arXiv subjects

Debdeep Pati

Publications and source records attributed to Debdeep Pati.

At least 55 records · Page 3Linked to original sources

arXiv2020

Statistical optimality and stability of tangent transform algorithms in logit models

A systematic approach to finding variational approximation in an otherwise intractable non-conjugate model is to exploit the general principle of convex duality by minorizing the marginal likelihood that renders the problem tractable. While such approaches are popular in the context of variational inference in non-conjugate Bayesian models, theoretical guarantees on statistical optimality and algorithmic convergence are lacking. Focusing on logistic regression models, we provide mild conditions on the data generating process to derive non-asymptotic upper bounds to the risk incurred by the variational optima. We demonstrate that these assumptions can be completely relaxed if one considers a slight variation of the algorithm by raising the likelihood to a fractional power. Next, we utilize the theory of dynamical systems to provide convergence guarantees for such algorithms in logistic and multinomial logit regression. In particular, we establish local asymptotic stability of the algorithm without any assumptions on the data-generating process. We explore a special case involving a semi-orthogonal design under which a global convergence is obtained. The theory is further illustrated using several numerical studies.

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arXiv2020

Statistical Guarantees and Algorithmic Convergence Issues of Variational Boosting

We provide statistical guarantees for Bayesian variational boosting by proposing a novel small bandwidth Gaussian mixture variational family. We employ a functional version of Frank-Wolfe optimization as our variational algorithm and study frequentist properties of the iterative boosting updates. Comparisons are drawn to the recent literature on boosting, describing how the choice of the variational family and the discrepancy measure affect both convergence and finite-sample statistical properties of the optimization routine. Specifically, we first demonstrate stochastic boundedness of the boosting iterates with respect to the data generating distribution. We next integrate this within our algorithm to provide an explicit convergence rate, ending with a result on the required number of boosting updates.

stat.ML↗

arXiv2020

Evidence bounds in singular models: probabilistic and variational perspectives

The marginal likelihood or evidence in Bayesian statistics contains an intrinsic penalty for larger model sizes and is a fundamental quantity in Bayesian model comparison. Over the past two decades, there has been steadily increasing activity to understand the nature of this penalty in singular statistical models, building on pioneering work by Sumio Watanabe. Unlike regular models where the Bayesian information criterion (BIC) encapsulates a first-order expansion of the logarithm of the marginal likelihood, parameter counting gets trickier in singular models where a quantity called the real log canonical threshold (RLCT) summarizes the effective model dimensionality. In this article, we offer a probabilistic treatment to recover non-asymptotic versions of established evidence bounds as well as prove a new result based on the Gibbs variational inequality. In particular, we show that mean-field variational inference correctly recovers the RLCT for any singular model in its canonical or normal form. We additionally exhibit sharpness of our bound by analyzing the dynamics of a general purpose coordinate ascent algorithm (CAVI) popularly employed in variational inference.

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arXiv2020

Dynamics of coordinate ascent variational inference: A case study in 2D Ising models

Variational algorithms have gained prominence over the past two decades as a scalable computational environment for Bayesian inference. In this article, we explore tools from the dynamical systems literature to study convergence of coordinate ascent algorithms for mean field variational inference. Focusing on the Ising model defined on two nodes, we fully characterize the dynamics of the sequential coordinate ascent algorithm and its parallel version. We observe that in the regime where the objective function is convex, both the algorithms are stable and exhibit convergence to the unique fixed point. Our analyses reveal interesting {\em discordances} between these two versions of the algorithm in the region when the objective function is non-convex. In fact, the parallel version exhibits a periodic oscillatory behavior which is absent in the sequential version. Drawing intuition from the Markov chain Monte Carlo literature, we {\em empirically} show that a parameter expansion of the Ising model, popularly called as the Edward--Sokal coupling, leads to an enlargement of the regime of convergence to the global optima.

math.DS↗

arXiv2020

Bayesian Hierarchical Modeling on Covariance Valued Data

Analysis of structural and functional connectivity (FC) of human brains is of pivotal importance for diagnosis of cognitive ability. The Human Connectome Project (HCP) provides an excellent source of neural data across different regions of interest (ROIs) of the living human brain. Individual specific data were available from an existing analysis (Dai et al., 2017) in the form of time varying covariance matrices representing the brain activity as the subjects perform a specific task. As a preliminary objective of studying the heterogeneity of brain connectomics across the population, we develop a probabilistic model for a sample of covariance matrices using a scaled Wishart distribution. We stress here that our data units are available in the form of covariance matrices, and we use the Wishart distribution to create our likelihood function rather than its more common usage as a prior on covariance matrices. Based on empirical explorations suggesting the data matrices to have low effective rank, we further model the center of the Wishart distribution using an orthogonal factor model type decomposition. We encourage shrinkage towards a low rank structure through a novel shrinkage prior and discuss strategies to sample from the posterior distribution using a combination of Gibbs and slice sampling. We extend our modeling framework to a dynamic setting to detect change points. The efficacy of the approach is explored in various simulation settings and exemplified on several case studies including our motivating HCP data. We extend our modeling framework to a dynamic setting to detect change points.

