stationary assumption about the environment, i.e., both user preferences and the dependency among users are assumed static over time. Unfortunately, this assumption hardly holds in practice due to users' ever-changing interests and dependence relations, which inevitably costs a recommender system sub-optimal performance in practice. In this work, we develop a collaborative dynamic bandit solution to handle a changing environment for recommendation. We explicitly model the underlying changes in both user preferences and their dependency relation as a stochastic process. Individual user's preference is modeled by a mixture of globally shared contextual bandit models with a Dirichlet Process prior. Collaboration among users is thus achieved via Bayesian inference over the global bandit models. Model selection and arm selection for each user are done via Thompson sampling to balance exploitation and exploration. Our solution is proved to maintain a standard $\tilde O(\sqrt{T})$ sublinear regret even in such a challenging environment. And extensive empirical evaluations on both synthetic and real-world datasets further confirmed the necessity of modeling a changing environment and our algorithm's practical advantages against several state-of-the-art online learning solutions.
Fully convolutional U-shaped neural networks have largely been the dominant approach for pixel-wise image segmentation. In this work, we tackle two defects that hinder their deployment in real-world applications: 1) Predictions lack uncertainty quantification that may be crucial to many decision-making systems; 2) Large memory storage and computational consumption demanding extensive hardware resources. To address these issues and improve their practicality we demonstrate a few-parameter compact Bayesian convolutional architecture, that achieves a marginal improvement in accuracy in comparison to related work using significantly fewer parameters and compute operations. The architecture combines parameter-efficient operations such as separable convolutions, bilinear interpolation, multi-scale feature propagation and Bayesian inference for per-pixel uncertainty quantification through Monte Carlo Dropout. The best performing configurations required fewer than 2.5 million parameters on diverse challenging datasets with few observations.
Gaussian variational inference and the Laplace approximation are popular alternatives to Markov chain Monte Carlo that formulate Bayesian posterior inference as an optimization problem, enabling the use of simple and scalable stochastic optimization algorithms. However, a key limitation of both methods is that the solution to the optimization problem is typically not tractable to compute; even in simple settings the problem is nonconvex. Thus, recently developed statistical guarantees -- which all involve the (data) asymptotic properties of the global optimum -- are not reliably obtained in practice. In this work, we provide two major contributions: a theoretical analysis of the asymptotic convexity properties of variational inference with a Gaussian family and the maximum a posteriori (MAP) problem required by the Laplace approximation; and two algorithms -- consistent Laplace approximation (CLA) and consistent stochastic variational inference (CSVI) -- that exploit these properties to find the optimal approximation in the asymptotic regime. Both CLA and CSVI involve a tractable initialization procedure that finds the local basin of the optimum, and CSVI further includes a scaled gradient descent algorithm that provably stays locally confined to that basin. Experiments on nonconvex synthetic and real-data examples show that compared with standard variational and Laplace approximations, both CSVI and CLA improve the likelihood of obtaining the global optimum of their respective optimization problems.
Purpose: we evaluated the generalization capability of deep neural networks (DNNs), trained to classify chest X-rays as Covid-19, normal or pneumonia, using a relatively small and mixed dataset. Methods: we proposed a DNN to perform lung segmentation and classification, stacking a segmentation module (U-Net), an original intermediate module and a classification module (DenseNet201). To evaluate generalization, we tested the DNN with an external dataset (from distinct localities) and used Bayesian inference to estimate probability distributions of performance metrics. Results: our DNN achieved 0.917 AUC on the external test dataset, and a DenseNet without segmentation, 0.906. Bayesian inference indicated mean accuracy of 76.1% and [0.695, 0.826] 95% HDI (highest density interval, which concentrates 95% of the metric's probability mass) with segmentation and, without segmentation, 71.7% and [0.646, 0.786]. Conclusion: employing a novel DNN evaluation technique, which uses LRP and Brixia scores, we discovered that areas where radiologists found strong Covid-19 symptoms are the most important for the stacked DNN classification. External validation showed smaller accuracies than internal, indicating difficulty in generalization, which is positively affected by segmentation. Finally, the performance in the external dataset and the analysis with LRP suggest that DNNs can be trained in small and mixed datasets and still successfully detect Covid-19.
Uncertainty quantification provides quantitative measures on the reliability of candidate solutions of ill-posed inverse problems. Due to their sequential nature, Monte Carlo sampling methods require large numbers of sampling steps for accurate Bayesian inference and are often computationally infeasible for large-scale inverse problems, such as seismic imaging. Our main contribution is a data-driven variational inference approach where we train a normalizing flow (NF), a type of invertible neural net, capable of cheaply sampling the posterior distribution given previously unseen seismic data from neighboring surveys. To arrive at this result, we train the NF on pairs of low- and high-fidelity migrated images. In our numerical example, we obtain high-fidelity images from the Parihaka dataset and low-fidelity images are derived from these images through the process of demigration, followed by adding noise and migration. During inference, given shot records from a new neighboring seismic survey, we first compute the reverse-time migration image. Next, by feeding this low-fidelity migrated image to the NF we gain access to samples from the posterior distribution virtually for free. We use these samples to compute a high-fidelity image including a first assessment of the image's reliability. To our knowledge, this is the first attempt to train a conditional network on what we know from neighboring images to improve the current image and assess its reliability.
