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One of the common tasks in unsupervised learning is dimensionality reduction, where the goal is to find meaningful low-dimensional structures hidden in high-dimensional data. Sometimes referred to as manifold learning, this problem is closely related to the problem of localization, which aims at embedding a weighted graph into a low-dimensional Euclidean space. Several methods have been proposed for localization, and also manifold learning. Nonetheless, the robustness property of most of them is little understood. In this paper, we obtain perturbation bounds for classical scaling and trilateration, which are then applied to derive performance bounds for Isomap, Landmark Isomap, and Maximum Variance Unfolding. A new perturbation bound for procrustes analysis plays a key role.

Sufficient dimension reduction (SDR) is a very useful concept for exploratory analysis and data visualization in regression, especially when the number of covariates is large. Many SDR methods have been proposed for regression with a continuous response, where the central subspace (CS) is the target of estimation. Various conditions, such as the linearity condition and the constant covariance condition, are imposed so that these methods can estimate at least a portion of the CS. In this paper we study SDR for regression and discriminant analysis with categorical response. Motivated by the exploratory analysis and data visualization aspects of SDR, we propose a new geometric framework to reformulate the SDR problem in terms of manifold optimization and introduce a new concept called Maximum Separation Subspace (MASES). The MASES naturally preserves the “sufficiency” in SDR without imposing additional conditions on the predictor distribution, and directly inspires a semi-parametric estimator. Numerical studies show MASES exhibits superior performance as compared with competing SDR methods in specific settings.

Michael Schweinberger, Jonathan Stewart.

Source: The Annals of Statistics, Volume 48, Number 1, 374--396.

Abstract:
Statistical inference for exponential-family models of random graphs with dependent edges is challenging. We stress the importance of additional structure and show that additional structure facilitates statistical inference. A simple example of a random graph with additional structure is a random graph with neighborhoods and local dependence within neighborhoods. We develop the first concentration and consistency results for maximum likelihood and $M$-estimators of a wide range of canonical and curved exponential-family models of random graphs with local dependence. All results are nonasymptotic and applicable to random graphs with finite populations of nodes, although asymptotic consistency results can be obtained as well. In addition, we show that additional structure can facilitate subgraph-to-graph estimation, and present concentration results for subgraph-to-graph estimators. As an application, we consider popular curved exponential-family models of random graphs, with local dependence induced by transitivity and parameter vectors whose dimensions depend on the number of nodes.

Alex Diana, Eleni Matechou, Jim Griffin, Alison Johnston.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 473--493.

Abstract:
Environmental changes in recent years have been linked to phenological shifts which in turn are linked to the survival of species. The work in this paper is motivated by capture-recapture data on blackcaps collected by the British Trust for Ornithology as part of the Constant Effort Sites monitoring scheme. Blackcaps overwinter abroad and migrate to the UK annually for breeding purposes. We propose a novel Bayesian nonparametric approach for expressing the bivariate density of individual arrival and departure times at different sites across a number of years as a mixture model. The new model combines the ideas of the hierarchical and the dependent Dirichlet process, allowing the estimation of site-specific weights and year-specific mixture locations, which are modelled as functions of environmental covariates using a multivariate extension of the Gaussian process. The proposed modelling framework is extremely general and can be used in any context where multivariate density estimation is performed jointly across different groups and in the presence of a continuous covariate.

Carolyn M. Rutter, Jonathan Ozik, Maria DeYoreo, Nicholson Collier.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2189--2212.

Abstract:
Microsimulation models (MSMs) are used to inform policy by predicting population-level outcomes under different scenarios. MSMs simulate individual-level event histories that mark the disease process (such as the development of cancer) and the effect of policy actions (such as screening) on these events. MSMs often have many unknown parameters; calibration is the process of searching the parameter space to select parameters that result in accurate MSM prediction of a wide range of targets. We develop Incremental Mixture Approximate Bayesian Computation (IMABC) for MSM calibration which results in a simulated sample from the posterior distribution of model parameters given calibration targets. IMABC begins with a rejection-based ABC step, drawing a sample of points from the prior distribution of model parameters and accepting points that result in simulated targets that are near observed targets. Next, the sample is iteratively updated by drawing additional points from a mixture of multivariate normal distributions and accepting points that result in accurate predictions. Posterior estimates are obtained by weighting the final set of accepted points to account for the adaptive sampling scheme. We demonstrate IMABC by calibrating CRC-SPIN 2.0, an updated version of a MSM for colorectal cancer (CRC) that has been used to inform national CRC screening guidelines.

