We develop heuristic interpolation methods for the function $t \mapsto \operatorname{trace}\left( (\mathbf{A} + t \mathbf{B})^{-1} \right)$, where the matrices $\mathbf{A}$ and $\mathbf{B}$ are symmetric and positive definite and $t$ is a real variable. This function is featured in many applications in statistics, machine learning, and computational physics. The presented interpolation functions are based on the modification of a sharp upper bound that we derive for this function, which is a new trace inequality for matrices. We demonstrate the accuracy and performance of the proposed method with numerical examples, namely, the marginal maximum likelihood estimation for linear Gaussian process regression and the estimation of the regularization parameter of ridge regression with the generalized cross-validation method.

This thesis focuses on the research and development of the Hemodynamic Tissue Signature (HTS) method: an unsupervised machine learning approach to describe the vascular heterogeneity of glioblastomas by means of perfusion MRI analysis. The HTS builds on the concept of habitats. An habitat is defined as a sub-region of the lesion with a particular MRI profile describing a specific physiological behavior. The HTS method delineates four habitats within the glioblastoma: the High Angiogenic Tumor (HAT) habitat, as the most perfused region of the enhancing tumor; the Low Angiogenic Tumor (LAT) habitat, as the region of the enhancing tumor with a lower angiogenic profile; the potentially Infiltrated Peripheral Edema (IPE) habitat, as the non-enhancing region adjacent to the tumor with elevated perfusion indexes; and the Vasogenic Peripheral Edema (VPE) habitat, as the remaining edema of the lesion with the lowest perfusion profile. The results of this thesis have been published in ten scientific contributions, including top-ranked journals and conferences in the areas of Medical Informatics, Statistics and Probability, Radiology & Nuclear Medicine, Machine Learning and Data Mining and Biomedical Engineering. An industrial patent registered in Spain (ES201431289A), Europe (EP3190542A1) and EEUU (US20170287133A1) was also issued, summarizing the efforts of the thesis to generate tangible assets besides the academic revenue obtained from research publications. Finally, the methods, technologies and original ideas conceived in this thesis led to the foundation of ONCOANALYTICS CDX, a company framed into the business model of companion diagnostics for pharmaceutical compounds, thought as a vehicle to facilitate the industrialization of the ONCOhabitats technology.

Independent sociological polls are forbidden in Belarus. Online polls performed without sound scientific rigour do not yield representative results. Yet, both inside and outside Belarus it is of great importance to obtain precise estimates of the ratings of all candidates. These ratings could function as reliable proxies for the election's outcomes. We conduct an independent poll based on the combination of the data collected via Viber and on the streets of Belarus. The Viber and the street data samples consist of almost 45000 and 1150 unique observations respectively. Bayesian regressions with poststratification were build to estimate ratings of the candidates and rates of early voting turnout for the population as a whole and within various focus subgroups. We show that both the officially announced results of the election and early voting rates are highly improbable. With a probability of at least 95%, Sviatlana Tikhanouskaya's rating lies between 75% and 80%, whereas Aliaksandr Lukashenka's rating lies between 13% and 18% and early voting rate predicted by the method ranges from 9% to 13% of those who took part in the election. These results contradict the officially announced outcomes, which are 10.12%, 80.11%, and 49.54% respectively and lie far outside even the 99.9% credible intervals predicted by our model. The only marginal groups of people where the upper bounds of the 99.9% credible intervals of the rating of Lukashenka are above 50% are people older than 60 and uneducated people. For all other marginal subgroups, including rural residents, even the upper bounds of 99.9% credible intervals for Lukashenka are far below 50%. The same is true for the population as a whole. Thus, with a probability of at least 99.9% Lukashenka could not have had enough electoral support to win the 2020 presidential election in Belarus.

This paper shows that dropout training in Generalized Linear Models is the minimax solution of a two-player, zero-sum game where an adversarial nature corrupts a statistician's covariates using a multiplicative nonparametric errors-in-variables model. In this game---known as a Distributionally Robust Optimization problem---nature's least favorable distribution is dropout noise, where nature independently deletes entries of the covariate vector with some fixed probability $\delta$. Our decision-theoretic analysis shows that dropout training---the statistician's minimax strategy in the game---indeed provides out-of-sample expected loss guarantees for distributions that arise from multiplicative perturbations of in-sample data. This paper also provides a novel, parallelizable, Unbiased Multi-Level Monte Carlo algorithm to speed-up the implementation of dropout training. Our algorithm has a much smaller computational cost compared to the naive implementation of dropout, provided the number of data points is much smaller than the dimension of the covariate vector.

Finite mixture models, typically Gaussian mixtures, are well known and widely used as model-based clustering. In practical situations, there are many non-Gaussian data that are heavy-tailed and/or asymmetric. Normal inverse Gaussian (NIG) distributions are normal-variance mean which mixing densities are inverse Gaussian distributions and can be used for both haavy-tail and asymmetry. For NIG mixture models, both expectation-maximization method and variational Bayesian (VB) algorithms have been proposed. However, the existing VB algorithm for NIG mixture have a disadvantage that the shape of the mixing density is limited. In this paper, we propose another VB algorithm for NIG mixture that improves on the shortcomings. We also propose an extension of Dirichlet process mixture models to overcome the difficulty in determining the number of clusters in finite mixture models. We evaluated the performance with artificial data and found that it outperformed Gaussian mixtures and existing implementations for NIG mixtures, especially for highly non-normative data.

