Nonparametric Bayesian Mixed-effect Model: a Sparse Gaussian Process Approach

Multi-task learning models using Gaussian processes (GP) have been developed and successfully applied in various applications. The main difficulty with this approach is the computational cost of inference using the union of examples from all tasks. Therefore sparse solutions, that avoid using the entire data directly and instead use a set of informative "representatives" are desirable. The paper investigates this problem for the grouped mixed-effect GP model where each individual response is given by a fixed-effect, taken from one of a set of unknown groups, plus a random individual effect function that captures variations among individuals. Such models have been widely used in previous work but no sparse solutions have been developed. The paper presents the first sparse solution for such problems, showing how the sparse approximation can be obtained by maximizing a variational lower bound on the marginal likelihood, generalizing ideas from single-task Gaussian processes to handle the mixed-effect model as well as grouping. Experiments using artificial and real data validate the approach showing that it can recover the performance of inference with the full sample, that it outperforms baseline methods, and that it outperforms state of the art sparse solutions for other multi-task GP formulations.Comment: Preliminary version appeared in ECML201

Variational bridge constructs for approximate Gaussian process regression

This paper introduces a method to approximate Gaussian process regression by representing the problem as a stochastic differential equation and using variational inference to approximate solutions. The approximations are compared with full GP regression and generated paths are demonstrated to be indistinguishable from GP samples. We show that the approach extends easily to non-linear dynamics and discuss extensions to which the approach can be easily applied

Bayesian optimization for fitting 3D morphable models of brain structures

Localize target areas in deep brain stimulation is a difficult task, due to the shape variability that brain structures exhibit between patients. The main problem in this process is that the fitting procedure is carried out by a registration method that lacks of accuracy. In this paper we proposed a novel method for 3D brain structure fitting based on Bayesian optimization. We use a morphable model in order to capture the shape variability in a given set of brain structures. Then from the trained model, we perform a Bayesian optimization task with the aim to find the best shape parameters that deform the trained model, and fits accurately to a given brain structure. The experimental results show that by using an optimization framework based on Bayesian optimization, the model performs an accurate fitting over cortical brain structures (thalamus, amygdala and ventricle) in comparison with common fitting methods, such as iterative closest point

Inequality in the face of death: The income gradient in mortality of the spanish pre-recession working-age population

The purpose of this paper is to evaluate the association between socioeconomic status (SES) and mortality over a three-year period for working-age Spaniards (2007–2009), paying particular attention to the effect of income level. The analysis is relatively new in Spain, and the studies are limited. Neither income nor wealth are included in existing Spanish mortality studies. The main reason for this limitation is the nature of the data sets used, mainly Census Records. We overcome this problem by using data on 693, 994 individuals taken from a Social Security sampling and used to estimate the probabilities of death for each income decile and the mortality rate ratios in three different models: (1) using only income, controlled by age and sex, (2) adding socio-economic and geographical variables, and (3) adding level of education. However, the data used here also have some limitations. They do not include government employees, the military or the Department of Justice personnel, whose exclusion we believe causes an under-representation of highly educated people in our sample. The results confirm that there is a non-linear relationship between mortality and income. This non-linear relationship implies that income redistribution resulting from progressive taxation systems could lead to higher reductions in mortality for low-income groups than the reductions induced in the mortality of the high-income population, thus reducing overall mortality. © 2021 by the authors. Licensee MDPI, Basel, Switzerland

Open String Thermodynamics and D-Branes

We study the thermodynamics of open superstrings in the presence of $p$-dimensional D-branes. We get some finite temperature dualities relating the one-loop canonical free energy of open strings to the self-energy of D-branes at dual temperature. For the open bosonic string the inverse dual temperature is, as expected, the dual length under T-duality, $4\pi^{2}\alpha^{'}/\beta$. On the contrary, for the $SO(N)$, type-I superstring the dual temperature is given by $\beta$-duality, $2\pi^{2}\alpha^{'}/\beta$. We also study the emergence of the Hagedorn singularity in the dual description as triggered by the coupling of the D-brane to unphysical tachyons as well as the high temperature limit.Comment: 16 pages, harvmac (b), epsf, 2 figures included. Minor changes; discussion in section 4 has been expanded and two footnotes and a reference adde

Metanálisis de la obra de Louise Bourgeois a través del dibujo infantil

Sin resume

Differentially private regression and classification with sparse Gaussian processes

A continuing challenge for machine learning is providing methods to perform computation on data while ensuring the data remains private. In this paper we build on the provable privacy guarantees of differential privacy which has been combined with Gaussian processes through the previously published \emph{cloaking method}. In this paper we solve several shortcomings of this method, starting with the problem of predictions in regions with low data density. We experiment with the use of inducing points to provide a sparse approximation and show that these can provide robust differential privacy in outlier areas and at higher dimensions. We then look at classification, and modify the Laplace approximation approach to provide differentially private predictions. We then combine this with the sparse approximation and demonstrate the capability to perform classification in high dimensions. We finally explore the issue of hyperparameter selection and develop a method for their private selection. This paper and associated libraries provide a robust toolkit for combining differential privacy and GPs in a practical manner

A Parzen-based distance between probability measures as an alternative of summary statistics in Approximate Bayesian Computation

Approximate Bayesian Computation (ABC) are likelihood-free Monte Carlo methods. ABC methods use a comparison between simulated data, using different parameters drew from a prior distribution, and observed data. This comparison process is based on computing a distance between the summary statistics from the simulated data and the observed data. For complex models, it is usually difficult to define a methodology for choosing or constructing the summary statistics. Recently, a nonparametric ABC has been proposed, that uses a dissimilarity measure between discrete distributions based on empirical kernel embeddings as an alternative for summary statistics. The nonparametric ABC outperforms other methods including ABC, kernel ABC or synthetic likelihood ABC. However, it assumes that the probability distributions are discrete, and it is not robust when dealing with few observations. In this paper, we propose to apply kernel embeddings using an smoother density estimator or Parzen estimator for comparing the empirical data distributions, and computing the ABC posterior. Synthetic data and real data were used to test the Bayesian inference of our method. We compare our method with respect to state-of-the-art methods, and demonstrate that our method is a robust estimator of the posterior distribution in terms of the number of observations

Derivative curve estimation in longitudinal studies using P-splines

The estimation of curve derivatives is of interest in many disciplines. It allows the extraction of important characteristics to gain insight about the underlying process. In the context of longitudinal data, the derivative allows the description of biological features of the individuals or finding change regions of interest. Although there are several approaches to estimate subject-specific curves and their derivatives, there are still open problems due to the complicated nature of these time course processes. In this article, we illustrate the use of P-spline models to estimate derivatives in the context of longitudinal data. We also propose a new penalty acting at the population and the subject-specific levels to address under-smoothing and boundary problems in derivative estimation. The practical performance of the proposal is evaluated through simulations, and comparisons with an alternative method are reported. Finally, an application to longitudinal height measurements of 125 football players in a youth professional academy is presented, where the goal is to analyse their growth and maturity patterns over time.RYC2019-027534-I The Medical Services of Athletic Clu