arXiv subjects

Natesh S. Pillai

Publications and source records attributed to Natesh S. Pillai.

At least 37 records · Page 2Linked to original sources

arXiv2016

More Powerful Multiple Testing in Randomized Experiments with Non-Compliance

Two common concerns raised in analyses of randomized experiments are (i) appropriately handling issues of non-compliance, and (ii) appropriately adjusting for multiple tests (e.g., on multiple outcomes or subgroups). Although simple intention-to-treat (ITT) and Bonferroni methods are valid in terms of type I error, they can each lead to a substantial loss of power; when employing both simultaneously, the total loss may be severe. Alternatives exist to address each concern. Here we propose an analysis method for experiments involving both features that merges posterior predictive $p$-values for complier causal effects with randomization-based multiple comparisons adjustments; the results are valid familywise tests that are doubly advantageous: more powerful than both those based on standard ITT statistics and those using traditional multiple comparison adjustments. The operating characteristics and advantages of our method are demonstrated through a series of simulated experiments and an analysis of the United States Job Training Partnership Act (JTPA) Study, where our methods lead to different conclusions regarding the significance of estimated JTPA effects.

stat.ME↗

arXiv2016

Sub-optimality of some continuous shrinkage priors

Two-component mixture priors provide a traditional way to induce sparsity in high-dimensional Bayes models. However, several aspects of such a prior, including computational complexities in high-dimensions, interpretation of exact zeros and non-sparse posterior summaries under standard loss functions, has motivated an amazing variety of continuous shrinkage priors, which can be expressed as global-local scale mixtures of Gaussians. Interestingly, we demonstrate that many commonly used shrinkage priors, including the Bayesian Lasso, do not have adequate posterior concentration in high-dimensional settings.

math.ST↗

arXiv2016

Kac's Walk on $n$-sphere mixes in $n\log n$ steps

Determining the mixing time of Kac's random walk on the sphere $\mathrm{S}^{n-1}$ is a long-standing open problem. We show that the total variation mixing time of Kac's walk on $\mathrm{S}^{n-1}$ is between $\frac{1}{2} \, n \log(n)$ and $200 \,n \log(n)$. Our bound is thus optimal up to a constant factor, improving on the best-known upper bound of $O(n^{5} \log(n)^{2})$ due to Jiang. Our main tool is a `non-Markovian' coupling recently introduced by the second author for obtaining the convergence rates of certain high dimensional Gibbs samplers in continuous state spaces.

math.PR↗

arXiv2016

Ratios and Cauchy Distribution

It is well known that the ratio of two independent standard Gaussian random variables follows a Cauchy distribution. Any convex combination of independent standard Cauchy random variables also follows a Cauchy distribution. In a recent joint work, the author proved a surprising multivariate generalization of the above facts. Fix $m > 1$ and let $Σ$ be a $m\times m$ positive semi-definite matrix. Let $X,Y \sim \mathrm{N}(0,Σ)$ be independent vectors. Let $\vec{w}=(w_1, \dots, w_m)$ be a vector of non-negative numbers with $\sum_{j=1}^m w_j = 1.$ The author proved recently that the random variable $$ Z = \sum_{j=1}^m w_j\frac{X_j}{Y_j}\; $$ also has the standard Cauchy distribution. In this note, we provide some more understanding of this result and give a number of natural generalizations. In particular, we observe that if $(X,Y)$ have the same marginal distribution, they need neither be independent nor be jointly normal for $Z$ to be Cauchy distributed. In fact, our calculations suggest that joint normality of $(X,Y)$ may be the only instance in which they can be independent. Our results also give a method to construct copulas of Cauchy distributions.

math.ST↗

arXiv2016

Maximum Likelihood Estimation for Single Particle, Passive Microrheology Data with Drift

Volume limitations and low yield thresholds of biological fluids have led to widespread use of passive microparticle rheology. The mean-squared-displacement (MSD) statistics of bead position time series (bead paths) are either applied directly to determine the creep compliance [Xu et al (1998)] or transformed to determine dynamic storage and loss moduli [Mason & Weitz (1995)]. A prevalent hurdle arises when there is a non-diffusive experimental drift in the data. Commensurate with the magnitude of drift relative to diffusive mobility, quantified by a Péclet number, the MSD statistics are distorted, and thus the path data must be "corrected" for drift. The standard approach is to estimate and subtract the drift from particle paths, and then calculate MSD statistics. We present an alternative, parametric approach using maximum likelihood estimation that simultaneously fits drift and diffusive model parameters from the path data; the MSD statistics (and consequently the compliance and dynamic moduli) then follow directly from the best-fit model. We illustrate and compare both methods on simulated path data over a range of Péclet numbers, where exact answers are known. We choose fractional Brownian motion as the numerical model because it affords tunable, sub-diffusive MSD statistics consistent with typical 30 second long, experimental observations of microbeads in several biological fluids. Finally, we apply and compare both methods on data from human bronchial epithelial cell culture mucus.