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arXiv2020

Nonasymptotic Laplace approximation under model misspecification

We present non-asymptotic two-sided bounds to the log-marginal likelihood in Bayesian inference. The classical Laplace approximation is recovered as the leading term. Our derivation permits model misspecification and allows the parameter dimension to grow with the sample size. We do not make any assumptions about the asymptotic shape of the posterior, and instead require certain regularity conditions on the likelihood ratio and that the posterior to be sufficiently concentrated.

math.ST↗

arXiv2020

Mass-shifting phenomenon of truncated multivariate normal priors

We show that lower-dimensional marginal densities of dependent zero-mean normal distributions truncated to the positive orthant exhibit a mass-shifting phenomenon. Despite the truncated multivariate normal density having a mode at the origin, the marginal density assigns increasingly small mass near the origin as the dimension increases. The phenomenon accentuates with stronger correlation between the random variables. A precise quantification characterizing the role of the dimension as well as the dependence is provided. This surprising behavior has serious implications towards Bayesian constrained estimation and inference, where the prior, in addition to having a full support, is required to assign a substantial probability near the origin to capture at parts of the true function of interest. Without further modification, we show that truncated normal priors are not suitable for modeling at regions and propose a novel alternative strategy based on shrinking the coordinates using a multiplicative scale parameter. The proposed shrinkage prior is empirically shown to guard against the mass shifting phenomenon while retaining computational efficiency.

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arXiv2019

Bayesian Copula Density Deconvolution for Zero-Inflated Data in Nutritional Epidemiology

Estimating the marginal and joint densities of the long-term average intakes of different dietary components is an important problem in nutritional epidemiology. Since these variables cannot be directly measured, data are usually collected in the form of 24-hour recalls of the intakes, which show marked patterns of conditional heteroscedasticity. Significantly compounding the challenges, the recalls for episodically consumed dietary components also include exact zeros. The problem of estimating the density of the latent long-time intakes from their observed measurement error contaminated proxies is then a problem of deconvolution of densities with zero-inflated data. We propose a Bayesian semiparametric solution to the problem, building on a novel hierarchical latent variable framework that translates the problem to one involving continuous surrogates only. Crucial to accommodating important aspects of the problem, we then design a copula-based approach to model the involved joint distributions, adopting different modeling strategies for the marginals of the different dietary components. We design efficient Markov chain Monte Carlo algorithms for posterior inference and illustrate the efficacy of the proposed method through simulation experiments. Applied to our motivating nutritional epidemiology problems, compared to other approaches, our method provides more realistic estimates of the consumption patterns of episodically consumed dietary components.

stat.ME↗

arXiv2019

The Soft Multivariate Truncated Normal Distribution with Applications to Bayesian Constrained Estimation

We propose a new distribution, called the soft tMVN distribution, which provides a smooth approximation to the truncated multivariate normal (tMVN) distribution with linear constraints. An efficient blocked Gibbs sampler is developed to sample from the soft tMVN distribution in high dimensions. We provide theoretical support to the approximation capability of the soft tMVN and provide further empirical evidence thereof. The soft tMVN distribution can be used to approximate simulations from a multivariate truncated normal distribution with linear constraints, or itself as a prior in shape-constrained problems.

