arXiv subjects

Debdeep Pati

Publications and source records attributed to Debdeep Pati.

77 records · Page 5Linked to original sources

arXiv2014

Dirichlet-Laplace priors for optimal shrinkage

Penalized regression methods, such as $L_1$ regularization, are routinely used in high-dimensional applications, and there is a rich literature on optimality properties under sparsity assumptions. In the Bayesian paradigm, sparsity is routinely induced through two-component mixture priors having a probability mass at zero, but such priors encounter daunting computational problems in high dimensions. This has motivated an amazing variety of continuous shrinkage priors, which can be expressed as global-local scale mixtures of Gaussians, facilitating computation. In sharp contrast to the frequentist literature, little is known about the properties of such priors and the convergence and concentration of the corresponding posterior distribution. In this article, we propose a new class of Dirichlet--Laplace (DL) priors, which possess optimal posterior concentration and lead to efficient posterior computation exploiting results from normalized random measure theory. Finite sample performance of Dirichlet--Laplace priors relative to alternatives is assessed in simulated and real data examples.

math.ST↗

arXiv2013

Adaptive Posterior Convergence Rates in Bayesian Density Deconvolution with Supersmooth Errors

Bayesian density deconvolution using nonparametric prior distributions is a useful alternative to the frequentist kernel based deconvolution estimators due to its potentially wide range of applicability, straightforward uncertainty quantification and generalizability to more sophisticated models. This article is the first substantive effort to theoretically quantify the behavior of the posterior in this recent line of research. In particular, assuming a known supersmooth error density, a Dirichlet process mixture of Normals on the true density leads to a posterior convergence rate same as the minimax rate $(\log n)^{-η/β}$ adaptively over the smoothness $η$ of an appropriate Hölder space of densities, where $β$ is the degree of smoothness of the error distribution. Our main contribution is achieving adaptive minimax rates with respect to the $L_p$ norm for $2 \leq p \leq \infty$ under mild regularity conditions on the true density. En route, we develop tight concentration bounds for a class of kernel based deconvolution estimators which might be of independent interest.

math.ST↗

arXiv2012

Bayesian shrinkage

Penalized regression methods, such as $L_1$ regularization, are routinely used in high-dimensional applications, and there is a rich literature on optimality properties under sparsity assumptions. In the Bayesian paradigm, sparsity is routinely induced through two-component mixture priors having a probability mass at zero, but such priors encounter daunting computational problems in high dimensions. This has motivated an amazing variety of continuous shrinkage priors, which can be expressed as global-local scale mixtures of Gaussians, facilitating computation. In sharp contrast to the corresponding frequentist literature, very little is known about the properties of such priors. Focusing on a broad class of shrinkage priors, we provide precise results on prior and posterior concentration. Interestingly, we demonstrate that most commonly used shrinkage priors, including the Bayesian Lasso, are suboptimal in high-dimensional settings. A new class of Dirichlet Laplace (DL) priors are proposed, which are optimal and lead to efficient posterior computation exploiting results from normalized random measure theory. Finite sample performance of Dirichlet Laplace priors relative to alternatives is assessed in simulations.

math.ST↗

arXiv2012

Bayesian hierarchical modeling of simply connected 2D shapes

Models for distributions of shapes contained within images can be widely used in biomedical applications ranging from tumor tracking for targeted radiation therapy to classifying cells in a blood sample. Our focus is on hierarchical probability models for the shape and size of simply connected 2D closed curves, avoiding the need to specify landmarks through modeling the entire curve while borrowing information across curves for related objects. Prevalent approaches follow a fundamentally different strategy in providing an initial point estimate of the curve and/or locations of landmarks, which are then fed into subsequent statistical analyses. Such two-stage methods ignore uncertainty in the first stage, and do not allow borrowing of information across objects in estimating object shapes and sizes. Our fully Bayesian hierarchical model is based on multiscale deformations within a linear combination of cyclic basis characterization, which facilitates automatic alignment of the different curves accounting for uncertainty. The characterization is shown to be highly flexible in representing 2D closed curves, leading to a nonparametric Bayesian prior with large support. Efficient Markov chain Monte Carlo methods are developed for simultaneous analysis of many objects. The methods are evaluated through simulation examples and applied to yeast cell imaging data.

stat.ME↗

arXiv2011

Posterior convergence rates in non-linear latent variable models

Non-linear latent variable models have become increasingly popular in a variety of applications. However, there has been little study on theoretical properties of these models. In this article, we study rates of posterior contraction in univariate density estimation for a class of non-linear latent variable models where unobserved U(0,1) latent variables are related to the response variables via a random non-linear regression with an additive error. Our approach relies on characterizing the space of densities induced by the above model as kernel convolutions with a general class of continuous mixing measures. The literature on posterior rates of contraction in density estimation almost entirely focuses on finite or countably infinite mixture models. We develop approximation results for our class of continuous mixing measures. Using an appropriate Gaussian process prior on the unknown regression function, we obtain the optimal frequentist rate up to a logarithmic factor under standard regularity conditions on the true density.

math.ST↗