cond-mat.soft↗

arXiv2016

Elementary Bounds On Mixing Times for Decomposable Markov Chains

Many finite-state reversible Markov chains can be naturally decomposed into "projection" and "restriction" chains. In this paper we provide bounds on the total variation mixing times of the original chain in terms of the mixing properties of these related chains. This paper is in the tradition of existing bounds on Poincare and log-Sobolev constants of Markov chains in terms of similar decompositions [JSTV04, MR02, MR06, MY09]. Our proofs are simple, relying largely on recent results relating hitting and mixing times of reversible Markov chains [PS13, Oli12]. We describe situations in which our results give substantially better bounds than those obtained by applying existing decomposition results and provide examples for illustration.

math.PR↗

arXiv2015

Model comparison and assessment for single particle tracking in biological fluids

State-of-the-art techniques in passive particle-tracking microscopy provide high-resolution path trajectories of diverse foreign particles in biological fluids. For particles on the order of 1 micron diameter, these paths are generally inconsistent with simple Brownian motion. Yet, despite an abundance of data confirming these findings and their wide-ranging scientific implications, stochastic modeling of the complex particle motion has received comparatively little attention. Even among posited models, there is virtually no literature on likelihood-based inference, model comparisons, and other quantitative assessments. In this article, we develop a rigorous and computationally efficient Bayesian methodology to address this gap. We analyze two of the most prevalent candidate models for 30 second paths of 1 micron diameter tracer particles in human lung mucus: fractional Brownian motion (fBM) and a Generalized Langevin Equation (GLE) consistent with viscoelastic theory. Our model comparisons distinctly favor GLE over fBM, with the former describing the data remarkably well up to the timescales for which we have reliable information.

stat.AP↗

arXiv2015

An unexpected encounter with Cauchy and Lévy

The Cauchy distribution is usually presented as a mathematical curiosity, an exception to the Law of Large Numbers, or even as an "Evil" distribution in some introductory courses. It therefore surprised us when Drton and Xiao (2016) proved the following result for $m=2$ and conjectured it for $m\ge 3$. Let $X= (X_1,..., X_m)$ and $Y = (Y_1, ...,Y_m)$ be i.i.d $N(0,Σ)$, where $Σ=\{σ_{ij}\}\ge 0$ is an $m\times m$ and \textit{arbitrary} covariance matrix with $σ_{jj}>0$ for all $1\leq j\leq m$. Then $$Z = \sum_{j=1}^m w_j \frac{X_j}{Y_j} \ \sim \mathrm{Cauchy}(0,1),$$ as long as $w=(w_1,..., w_m) $ is independent of $(X, Y)$, $w_j\ge 0, j=1,..., m$, and $\sum_{j=1}^m w_j=1$. In this note, we present an elementary proof of this conjecture for any $m \geq 2$ by linking $Z$ to a geometric characterization of Cauchy(0,1) given in Willams (1969). This general result is essential to the large sample behavior of Wald tests in many applications such as factor models and contingency tables. It also leads to other unexpected results such as $$ \sum_{i=1}^m\sum_{j=1}^m \frac{w_iw_jσ_{ij}}{X_iX_j} \sim {\text{Lévy}}(0, 1). $$ This generalizes the "super Cauchy phenomenon" that the average of $m$ i.i.d. standard Lévy variables (i.e., inverse chi-squared variables with one degree of freedom) has the same distribution as that of a single standard Lévy variable multiplied by $m$ (which is obtained by taking $w_j=1/m$ and $Σ$ to be the identity matrix).

math.ST↗

arXiv2015

Accelerating Asymptotically Exact MCMC for Computationally Intensive Models via Local Approximations

