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Some Contributions to Inference and Model/Variable Selection in High-Dimensional Problems
(2026-08-19) Paul, Sayantan
In today’s world, we often come across situations where we need to infer about several parameters simultaneously based on a single dataset. This is known as the problem of simultaneous statistical inference and the need for such inference arises for data from the fields of biology, astronomy, genomics, bioinformatics, medicine, economics, finance, and image processing, just to name a few. As the name suggests, instead of being concerned about the optimality of inference for the individual parameters, the main interest here is in proposing procedures which can provide satisfactory results in the overall inferential problem involving all parameters of interest. The need for simultaneous inference may arise in the contexts of hypothesis testing, point estimation of several parameters, and building confidence intervals. Such inference becomes challenging when the number of parameters increases with sample size at the same or at a higher rate, i.e., when the problem becomes the high-dimensional. Equally challenging are the problems of model selection or variable selection for data of such kind.
In this thesis, our interest is in simultaneous statistical inference and model/variable selection in the high-dimensional setting. One of our main goals would be to propose new inference and model/variable selection procedures and study their properties through theory and simulations. We would also employ some of our proposed methods in suitable real-life examples to understand how they perform in actual applications. We would also like to theoretically address interesting questions about existing procedures widely used in such contexts. An important focus of this work will be the special situations when the number of significant (which may mean nonzero or of sufficiently large magnitude in appropriate contexts) parameters is small compared to the total number of parameters under consideration. These are the so-called “sparse” situations. The assumption of sparsity is natural and very common in the literature, e.g., in the high-dimensional regression setting and in the problem of inference on a high dimensional mean vector. Our approaches here will be built under the Bayesian setting where the unknown parameters are assumed to follow some prior probability distributions. A big focus of our theoretical work will be on studying the optimality (in the frequentist or Bayesian decision theoretic sense) of the new methods proposed or of existing methods.
A natural Bayesian approach under sparse settings is to model the parameters by two-group spikeand-slab priors. These priors are expressed as mixtures of a distribution degenerate at 0 (or highly concentrated near 0) and a distribution with high spread. However, as noted by many authors, inference with such priors can be computationally very challenging in high-dimensional situations and complex parametric frameworks. As an alternative to two-group mixture priors, there have been proposals in the literature to consider unimodal, continuous priors having sufficient mass around zero while possessing sufficiently heavy tails. Such priors are named as one-group “global-local” priors, the name “global-local” originating from the use of two sets of parameters to simultaneously enforce and modulate shrinkage at the global level and at the levels of individual parameters respectively. These are called the global shrinkage parameter and the local shrinkage parameters respectively.
Although serious efforts have gone into studying optimality of inference using one-group priors in sparse parametric settings, several interesting questions in this connection are still unanswered and several areas somewhat less trodden in the current literature. This thesis is a modest attempt to address some of these in the context of specific models. A large part of our work will focus on inference on the famous normal means model and the linear regression model. The thesis concludes with a study of inference on non-normal count data. Our study is based on a broad class of one-group global-local shrinkage priors covering a lot of popularly used priors including the horseshoe.
When the mean parameter of the sparse normal means model is modeled by a spike-and-slab prior, it has been shown in the literature that the corresponding posterior distribution contracts around the truth at a near minimax rate when the level of sparsity is unknown. This raises the question of whether the same phenomenon works when one uses a one-group prior instead of its two-group counterpart for the same problem. We provide an answer to this question in Chapter 2 of the thesis. In order to handle the unknown level of sparsity, we either estimate the global shrinkage parameter based on the data, an empirical Bayes approach, or model it by a non-degenerate absolutely continuous prior distribution on a suitable support in a full Bayes approach. We establish that the posterior distributions of the mean parameter vector, when modelled by broad classes of one-group priors, contract around the truth at a near minimax rate, for both the empirical Bayes and full Bayes approaches. Another interesting question is whether one-group priors can be used to form a good decision rule for the simultaneous testing problem of whether the means are zero or not when the means are truly generated from a two-group prior. In the latter half of this chapter, we are interested in answering this question, assuming that the level of sparsity is unknown. Considering a full Bayes approach, we are to establish that the Bayes risk of a decision rule using a broad class of one-group priors attains the Bayes risk of the optimal rule for the two-group settings asymptotically for a wide range of sparsity levels. The loss function considered is the additive symmetric 0 − 1 loss measuring the number of misclassifications made by a multiple testing rule.