Successful application of machine learning models to real-world prediction problems, e.g. financial forecasting and personalized medicine, has proved to be challenging, because such settings require limiting and quantifying the uncertainty in the model predictions, i.e. providing valid and accurate prediction intervals. Conformal Prediction is a distribution-free approach to construct valid prediction intervals in finite samples. However, the prediction intervals constructed by Conformal Prediction are often (because of over-fitting, inappropriate measures of nonconformity, or other issues) overly conservative and hence inadequate for the application(s) at hand. This paper proposes an AutoML framework called Automatic Machine Learning for Conformal Prediction (AutoCP). Unlike the familiar AutoML frameworks that attempt to select the best prediction model, AutoCP constructs prediction intervals that achieve the user-specified target coverage rate while optimizing the interval length to be accurate and less conservative. We tested AutoCP on a variety of datasets and found that it significantly outperforms benchmark algorithms.

The COVID-19 pandemic provides new motivation for a classic problem in epidemiology: estimating the empirical rate of transmission during an outbreak (formally, the time-varying reproduction number) from case counts. While standard methods exist, they work best at coarse-grained national or state scales with abundant data, and struggle to accommodate the partial observability and sparse data common at finer scales (e.g., individual schools or towns). For example, case counts may be sparse when only a small fraction of infections are caught by a testing program. Or, whether an infected individual tests positive may depend on the kind of test and the point in time when they are tested. We propose a Bayesian framework which accommodates partial observability in a principled manner. Our model places a Gaussian process prior over the unknown reproduction number at each time step and models observations sampled from the distribution of a specific testing program. For example, our framework can accommodate a variety of kinds of tests (viral RNA, antibody, antigen, etc.) and sampling schemes (e.g., longitudinal or cross-sectional screening). Inference in this framework is complicated by the presence of tens or hundreds of thousands of discrete latent variables. To address this challenge, we propose an efficient stochastic variational inference method which relies on a novel gradient estimator for the variational objective. Experimental results for an example motivated by COVID-19 show that our method produces an accurate and well-calibrated posterior, while standard methods for estimating the reproduction number can fail badly.

Machine learning can precisely identify different cancer tumors at any stage by classifying cancerous and healthy samples based on their genomic profile. We have developed novel methods of MLAC (Machine Learning Against Cancer) achieving perfect results with perfect precision, sensitivity, and specificity. We have used the whole genome sequencing data acquired by next-generation RNA sequencing techniques in The Cancer Genome Atlas and Genotype-Tissue Expression projects for cancerous and healthy tissues respectively. Moreover, we have shown that unsupervised machine learning clustering has great potential to be used for cancer diagnosis. Indeed, a creative way to work with data and general algorithms has resulted in perfect classification i.e. all precision, sensitivity, and specificity are equal to 1 for most of the different tumor types even with a modest amount of data, and the same method works well on a series of cancers and results in great clustering of cancerous and healthy samples too. Our system can be used in practice because once the classifier is trained, it can be used to classify any new sample of new potential patients. One advantage of our work is that the aforementioned perfect precision and recall are obtained on samples of all stages including very early stages of cancer; therefore, it is a promising tool for diagnosis of cancers in early stages. Another advantage of our novel model is that it works with normalized values of RNA sequencing data, hence people's private sensitive medical data will remain hidden, protected, and safe. This type of analysis will be widespread and economical in the future and people can even learn to receive their RNA sequencing data and do their own preliminary cancer studies themselves which have the potential to help the healthcare systems. It is a great step forward toward good health that is the main base of sustainable societies.

Update: This post is part of a blog series on Meta-Learning that I'm working on. Check out part 1 and part 2. In my previous post, "Meta-Learning Is All You Need," I discussed the motivation for the meta-learning paradigm, explained the mathematical underpinning, and reviewed the three approaches to design a meta-learning algorithm (namely, black-box, optimization-based, and non-parametric). I also mentioned in the post that there are two views of the meta-learning problem: a deterministic view and a probabilistic view, according to Chelsea Finn. Note: The content of this post is primarily based on CS330's lecture 5 on Bayesian meta-learning. It is accessible to the public.

Numerous Bayesian Network (BN) structure learning algorithms have been proposed in the literature over the past few decades. Each publication makes an empirical or theoretical case for the algorithm proposed in that publication and results across studies are often inconsistent in their claims about which algorithm is 'best'. This is partly because there is no agreed evaluation approach to determine their effectiveness. Moreover, each algorithm is based on a set of assumptions, such as complete data and causal sufficiency, and tend to be evaluated with data that conforms to these assumptions, however unrealistic these assumptions may be in the real world. As a result, it is widely accepted that synthetic performance overestimates real performance, although to what degree this may happen remains unknown. This paper investigates the performance of 15 structure learning algorithms. We propose a methodology that applies the algorithms to data that incorporates synthetic noise, in an effort to better understand the performance of structure learning algorithms when applied to real data. Each algorithm is tested over multiple case studies, sample sizes, types of noise, and assessed with multiple evaluation criteria. This work involved approximately 10,000 graphs with a total structure learning runtime of seven months. It provides the first large-scale empirical validation of BN structure learning algorithms under different assumptions of data noise. The results suggest that traditional synthetic performance may overestimate real-world performance by anywhere between 10% and more than 50%. They also show that while score-based learning is generally superior to constraint-based learning, a higher fitting score does not necessarily imply a more accurate causal graph. To facilitate comparisons with future studies, we have made all data, raw results, graphs and BN models freely available online.