We construct a new framework for accelerating Markov chain Monte Carlo in posterior sampling problems where standard methods are limited by the computational cost of the likelihood, or of numerical models embedded therein. Our approach introduces local approximations of these models into the Metropolis-Hastings kernel, borrowing ideas from deterministic approximation theory, optimization, and experimental design. Previous efforts at integrating approximate models into inference typically sacrifice either the sampler's exactness or efficiency; our work seeks to address these limitations by exploiting useful convergence characteristics of local approximations. We prove the ergodicity of our approximate Markov chain, showing that it samples asymptotically from the \emph{exact} posterior distribution of interest. We describe variations of the algorithm that employ either local polynomial approximations or local Gaussian process regressors. Our theoretical results reinforce the key observation underlying this paper: when the likelihood has some \emph{local} regularity, the number of model evaluations per MCMC step can be greatly reduced without biasing the Monte Carlo average. Numerical experiments demonstrate multiple order-of-magnitude reductions in the number of forward model evaluations used in representative ODE and PDE inference problems, with both synthetic and real data.

stat.ME↗

arXiv2015

Bayesian Nonparametric Weighted Sampling Inference

It has historically been a challenge to perform Bayesian inference in a design-based survey context. The present paper develops a Bayesian model for sampling inference in the presence of inverse-probability weights. We use a hierarchical approach in which we model the distribution of the weights of the nonsampled units in the population and simultaneously include them as predictors in a nonparametric Gaussian process regression. We use simulation studies to evaluate the performance of our procedure and compare it to the classical design-based estimator. We apply our method to the Fragile Family and Child Wellbeing Study. Our studies find the Bayesian nonparametric finite population estimator to be more robust than the classical design-based estimator without loss in efficiency, which works because we induce regularization for small cells and thus this is a way of automatically smoothing the highly variable weights.

stat.ME↗

arXiv2015

Ergodicity of Approximate MCMC Chains with Applications to Large Data Sets

In many modern applications, difficulty in evaluating the posterior density makes performing even a single MCMC step slow. This difficulty can be caused by intractable likelihood functions, but also appears for routine problems with large data sets. Many researchers have responded by running approximate versions of MCMC algorithms. In this note, we develop quantitative bounds for showing the ergodicity of these approximate samplers. We then use these bounds to study the bias-variance trade-off of approximate MCMC algorithms. We apply our results to simple versions of recently proposed algorithms, including a variant of the "austerity" framework of Korratikara et al.

math.ST↗

arXiv2015

Gaussian Process Regression with Location Errors

In this paper, we investigate Gaussian process regression models where inputs are subject to measurement error. In spatial statistics, input measurement errors occur when the geographical locations of observed data are not known exactly. Such sources of error are not special cases of "nugget" or microscale variation, and require alternative methods for both interpolation and parameter estimation. Gaussian process models do not straightforwardly extend to incorporate input measurement error, and simply ignoring noise in the input space can lead to poor performance for both prediction and parameter inference. We review and extend existing theory on prediction and estimation in the presence of location errors, and show that ignoring location errors may lead to Kriging that is not "self-efficient". We also introduce a Markov Chain Monte Carlo (MCMC) approach using the Hybrid Monte Carlo algorithm that obtains optimal (minimum MSE) predictions, and discuss situations that lead to multimodality of the target distribution and/or poor chain mixing. Through simulation study and analysis of global air temperature data, we show that appropriate methods for incorporating location measurement error are essential to valid inference in this regime.

stat.ME↗

arXiv2015

Degrees of freedom for combining regression with factor analysis

In the AGEMAP genomics study, researchers were interested in detecting genes related to age in a variety of tissue types. After not finding many age-related genes in some of the analyzed tissue types, the study was criticized for having low power. It is possible that the low power is due to the presence of important unmeasured variables, and indeed we find that a latent factor model appears to explain substantial variability not captured by measured covariates. We propose including the estimated latent factors in a multiple regression model. The key difficulty in doing so is assigning appropriate degrees of freedom to the estimated factors to obtain unbiased error variance estimators and enable valid hypothesis testing. When the number of responses is large relative to the sample size, treating the estimated factors like observed covariates leads to a downward bias in the variance estimates. Many ad-hoc solutions to this problem have been proposed in the literature without the backup of a careful theoretical analysis. Using recent results from random matrix theory, we derive a simple, easy to use expression for degrees of freedom. Our estimate gives a principled alternative to ad-hoc approaches in common use. Extensive simulation results show excellent agreement between the proposed estimator and its theoretical value. Applying our methodology to the AGEMAP genomics study, we found an order of magnitude increase in the number of significant genes. Although we focus on the AGEMAP study, the methods developed in this paper are widely applicable to other multivariate models, and thus are of independent interest.