One of the main goals in a multiple hypothesis testing problem is to propose a decision rule which can control some overall measure of type I error rate, e.g., the False Discovery Rate (FDR). Given that a testing rule controls the FDR at some desired level, the next obvious question is whether the same rule can provide any control over the False Negative Rate (FNR). In this context, researchers are often interested in the optimal multiple testing rules in terms of controlling sum of certain type I and type II error measures. This can be answered by studying the minimax risk of multiple testing rules with respect to the corresponding loss functions. Very recently, for the normal means model, the expressions for the minimax risks based on misclassification (or Hamming) loss and the loss defined as the sum of FDP and FNP have been derived. It has also been proved that the famous Benjamini-Hochberg (BH) procedure and an ℓ−value based procedure using spike-and-slab priors attain the minimax risk asymptotically adaptively over broad sparsity levels. This motivates us to study whether decision rules based on one-group priors, if any, can enjoy such asymptotic optimality. When the level of sparsity is known, by choosing the global shrinkage parameter appropriately based on the knowledge of sparsity, we prove that the corresponding decision rule based on the broad class of one-group priors mentioned earlier achieves the minimax risk for both loss functions stated earlier. When the level of sparsity is unknown, some empirical Bayes and full Bayes versions of our decision rules can also attain the minimax risk. These results are proved in Chapter 3 of the thesis.
Another important problem of interest is variable/model selection in a high-dimensional normal linear regression model. In this thesis, we are interested in a situation where covariates under study inregression model. In Chapter 4 of this thesis, motivated by the existing literature, we are interested in proposing a decision rule and resulting estimators, based on a general class of global-local priors, which have the “Oracle property”, in the sense that, they achieve variable selection consistency and optimal estimation rate, respectively. We propose a decision rule which declares a group to be active if the ratio of the ℓ2 norm of the posterior mean of the group regression coefficient to that of the least square estimate exceeds half. When the design matrix is block-orthogonal, we are able to establish that the global shrinkage parameter can be chosen in such a way that our proposed inference procedures have both selection consistency and optimal estimation rate, provided the level of sparsity is known. Even if the sparsity pattern is unknown, our modified decision rules, by either estimating the global shrinkage parameter from the data or by modeling a prior on it, can still enjoy the oracle property. In the simulation studies, our rules perform favorably compared to many existing methods in a variety of sparsity settings. Our methods, when applied to real datasets, also return encouraging results.
In Chapters 2 through 4 of the thesis, our main areas of interest were the situations when the observed data are generated from normal distributions of various parametric forms. However, depending on the problem of interest, the Gaussianity assumption is not always proper. In Chapter 5 of this thesis, we are interested in one such example where the data consists of counts of events, most counts being close to zero while some are moderate or large. The data is modelled by Poisson distribution with unknown means. Clearly the natural prior for such means would be a two-group mixture with large mass on the component concentrated near zero. Our interest is in finding at if one-group modelling is still a good alternative in this context, as seen in previously in Chapters 2 through 4 of this thesis. Specifically, one of the questions this leads us to is whether any decision rule for multiple testing based on one-group priors can approximate the optimal rule with respect to two-group priors in terms of risk when the sample size gets large. Towards that we first obtain the asymptotic expression for the optimal Bayes risk under two-group prior in appropriate asymptotic framework. The loss is taken to be additive symmetric 0 − 1 loss. Next, irrespective of the sparsity pattern to be known or unknown, we establish that the Bayes risks corresponding to our proposed decision rules based on one-group priors attain the optimal Bayes risk, up to some multiplicative constant. Finally, the theoretical results are verified using simulation studies followed by a real data analysis. Many of the theoretical results derived in this thesis are the first of their kind in the literature in their specific contexts. Last but not the least, our theoretical results, simulations and to some extent real data analyses reinforce the logic of using appropriately chosen one-group priors as alternatives to their two-group counterparts in high-dimensional sparse parametric settings.