stat.CO↗

arXiv2019

Bayesian Graph Selection Consistency Under Model Misspecification

Gaussian graphical models are a popular tool to learn the dependence structure in the form of a graph among variables of interest. Bayesian methods have gained in popularity in the last two decades due to their ability to simultaneously learn the covariance and the graph and characterize uncertainty in the selection. For scalability of the Markov chain Monte Carlo algorithms, decomposability is commonly imposed on the graph space. A wide variety of graphical conjugate priors are proposed jointly on the covariance matrix and the graph with improved algorithms to search along the space of decomposable graphs, rendering the methods extremely popular in the context of multivariate dependence modeling. {\it An open problem} in Bayesian decomposable structure learning is whether the posterior distribution is able to select a meaningful decomposable graph that it is ``close'' in an appropriate sense to the true non-decomposable graph, when the dimension of the variables increases with the sample size. In this article, we explore specific conditions on the true precision matrix and the graph which results in an affirmative answer to this question using a commonly used hyper-inverse Wishart prior on the covariance matrix and a suitable complexity prior on the graph space, both in the well-specified and misspecified settings. In absence of structural sparsity assumptions, our strong selection consistency holds in a high dimensional setting where $p = O(n^α)$ for $α< 1/3$. We show when the true graph is non-decomposable, the posterior distribution on the graph concentrates on a set of graphs that are {\it minimal triangulations} of the true graph.

math.ST↗

arXiv2019

Efficient Bayesian shape-restricted function estimation with constrained Gaussian process priors

This article revisits the problem of Bayesian shape-restricted inference in the light of a recently developed approximate Gaussian process that admits an equivalent formulation of the shape constraints in terms of the basis coefficients. We propose a strategy to efficiently sample from the resulting constrained posterior by absorbing a smooth relaxation of the constraint in the likelihood and using circulant embedding techniques to sample from the unconstrained modified prior. We additionally pay careful attention to mitigate the computational complexity arising from updating hyperparameters within the covariance kernel of the Gaussian process. The developed algorithm is shown to be accurate and highly efficient in simulated and real data examples.

stat.ME↗

arXiv2018

Revisiting the proton-radius problem using constrained Gaussian processes

Background: The "proton radius puzzle" refers to an eight-year old problem that highlights major inconsistencies in the extraction of the charge radius of the proton from muonic Lamb-shift experiments as compared against experiments using elastic electron scattering. For the latter, the determination of the charge radius involves an extrapolation of the experimental form factor to zero momentum transfer. Purpose: To estimate the proton radius by introducing a novel non-parametric approach to model the electric form factor of the proton. Methods: Within a Bayesian paradigm, we develop a model flexible enough to fit the data without any parametric assumptions on the form factor. The Bayesian estimation is guided by imposing only two physical constraints on the form factor: (a) its value at zero momentum transfer (normalization) and (b) its overall shape, assumed to be a monotonically decreasing function of the momentum transfer. Variants of these assumptions are explored to assess the impact of these constraints. Results: So far our results are inconclusive in regard to the proton puzzle, as they depend on both, the assumed constrains and the range of experimental data used. For example, if only low momentum-transfer data is used, adopting only the normalization constraint provides a value compatible with the smaller muonic result, while imposing only the shape constraint favors the larger electronic value. Conclusions: We have presented a novel technique to estimate the proton radius from electron scattering data based on a non-parametric Gaussian process. We have shown the impact of the physical constraints imposed on the form factor and of the range of experimental data used. In this regard, we are hopeful that as this technique is refined and with the anticipated new results from the PRad experiment, we will get closer to resolve of the puzzle.

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arXiv2018

Shape-Constrained Univariate Density Estimation

While the problem of estimating a probability density function (pdf) from its observations is classical, the estimation under additional shape constraints is both important and challenging. We introduce an efficient, geometric approach for estimating pdfs given the number of its modes. This approach explores the space of constrained pdf's using an action of the diffeomorphism group that preserves their shapes. It starts with an initial template, with the desired number of modes and arbitrarily chosen heights at the critical points, and transforms it via: (1) composition by diffeomorphisms and (2) normalization to obtain the final density estimate. The search for optimal diffeomorphism is performed under the maximum-likelihood criterion and is accomplished by mapping diffeomorphisms to the tangent space of a Hilbert sphere, a vector space whose elements can be expressed using an orthogonal basis. This framework is first applied to shape-constrained univariate, unconditional pdf estimation and then extended to conditional pdf estimation. We derive asymptotic convergence rates of the estimator and demonstrate this approach using a synthetic dataset involving speed distribution for different traffic flow on Californian driveways.

