Transforming Gaussian processes with normalizing flows

Gaussian Processes (GP) can be used as flexible, non-parametric function priors. Inspired by the growing body of work on Normalizing Flows, we enlarge this class of priors through a parametric invertible transformation that can be made input-dependent. Doing so also allows us to encode interpretable prior knowledge (e.g., boundedness constraints). We derive a variational approximation to the resulting Bayesian inference problem, which is as fast as stochastic variational GP regression (Hensman et al., 2013; Dezfouli and Bonilla, 2015). This makes the model a computationally efficient alternative to other hierarchical extensions of GP priors (Lázaro-Gredilla,2012; Damianou and Lawrence,2013). The resulting algorithm’s computational and inferential performance is excellent, and we demonstrate this on a range of data sets. For example, even with only 5 inducing points and an input-dependent flow, our method is consistently competitive with a standard sparse GP fitted using 100 inducing points

Sparse Machine Learning Methods for Autonomous Decision Making

Sparse regression methods are used for the reconstruction of compressed signals, that are usually sparse in some bases; or in feature selection problem, where only few features are meaningful. This thesis overviews the existing Bayesian methods for dealing with sparsity, improves them and provides new models for these problems. The novel models decrease complexity, allow to model structure and provide uncertainty distributions in such applications as medicine and computer vision. The thesis starts with exploring Bayesian sparsity for the problem of compressive back- ground subtraction. Sparsity naturally arises in this problem as foreground usually occupies only small part of the video frame. The use of Bayesian compressive sensing improves the solutions in independent and multi-task scenarios. It also raises an important problem of exploring the structure of the data, as foreground pixels are usually clustered in groups. The problem of structure modelling in sparse problems is addressed with hierarchical Gaussian processes, that are the Bayesian way of imposing structure without specifying its exact patterns. Full Bayesian inference based on expectation propagation is provided for offline and online algorithms. The experiments demonstrate the applicability of these methods for the compressed background subtraction and brain activity localisation problems. The majority of sparse Bayesian methods are computationally intensive. This thesis proposes a novel sparse regression method based on the Bayesian neural networks. It makes the prediction operation fast and additionally estimates the uncertainty of predictions, while requiring a longer training phase. The results are demonstrated in the active learning scenario, where the estimated uncertainty is used for experiment design. Sparse methods are also used as part of other methods such as Gaussian processes that suffer from high computational complexity. The use of active sparse subsets of data improves the performance on large datasets. The thesis proposes a method of dealing with the complexity problem for online data updates using Bayesian filtering

Machine learning for efficient recognition of anatomical structures and abnormalities in biomedical images

Three studies have been carried out to investigate new approaches to efficient image segmentation and anomaly detection. The first study investigates the use of deep learning in patch based segmentation. Current approaches to patch based segmentation use low level features such as the sum of squared differences between patches. We argue that better segmentation can be achieved by harnessing the power of deep neural networks. Currently these networks make extensive use of convolutional layers. However, we argue that in the context of patch based segmentation, convolutional layers have little advantage over the canonical artificial neural network architecture. This is because a patch is small, and does not need decomposition and thus will not benefit from convolution. Instead, we make use of the canonical architecture in which neurons only compute dot products, but also incorporate modern techniques of deep learning. The resulting classifier is much faster and less memory-hungry than convolution based networks. In a test application to the segmentation of hippocampus in human brain MR images, we significantly outperformed prior art with a median Dice score up to 90.98% at a near real-time speed (<1s). The second study is an investigation into mouse phenotyping, and develops a high-throughput framework to detect morphological abnormality in mouse embryo micro-CT images. Existing work in this line is centred on, either the detection of phenotype-specific features or comparative analytics. The former approach lacks generality and the latter can often fail, for example, when the abnormality is not associated with severe volume variation. Both these approaches often require image segmentation as a pre-requisite, which is very challenging when applied to embryo phenotyping. A new approach to this problem in which non-rigid registration is combined with robust principal component analysis (RPCA), is proposed. The new framework is able to efficiently perform abnormality detection in a batch of images. It is sensitive to both volumetric and non-volumetric variations, and does not require image segmentation. In a validation study, it successfully distinguished the abnormal VSD and polydactyly phenotypes from the normal, respectively, at 85.19% and 88.89% specificities, with 100% sensitivity in both cases. The third study investigates the RPCA technique in more depth. RPCA is an extension of PCA that tolerates certain levels of data distortion during feature extraction, and is able to decompose images into regular and singular components. It has previously been applied to many computer vision problems (e.g. video surveillance), attaining excellent performance. However these applications commonly rest on a critical condition: in the majority of images being processed, there is a background with very little variation. By contrast in biomedical imaging there is significant natural variation across different images, resulting from inter-subject variability and physiological movements. Non-rigid registration can go some way towards reducing this variance, but cannot eliminate it entirely. To address this problem we propose a modified framework (RPCA-P) that is able to incorporate natural variation priors and adjust outlier tolerance locally, so that voxels associated with structures of higher variability are compensated with a higher tolerance in regularity estimation. An experimental study was applied to the same mouse embryo micro-CT data, and notably improved the detection specificity to 94.12% for the VSD and 90.97% for the polydactyly, while maintaining the sensitivity at 100%.Open Acces

