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The Continuous-Time Hidden Markov Model (CT-HMM) is an attractive approach to modeling disease progression due to its ability to describe noisy observations arriving irregularly in time. However, the lack of an efficient parameter learning algorithm for CT-HMM restricts its use to very small models or requires unrealistic constraints on the state transitions. In this paper, we present the first complete characterization of efficient EM-based learning methods for CT-HMM models, as well as the first solution to decoding the optimal state transition sequence and the corresponding state dwelling time. We show that EM-based learning consists of two challenges: the estimation of posterior state probabilities and the computation of end-state conditioned statistics. We solve the first challenge by reformulating the estimation problem as an equivalent discrete time-inhomogeneous hidden Markov model. The second challenge is addressed by adapting three distinct approaches from the continuous time Markov chain (CTMC) literature to the CT-HMM domain. Additionally, we further improve the efficiency of the most efficient method by a factor of the number of states. Then, for decoding, we incorporate a state-of-the-art method from the (CTMC) literature, and extend the end-state conditioned optimal state sequence decoding to the CT-HMM case with the computation of the expected state dwelling time. We demonstrate the use of CT-HMMs with more than 100 states to visualize and predict disease progression using a glaucoma dataset and an Alzheimer's disease dataset, and to decode and visualize the most probable state transition trajectory for individuals on the glaucoma dataset, which helps to identify progressing phenotypes in a comprehensive way. Finally, we apply the CT-HMM modeling and decoding strategy to investigate the progression of language acquisition and development.

We study a regression problem on a compact manifold M. In order to take advantage of the underlying geometry and topology of the data, the regression task is performed on the basis of the first several eigenfunctions of the Laplace-Beltrami operator of the manifold, that are regularized with topological penalties. The proposed penalties are based on the topology of the sub-level sets of either the eigenfunctions or the estimated function. The overall approach is shown to yield promising and competitive performance on various applications to both synthetic and real data sets. We also provide theoretical guarantees on the regression function estimates, on both its prediction error and its smoothness (in a topological sense). Taken together, these results support the relevance of our approach in the case where the targeted function is "topologically smooth".

We consider the Bayesian analysis of models in which the unknown distribution of the outcomes is specified up to a set of conditional moment restrictions. The nonparametric exponentially tilted empirical likelihood function is constructed to satisfy a sequence of unconditional moments based on an increasing (in sample size) vector of approximating functions (such as tensor splines based on the splines of each conditioning variable). For any given sample size, results are robust to the number of expanded moments. We derive Bernstein-von Mises theorems for the behavior of the posterior distribution under both correct and incorrect specification of the conditional moments, subject to growth rate conditions (slower under misspecification) on the number of approximating functions. A large-sample theory for comparing different conditional moment models is also developed. The central result is that the marginal likelihood criterion selects the model that is less misspecified. We also introduce sparsity-based model search for high-dimensional conditioning variables, and provide efficient MCMC computations for high-dimensional parameters. Along with clarifying examples, the framework is illustrated with real-data applications to risk-factor determination in finance, and causal inference under conditional ignorability.

Finite linear least squares is one of the core problems of numerical linear algebra, with countless applications across science and engineering. Consequently, there is a rich and ongoing literature on algorithms for solving linear least squares problems. In this paper, we explore a variant in which the system's matrix has one infinite dimension (i.e., it is a quasimatrix). We call such problems semi-infinite linear regression problems. As we show, the semi-infinite case arises in several applications, such as supervised learning and function approximation, and allows for novel interpretations of existing algorithms. We explore semi-infinite linear regression rigorously and algorithmically. To that end, we give a formal framework for working with quasimatrices, and generalize several algorithms designed for the finite problem to the infinite case. Finally, we suggest the use of various sampling methods for obtaining an approximate solution.

In this paper several related estimation problems are addressed from a Bayesian point of view and optimal estimators are obtained for each of them when some natural loss functions are considered. Namely, we are interested in estimating a regression curve. Simultaneously, the estimation problems of a conditional distribution function, or a conditional density, or even the conditional distribution itself, are considered. All these problems are posed in a sufficiently general framework to cover continuous and discrete, univariate and multivariate, parametric and non-parametric cases, without the need to use a specific prior distribution. The loss functions considered come naturally from the quadratic error loss function comonly used in estimating a real function of the unknown parameter. The cornerstone of the mentioned Bayes estimators is the posterior predictive distribution. Some examples are provided to illustrate these results.