stat.ME↗

arXiv2018

Probabilistic community detection with unknown number of communities

A fundamental problem in network analysis is clustering the nodes into groups which share a similar connectivity pattern. Existing algorithms for community detection assume the knowledge of the number of clusters or estimate it a priori using various selection criteria and subsequently estimate the community structure. Ignoring the uncertainty in the first stage may lead to erroneous clustering, particularly when the community structure is vague. We instead propose a coherent probabilistic framework for simultaneous estimation of the number of communities and the community structure, adapting recently developed Bayesian nonparametric techniques to network models. An efficient Markov chain Monte Carlo (MCMC) algorithm is proposed which obviates the need to perform reversible jump MCMC on the number of clusters. The methodology is shown to outperform recently developed community detection algorithms in a variety of synthetic data examples and in benchmark real-datasets. Using an appropriate metric on the space of all configurations, we develop non-asymptotic Bayes risk bounds even when the number of clusters is unknown. Enroute, we develop concentration properties of non-linear functions of Bernoulli random variables, which may be of independent interest.

stat.ME↗

arXiv2018

$α$-Variational Inference with Statistical Guarantees

We propose a family of variational approximations to Bayesian posterior distributions, called $α$-VB, with provable statistical guarantees. The standard variational approximation is a special case of $α$-VB with $α=1$. When $α\in(0,1]$, a novel class of variational inequalities are developed for linking the Bayes risk under the variational approximation to the objective function in the variational optimization problem, implying that maximizing the evidence lower bound in variational inference has the effect of minimizing the Bayes risk within the variational density family. Operating in a frequentist setup, the variational inequalities imply that point estimates constructed from the $α$-VB procedure converge at an optimal rate to the true parameter in a wide range of problems. We illustrate our general theory with a number of examples, including the mean-field variational approximation to (low)-high-dimensional Bayesian linear regression with spike and slab priors, mixture of Gaussian models, latent Dirichlet allocation, and (mixture of) Gaussian variational approximation in regular parametric models.

math.ST↗

arXiv2017

On Statistical Optimality of Variational Bayes

The article addresses a long-standing open problem on the justification of using variational Bayes methods for parameter estimation. We provide general conditions for obtaining optimal risk bounds for point estimates acquired from mean-field variational Bayesian inference. The conditions pertain to the existence of certain test functions for the distance metric on the parameter space and minimal assumptions on the prior. A general recipe for verification of the conditions is outlined which is broadly applicable to existing Bayesian models with or without latent variables. As illustrations, specific applications to Latent Dirichlet Allocation and Gaussian mixture models are discussed.

math.ST↗

arXiv2017

A Two-Step Geometric Framework For Density Modeling

We introduce a novel two-step approach for estimating a probability density function (pdf) given its samples, with the second and important step coming from a geometric formulation. The procedure involves obtaining an initial estimate of the pdf and then transforming it via a warping function to reach the final estimate. The initial estimate is intended to be computationally fast, albeit suboptimal, but its warping creates a larger, flexible class of density functions, resulting in substantially improved estimation. The search for optimal warping is accomplished by mapping diffeomorphic functions to the tangent space of a Hilbert sphere, a vector space whose elements can be expressed using an orthogonal basis. Using a truncated basis expansion, we estimate the optimal warping under a (penalized) likelihood criterion and, thus, the optimal density estimate. This framework is introduced for univariate, unconditional pdf estimation and then extended to conditional pdf estimation. The approach avoids many of the computational pitfalls associated with classical conditional-density estimation methods, without losing on estimation performance. We derive asymptotic convergence rates of the density estimator and demonstrate this approach using both synthetic datasets and real data, the latter relating to the association of a toxic metabolite on preterm birth.

stat.ME↗

arXiv2017

Frequentist coverage and sup-norm convergence rate in Gaussian process regression

Gaussian process (GP) regression is a powerful interpolation technique due to its flexibility in capturing non-linearity. In this paper, we provide a general framework for understanding the frequentist coverage of point-wise and simultaneous Bayesian credible sets in GP regression. As an intermediate result, we develop a Bernstein von-Mises type result under supremum norm in random design GP regression. Identifying both the mean and covariance function of the posterior distribution of the Gaussian process as regularized $M$-estimators, we show that the sampling distribution of the posterior mean function and the centered posterior distribution can be respectively approximated by two population level GPs. By developing a comparison inequality between two GPs, we provide exact characterization of frequentist coverage probabilities of Bayesian point-wise credible intervals and simultaneous credible bands of the regression function. Our results show that inference based on GP regression tends to be conservative; when the prior is under-smoothed, the resulting credible intervals and bands have minimax-optimal sizes, with their frequentist coverage converging to a non-degenerate value between their nominal level and one. As a byproduct of our theory, we show that the GP regression also yields minimax-optimal posterior contraction rate relative to the supremum norm, which provides a positive evidence to the long standing problem on optimal supremum norm contraction rate in GP regression.

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