Jorge A. León.

Source: Bernoulli, Volume 26, Number 3, 2436--2462.

Abstract:
Let $B^{H}$ be a fractional Brownian motion with Hurst parameter $Hin (0,1/2)$ and $p:mathbb{R} ightarrow mathbb{R}$ a polynomial function. The main purpose of this paper is to introduce a Stratonovich type stochastic integral with respect to $B^{H}$, whose domain includes the process $p(B^{H})$. That is, an integral that allows us to integrate $p(B^{H})$ with respect to $B^{H}$, which does not happen with the symmetric integral given by Russo and Vallois ( Probab. Theory Related Fields 97 (1993) 403–421) in general. Towards this end, we combine the approaches utilized by León and Nualart ( Stochastic Process. Appl. 115 (2005) 481–492), and Russo and Vallois ( Probab. Theory Related Fields 97 (1993) 403–421), whose aims are to extend the domain of the divergence operator for Gaussian processes and to define some stochastic integrals, respectively. Then, we study the relation between this Stratonovich integral and the extension of the divergence operator (see León and Nualart ( Stochastic Process. Appl. 115 (2005) 481–492)), an Itô formula and the existence of a unique solution of some Stratonovich stochastic differential equations. These last results have been analyzed by Alòs, León and Nualart ( Taiwanese J. Math. 5 (2001) 609–632), where the Hurst paramert $H$ belongs to the interval $(1/4,1/2)$.

Gábor Lugosi, Shahar Mendelson, Nikita Zhivotovskiy.

Source: Bernoulli, Volume 26, Number 3, 2253--2274.

Abstract:
We present results on the concentration properties of the spectral norm $|A_{p}|$ of the adjacency matrix $A_{p}$ of an Erdős–Rényi random graph $G(n,p)$. First, we consider the Erdős–Rényi random graph process and prove that $|A_{p}|$ is uniformly concentrated over the range $pin[Clog n/n,1]$. The analysis is based on delocalization arguments, uniform laws of large numbers, together with the entropy method to prove concentration inequalities. As an application of our techniques, we prove sharp sub-Gaussian moment inequalities for $|A_{p}|$ for all $pin[clog^{3}n/n,1]$ that improve the general bounds of Alon, Krivelevich, and Vu ( Israel J. Math. 131 (2002) 259–267) and some of the more recent results of Erdős et al. ( Ann. Probab. 41 (2013) 2279–2375). Both results are consistent with the asymptotic result of Füredi and Komlós ( Combinatorica 1 (1981) 233–241) that holds for fixed $p$ as $n oinfty$.

Sébastien Gadat, Sébastien Gerchinovitz, Clément Marteau.

Source: Bernoulli, Volume 26, Number 3, 1797--1831.

Abstract:
We consider the binary supervised classification problem with the Gaussian functional model introduced in ( Math. Methods Statist. 22 (2013) 213–225). Taking advantage of the Gaussian structure, we design a natural plug-in classifier and derive a family of upper bounds on its worst-case excess risk over Sobolev spaces. These bounds are parametrized by a separation distance quantifying the difficulty of the problem, and are proved to be optimal (up to logarithmic factors) through matching minimax lower bounds. Using the recent works of (In Advances in Neural Information Processing Systems (2014) 3437–3445 Curran Associates) and ( Ann. Statist. 44 (2016) 982–1009), we also derive a logarithmic lower bound showing that the popular $k$-nearest neighbors classifier is far from optimality in this specific functional setting.

Jing Lei.

Source: Bernoulli, Volume 26, Number 1, 767--798.