Conditional Variational Autoencoder for Learned Image Reconstruction

Learned image reconstruction techniques using deep neural networks have recently gained popularity and have delivered promising empirical results. However, most approaches focus on one single recovery for each observation, and thus neglect information uncertainty. In this work, we develop a novel computational framework that approximates the posterior distribution of the unknown image at each query observation. The proposed framework is very flexible: it handles implicit noise models and priors, it incorporates the data formation process (i.e., the forward operator), and the learned reconstructive properties are transferable between different datasets. Once the network is trained using the conditional variational autoencoder loss, it provides a computationally efficient sampler for the approximate posterior distribution via feed-forward propagation, and the summarizing statistics of the generated samples are used for both point-estimation and uncertainty quantification. We illustrate the proposed framework with extensive numerical experiments on positron emission tomography (with both moderate and low-count levels) showing that the framework generates high-quality samples when compared with state-of-the-art methods

Bayesian nonparametric models for biomedical data analysis

In this dissertation, we develop nonparametric Bayesian models for biomedical data analysis. In particular, we focus on inference for tumor heterogeneity and inference for missing data. First, we present a Bayesian feature allocation model for tumor subclone reconstruction using mutation pairs. The key innovation lies in the use of short reads mapped to pairs of proximal single nucleotide variants (SNVs). In contrast, most existing methods use only marginal reads for unpaired SNVs. In the same context of using mutation pairs, in order to recover the phylogenetic relationship of subclones, we then develop a Bayesian treed feature allocation model. In contrast to commonly used feature allocation models, we allow the latent features to be dependent, using a tree structure to introduce dependence. Finally, we propose a nonparametric Bayesian approach to monotone missing data in longitudinal studies with non-ignorable missingness. In contrast to most existing methods, our method allow for incorporating information from auxiliary covariates and is able to capture complex structures among the response, missingness and auxiliary covariates. Our models are validated through simulation studies and are applied to real-world biomedical datasets.Statistic

Scalable approximate inference methods for Bayesian deep learning

This thesis proposes multiple methods for approximate inference in deep Bayesian neural networks split across three parts. The first part develops a scalable Laplace approximation based on a block- diagonal Kronecker factored approximation of the Hessian. This approximation accounts for parameter correlations – overcoming the overly restrictive independence assumption of diagonal methods – while avoiding the quadratic scaling in the num- ber of parameters of the full Laplace approximation. The chapter further extends the method to online learning where datasets are observed one at a time. As the experiments demonstrate, modelling correlations between the parameters leads to improved performance over the diagonal approximation in uncertainty estimation and continual learning, in particular in the latter setting the improvements can be substantial. The second part explores two parameter-efficient approaches for variational inference in neural networks, one based on factorised binary distributions over the weights, one extending ideas from sparse Gaussian processes to neural network weight matrices. The former encounters similar underfitting issues as mean-field Gaussian approaches, which can be alleviated by a MAP-style method in a hierarchi- cal model. The latter, based on an extension of Matheron’s rule to matrix normal distributions, achieves comparable uncertainty estimation performance to ensembles with the accuracy of a deterministic network while using only 25% of the number of parameters of a single ResNet-50. The third part introduces TyXe, a probabilistic programming library built on top of Pyro to facilitate turning PyTorch neural networks into Bayesian ones. In contrast to existing frameworks, TyXe avoids introducing a layer abstraction, allowing it to support arbitrary architectures. This is demonstrated in a range of applications, from image classification with torchvision ResNets over node labelling with DGL graph neural networks to incorporating uncertainty into neural radiance fields with PyTorch3d

A review of probabilistic forecasting and prediction with machine learning

Predictions and forecasts of machine learning models should take the form of probability distributions, aiming to increase the quantity of information communicated to end users. Although applications of probabilistic prediction and forecasting with machine learning models in academia and industry are becoming more frequent, related concepts and methods have not been formalized and structured under a holistic view of the entire field. Here, we review the topic of predictive uncertainty estimation with machine learning algorithms, as well as the related metrics (consistent scoring functions and proper scoring rules) for assessing probabilistic predictions. The review covers a time period spanning from the introduction of early statistical (linear regression and time series models, based on Bayesian statistics or quantile regression) to recent machine learning algorithms (including generalized additive models for location, scale and shape, random forests, boosting and deep learning algorithms) that are more flexible by nature. The review of the progress in the field, expedites our understanding on how to develop new algorithms tailored to users' needs, since the latest advancements are based on some fundamental concepts applied to more complex algorithms. We conclude by classifying the material and discussing challenges that are becoming a hot topic of research.Comment: 83 pages, 5 figure