stat.ME↗

arXiv2015

Hypothesis testing for high-dimensional sparse binary regression

In this paper, we study the detection boundary for minimax hypothesis testing in the context of high-dimensional, sparse binary regression models. Motivated by genetic sequencing association studies for rare variant effects, we investigate the complexity of the hypothesis testing problem when the design matrix is sparse. We observe a new phenomenon in the behavior of detection boundary which does not occur in the case of Gaussian linear regression. We derive the detection boundary as a function of two components: a design matrix sparsity index and signal strength, each of which is a function of the sparsity of the alternative. For any alternative, if the design matrix sparsity index is too high, any test is asymptotically powerless irrespective of the magnitude of signal strength. For binary design matrices with the sparsity index that is not too high, our results are parallel to those in the Gaussian case. In this context, we derive detection boundaries for both dense and sparse regimes. For the dense regime, we show that the generalized likelihood ratio is rate optimal; for the sparse regime, we propose an extended Higher Criticism Test and show it is rate optimal and sharp. We illustrate the finite sample properties of the theoretical results using simulation studies.

math.ST↗

arXiv2015

Mixing times for a constrained Ising process on the torus at low density

We study a kinetically constrained Ising process (KCIP) associated with a graph G and density parameter p; this process is an interacting particle system with state space $\{0,1\}^{G}$. The stationary distribution of the KCIP Markov chain is the Binomial($|G|, p$) distribution on the number of particles, conditioned on having at least one particle. The `constraint' in the name of the process refers to the rule that a vertex cannot change its state unless it has at least one neighbour in state `1'. The KCIP has been proposed by statistical physicists as a model for the glass transition, and more recently as a simple algorithm for data storage in computer networks. In this note, we study the mixing time of this process on the torus $G = \mathbb{Z}_{L}^{d}$, $d \geq 3$, in the low-density regime $p = \frac{c}{n}$ for arbitrary $0 < c < 1$; this regime is the subject of a conjecture of Aldous and is natural in the context of computer networks. Our results provide a counterexample to Aldous' conjecture, suggest a natural modifcation of the conjecture, and show that this modifcation is correct up to logarithmic factors. The methods developed in this paper also provide a strategy for tackling Aldous' conjecture for other graphs.

math.PR↗

arXiv2014

A location-mixture autoregressive model for online forecasting of lung tumor motion

Lung tumor tracking for radiotherapy requires real-time, multiple-step ahead forecasting of a quasi-periodic time series recording instantaneous tumor locations. We introduce a location-mixture autoregressive (LMAR) process that admits multimodal conditional distributions, fast approximate inference using the EM algorithm and accurate multiple-step ahead predictive distributions. LMAR outperforms several commonly used methods in terms of out-of-sample prediction accuracy using clinical data from lung tumor patients. With its superior predictive performance and real-time computation, the LMAR model could be effectively implemented for use in current tumor tracking systems.

stat.ME↗

arXiv2014

Finite Sample Properties of Adaptive Markov Chains via Curvature

Adaptive Markov chains are an important class of Monte Carlo methods for sampling from probability distributions. The time evolution of adaptive algorithms depends on past samples, and thus these algorithms are non-Markovian. Although there has been previous work establishing conditions for their ergodicity, not much is known theoretically about their finite sample properties. In this paper, using a notion of discrete Ricci curvature for Markov kernels introduced by Ollivier, we establish concentration inequalities and finite sample bounds for a class of adaptive Markov chains. After establishing some general results, we give quantitative bounds for `multi-level' adaptive algorithms such as the equi-energy sampler. We also provide the first rigorous proofs that the finite sample properties of an equi-energy sampler are superior to those of related parallel tempering and Metropolis-Hastings samplers after a learning period comparable to their mixing times.

math.PR↗

arXiv2014

Posterior contraction in sparse Bayesian factor models for massive covariance matrices

Sparse Bayesian factor models are routinely implemented for parsimonious dependence modeling and dimensionality reduction in high-dimensional applications. We provide theoretical understanding of such Bayesian procedures in terms of posterior convergence rates in inferring high-dimensional covariance matrices where the dimension can be larger than the sample size. Under relevant sparsity assumptions on the true covariance matrix, we show that commonly-used point mass mixture priors on the factor loadings lead to consistent estimation in the operator norm even when $p\gg n$. One of our major contributions is to develop a new class of continuous shrinkage priors and provide insights into their concentration around sparse vectors. Using such priors for the factor loadings, we obtain similar rate of convergence as obtained with point mass mixture priors. To obtain the convergence rates, we construct test functions to separate points in the space of high-dimensional covariance matrices using insights from random matrix theory; the tools developed may be of independent interest. We also derive minimax rates and show that the Bayesian posterior rates of convergence coincide with the minimax rates upto a $\sqrt{\log n}$ term.

math.ST↗