Autoencoders gained popularity in the deep learning revolution given their ability to compress data and provide dimensionality reduction. Although prominent deep learning methods have been used to enhance autoencoders, the need to provide robust uncertainty quantification remains a challenge. This has been addressed with variational autoencoders so far. Bayesian inference via Markov Chain Monte Carlo (MCMC) sampling has faced several limitations for large models; however, recent advances in parallel computing and advanced proposal schemes have opened routes less traveled. This paper presents Bayesian autoencoders powered by MCMC sampling implemented using parallel computing and Langevin-gradient proposal distribution. The results indicate that the proposed Bayesian autoencoder provides similar performance accuracy when compared to related methods in the literature. Furthermore, it provides uncertainty quantification in the reduced data representation. This motivates further applications of the Bayesian autoencoder framework for other deep learning models.
Tensor decomposition is a fundamental tool for multiway data analysis. While most decomposition algorithms operate a collection of static data and perform batch processes, many applications produce data in a streaming manner - every time a subset of entries are generated, and previously seen entries cannot be revisited. In such scenarios, traditional decomposition approaches will be inappropriate, because they cannot provide timely updates when new data come in, and they need to access the whole dataset many times for batch optimization. To address this issue, we propose POST, a PrObabilistic Streaming Tensor decomposition algorithm, which enables real-time updates and predictions upon receiving new tensor entries, and supports dynamic growth of all the modes. Compared with the state-of-the-art streaming decomposition approach MAST, POST is more flexible in that it can handle arbitrary orders of streaming entries, and hence is more widely applicable. In addition, as a Bayesian inference algorithm, POST can quantify the uncertainty of the latent embeddings via their posterior distributions, and the confidence levels of the missing entry value predictions. On several real-world datasets, POST exhibits better or comparable predictive performance than MAST and other static decomposition algorithms.
We present a framework for approximate Bayesian inference when only a limited number of noisy log-likelihood evaluations can be obtained due to computational constraints, which is becoming increasingly common for applications of complex models. We model the log-likelihood function using a Gaussian process (GP) and the main methodological innovation is to apply this model to emulate the progression that an exact Metropolis-Hastings (MH) sampler would take if it was applicable. Informative log-likelihood evaluation locations are selected using a sequential experimental design strategy until the MH accept/reject decision is done accurately enough according to the GP model. The resulting approximate sampler is conceptually simple and sample-efficient. It is also more robust to violations of GP modelling assumptions compared with earlier, related "Bayesian optimisation-like" methods tailored for Bayesian inference. We discuss some theoretical aspects and various interpretations of the resulting approximate MH sampler, and demonstrate its benefits in the context of Bayesian and generalised Bayesian likelihood-free inference for simulator-based statistical models.
Non-pharmaceutical interventions have been critical in the fight against the COVID-19 pandemic. However, these sanitary measures have been partially lifted due to socioeconomic factors causing a worrisome rebound of the epidemic in several countries. In this work, we assess the effectiveness of the mitigation implemented to constrain the spread of SARS-CoV-2 in the Mexican territory during 2020. We also investigate to what extent the initial deployment of the vaccine will help to mitigate the pandemic and reduce the need for social distancing and other mobility restrictions. Our modeling approach is based on a simple mechanistic Kermack-McKendrick-type model. To quantify the effect of NPIs, we perform a monthly Bayesian inference using officially published data. The results suggest that in the absence of the sanitary measures, the cumulative number of infections, hospitalizations, and deaths would have been at least twice the official number. Moreover, for low vaccine coverage levels, relaxing NPIs may dramatically increase the disease burden; therefore, safety measures are of critical importance at the early stages of vaccination. The simulations also suggest that it may be more desirable to employ a vaccine with low efficacy but reach a high coverage than a vaccine with high effectiveness but low coverage levels. This supports the hypothesis that single doses to more individuals will be more effective than two doses for every person.
To make informed decisions in natural environments that change over time, humans must update their beliefs as new observations are gathered. Studies exploring human inference as a dynamical process that unfolds in time have focused on situations in which the statistics of observations are history-independent. Yet temporal structure is everywhere in nature, and yields history-dependent observations. Do humans modify their inference processes depending on the latent temporal statistics of their observations? We investigate this question experimentally and theoretically using a change-point inference task. We show that humans adapt their inference process to fine aspects of the temporal structure in the statistics of stimuli. As such, humans behave qualitatively in a Bayesian fashion, but, quantitatively, deviate away from optimality. Perhaps more importantly, humans behave suboptimally in that their responses are not deterministic, but variable. We show that this variability itself is modulated by the temporal statistics of stimuli. To elucidate the cognitive algorithm that yields this behavior, we investigate a broad array of existing and new models that characterize different sources of suboptimal deviations away from Bayesian inference. While models with 'output noise' that corrupts the response-selection process are natural candidates, human behavior is best described by sampling-based inference models, in which the main ingredient is a compressed approximation of the posterior, represented through a modest set of random samples and updated over time. This result comes to complement a growing literature on sample-based representation and learning in humans.