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Predictive power inference (PPI and PPI++) is a recently developed statistical method for computing confidence intervals and tests. It combines observations with machine-learning predictions. We use this technique to measure the association between the thickness of retinal layers and the time from the onset of Multiple Sclerosis (MS) symptoms. Further, we correlate the former with the Expanded Disability Status Scale, a measure of the progression of MS. In both cases, the confidence intervals provided with PPI++ improve upon standard statistical methodology, showing the advantage of PPI++ for answering inference problems in healthcare. 

Learning nonparametric systems of Ordinary Differential Equations (ODEs) $$\dot x = f(t,x)$$ from noisy data is an emerging machine learning topic. We use the well-developed theory of Reproducing Kernel Hilbert Spaces (RKHS) to define candidates for $$f$$ for which the solution of the ODE exists and is unique. Learning $$f$$ consists of solving a constrained optimization problem in an RKHS. We propose a penalty method that iteratively uses the Representer theorem and Euler approximations to provide a numerical solution. We prove a generalization bound for the $L^2$ distance between $$x$$ and its estimator. Experiments are provided for the FitzHugh–Nagumo oscillator, the Lorenz system, and for predicting the Amyloid level in the cortex of aging subjects. In all cases, we show competitive results compared with the state-of-the-art. 

We consider the problem of active learning for level set estimation (LSE), where the goal is to localize all regions where a function of interest lies above/below a given threshold as quickly as possible. We present a finite-horizon search procedure to perform LSE in one dimension while optimally balancing both the final estimation error and the distance traveled during active learning for a fixed number of samples. A tuning parameter is used to trade off between the estimation accuracy and distance traveled. We show that the resulting optimization problem can be solved in closed form and that the resulting policy generalizes existing approaches to this problem. We then show how this approach can be used to perform level set estimation in two dimensions, under some additional assumptions, under the popular Gaussian process model. Empirical results on synthetic data indicate that as the cost of travel increases, our method's ability to treat distance nonmyopically allows it to significantly improve on the state of the art. On real air quality data, our approach achieves roughly one fifth the estimation error at less than half the cost of competing algorithms. 

We consider the problem of active learning in the context of spatial sampling for boundary estimation, where the goal is to estimate an unknown boundary as accurately and quickly as possible. We present a finite-horizon search procedure to optimally minimize both the final estimation error and the distance traveled for a fixed number of samples, where a tuning parameter is used to trade off between the estimation accuracy and distance traveled. We show that the resulting optimization problem can be solved in closed form and that the resulting policy generalizes existing approaches to this problem.