Abstract:
We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization can cover Euclidean spaces with large dimensionality, with the optimal dependence on the dimensionality. Our method also covers the important case of Gaussian processes in separable Hilbert spaces, with rate-optimal upper bounds for functional data distributions whose coordinates decay geometrically or polynomially. Moreover, our bounds of the expected value can be combined with mean-concentration results to yield improved exponential tail probability bounds for the Wasserstein error of empirical measures under Bernstein-type or log Sobolev-type conditions.

Barack Obama said the “rule of law is at risk” following the justice department’s decision to drop charges against former Trump advisor Mike Flynn, as he issued a stark warning about the long-term impact on the American way of life by his successor.

Maxim Rabinovich, Aaditya Ramdas, Michael I. Jordan, Martin J. Wainwright.

Source: Bayesian Analysis, Volume 15, Number 2, 505--532.

Abstract:
Slow mixing is the central hurdle is applications of Markov chains, especially those used for Monte Carlo approximations (MCMC). In the setting of Bayesian inference, it is often only of interest to estimate the stationary expectations of a small set of functions, and so the usual definition of mixing based on total variation convergence may be too conservative. Accordingly, we introduce function-specific analogs of mixing times and spectral gaps, and use them to prove Hoeffding-like function-specific concentration inequalities. These results show that it is possible for empirical expectations of functions to concentrate long before the underlying chain has mixed in the classical sense, and we show that the concentration rates we achieve are optimal up to constants. We use our techniques to derive confidence intervals that are sharper than those implied by both classical Markov-chain Hoeffding bounds and Berry-Esseen-corrected central limit theorem (CLT) bounds. For applications that require testing, rather than point estimation, we show similar improvements over recent sequential testing results for MCMC. We conclude by applying our framework to real-data examples of MCMC, providing evidence that our theory is both accurate and relevant to practice.

Jeong Eun Lee, Geoff K. Nicholls, Robin J. Ryder.

Source: Bayesian Analysis, Volume 14, Number 4, 1245--1269.

Abstract:
We develop and apply two calibration procedures for checking the coverage of approximate Bayesian credible sets, including intervals estimated using Monte Carlo methods. The user has an ideal prior and likelihood, but generates a credible set for an approximate posterior based on some approximate prior and likelihood. We estimate the realised posterior coverage achieved by the approximate credible set. This is the coverage of the unknown “true” parameter if the data are a realisation of the user’s ideal observation model conditioned on the parameter, and the parameter is a draw from the user’s ideal prior. In one approach we estimate the posterior coverage at the data by making a semi-parametric logistic regression of binary coverage outcomes on simulated data against summary statistics evaluated on simulated data. In another we use Importance Sampling from the approximate posterior, windowing simulated data to fall close to the observed data. We illustrate our methods on four examples.

Mengyang Gu.

Source: Bayesian Analysis, Volume 14, Number 3, 877--905.

Abstract:
Gaussian stochastic process (GaSP) has been widely used in two fundamental problems in uncertainty quantification, namely the emulation and calibration of mathematical models. Some objective priors, such as the reference prior, are studied in the context of emulating (approximating) computationally expensive mathematical models. In this work, we introduce a new class of priors, called the jointly robust prior, for both the emulation and calibration. This prior is designed to maintain various advantages from the reference prior. In emulation, the jointly robust prior has an appropriate tail decay rate as the reference prior, and is computationally simpler than the reference prior in parameter estimation. Moreover, the marginal posterior mode estimation with the jointly robust prior can separate the influential and inert inputs in mathematical models, while the reference prior does not have this property. We establish the posterior propriety for a large class of priors in calibration, including the reference prior and jointly robust prior in general scenarios, but the jointly robust prior is preferred because the calibrated mathematical model typically predicts the reality well. The jointly robust prior is used as the default prior in two new R packages, called “RobustGaSP” and “RobustCalibration”, available on CRAN for emulation and calibration, respectively.

Barry E. Stein
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Christina Zelano
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CJ Pike
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KM Harris
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WR Softky
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HG Kuhn
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The enterprise IT environment is complex. Many systems, technologies and practices developed at various times coexist in the same world. With expectations for technological advancements at their peak, we're tasked with enabling these systems to work together harmoniously to support the continuous sharing of information. Systems and data must connect as if all information were native to each.