Markov Chain Monte Carlo (MCMC) methods are a powerful tool for computation with complex probability distributions. However the performance of such methods is critically dependant on properly tuned parameters, most of which are difficult if not impossible to know a priori for a given target distribution. Adaptive MCMC methods aim to address this by allowing the parameters to be updated during sampling based on previous samples from the chain at the expense of requiring a new theoretical analysis to ensure convergence. In this work we extend the convergence theory of adaptive MCMC methods to a new class of methods built on a powerful class of parametric density estimators known as normalizing flows. In particular, we consider an independent Metropolis-Hastings sampler where the proposal distribution is represented by a normalizing flow whose parameters are updated using stochastic gradient descent. We explore the practical performance of this procedure on both synthetic settings and in the analysis of a physical field system and compare it against both adaptive and non-adaptive MCMC methods.

We address the problem of how to achieve optimal inference in distributed quantile regression without stringent scaling conditions. This is challenging due to the non-smooth nature of the quantile regression loss function, which invalidates the use of existing methodology. The difficulties are resolved through a double-smoothing approach that is applied to the local (at each data source) and global objective functions. Despite the reliance on a delicate combination of local and global smoothing parameters, the quantile regression model is fully parametric, thereby facilitating interpretation. In the low-dimensional regime, we discuss and compare several alternative confidence set constructions, based on inversion of Wald and score-type tests and resam-pling techniques, detailing an improvement that is effective for more extreme quantile coefficients. In high dimensions, a sparse framework is adopted, where the proposed doubly-smoothed objective function is complemented with an $\ell_1$-penalty. A thorough simulation study further elucidates our findings. Finally, we provide estimation theory and numerical studies for sparse quantile regression in the high-dimensional setting.

Understanding how treatment effects vary on individual characteristics is critical in the contexts of personalized medicine, personalized advertising and policy design. When the characteristics are of practical interest are only a subset of full covariate, non-parametric estimation is often desirable; but few methods are available due to the computational difficult. Existing non-parametric methods such as the inverse probability weighting methods have limitations that hinder their use in many practical settings where the values of propensity scores are close to 0 or 1. We propose the propensity score regression (PSR) that allows the non-parametric estimation of the heterogeneous treatment effects in a wide context. PSR includes two non-parametric regressions in turn, where it first regresses on the propensity scores together with the characteristics of interest, to obtain an intermediate estimate; and then, regress the intermediate estimates on the characteristics of interest only. By including propensity scores as regressors in the non-parametric manner, PSR is capable of substantially easing the computational difficulty while remain (locally) insensitive to any value of propensity scores. We present several appealing properties of PSR, including the consistency and asymptotical normality, and in particular the existence of an explicit variance estimator, from which the analytical behaviour of PSR and its precision can be assessed. Simulation studies indicate that PSR outperform existing methods in varying settings with extreme values of propensity scores. We apply our method to the national 2009 flu survey (NHFS) data to investigate the effects of seasonal influenza vaccination and having paid sick leave across different age groups.

We theoretically analyze the typical learning performance of $\ell_{1}$-regularized linear regression ($\ell_1$-LinR) for Ising model selection using the replica method from statistical mechanics. For typical random regular graphs in the paramagnetic phase, an accurate estimate of the typical sample complexity of $\ell_1$-LinR is obtained. Remarkably, despite the model misspecification, $\ell_1$-LinR is model selection consistent with the same order of sample complexity as $\ell_{1}$-regularized logistic regression ($\ell_1$-LogR), i.e., $M=\mathcal{O}\left(\log N\right)$, where $N$ is the number of variables of the Ising model. Moreover, we provide an efficient method to accurately predict the non-asymptotic behavior of $\ell_1$-LinR for moderate $M, N$, such as precision and recall. Simulations show a fairly good agreement between theoretical predictions and experimental results, even for graphs with many loops, which supports our findings. Although this paper mainly focuses on $\ell_1$-LinR, our method is readily applicable for precisely characterizing the typical learning performances of a wide class of $\ell_{1}$-regularized $M$-estimators including $\ell_1$-LogR and interaction screening.