Indian Statistical Institute 4th review committee report, 2021
(Govt of India, 2021) India. Ministry of Statistics and Programme Implementation
The 4th Review Committee of ISI was set up through a Gazette Notification dated 28 January 2020 and its subsequent amendments. The first meeting of the Review Committee was fixed on 23 March 2020, but had to be postponed by government orders, which were issued in the wake of the first wave of the COVID-19 pandemic. This was followed by the devastating second wave, from which the nation is still slowly recovering. Thus, in spite of the best intentions of the 4th ISI Review Committee, all the meetings had to be held virtually. Despite all the constraints, the members of the 4th Review Committee spent maximum time and effort to gather an understanding of the institute, its legacy, work culture, glorious past, and its current state in terms of quality of research, relevance and impact, strengths and weaknesses, and future potential. Based on the evidence gathered during the review period using several methods and its analysis, the 4th Review Committee is presenting its detailed report along with a set of recommendations, which if implemented in the spirit in which they were conceived, will have far-reaching and positive ramifications for the institute. The Review Committee strongly feels that it is possible to Reimagine, Reinvent and Reposition the Indian Statistical Institute, as a frontline globally recognized institute. The Review Committee was impressed with the rich history of the Institute - the towering stalwarts who provided a strong foundation to the Institute, the outstanding researchers who went on to make pioneering contributions in Statistics and a number of allied areas globally, the forward-looking teaching programs which ISI initiated, the first computer in India that was brought to the ISI and numerous other such achievements. Even in the present times, the Institute has a formidable lineup of some excellent researchers working in the Institute, who have brought numerous laurels to the Institute. While appreciating the immense past contributions of ISI, the Review Committee came to the conclusion that over the past few decades, ISI has gradually lost the sheer brilliance and deep engagement with the external world that was its USP in the 60s and 70s. While the Institute continues to produce several outstanding students each year, it appears to have failed to keep pace with the changing times and has not scaled its effort to a level, where not only will it remain relevant, but also remain competitive. Presently, the Institute has individual level brilliance, but limited collective excellence and contribution. Thus, the Institute has very low presence in mega-scale national and international initiatives and projects. The Review Committee, through interactions with multiple stakeholders, concluded that ISI continues to have a perception of uniqueness relying more on past successes than current initiatives, suffers from a lethargy to generate resources leading to an almost full reliance on government funds, and produces only a limited number of students; thus distancing itself from the changes happening in the external world. The Review Committee, while recognizing the high potential of ISI, is convinced that the Institute urgently needs to make sincere attempts at course correction and digital transformation of almost every aspect of its functioning, in order to remain relevant and rise to the preeminent top position despite the stiff competition from the newly emerged and emerging top institutions in the country. It should scale substantially by building innovative teaching, training and application programs in Statistics, Machine Learning, Data Sciences and other contemporary areas, hugely ramp up student strength, and conduct truly original research activities in the frontier areas. ISI has to channelize a part of its effort towards proactively and aggressively reaching out to the government and the industry, in order to provide effective and high-quality solutions to their problems. In this way, not only will the scientists benefit by doing real-life translational research but also the visibility of the Institute will be enhanced significantly. Moreover, through such initiatives, ISI should strive to become much more financially independent. The Review Committee believes that the ISI can rise to its true potential and effect a turnaround, if it can bring in certain fundamental changes, some of which are not just incremental, but truly radical in its structure, systems and processes backed up by much higher aspirational levels, from top leadership, down to every worker. The Review Committee was alarmed to observe that the ISI is severely constrained in terms of faculty numbers, funding, infrastructure and other resources necessary to create a vibrant education, research and innovation ecosystem. It was also deeply concerned to note that accountability amongst the workers of the Institute, including the scientific workers, is low, work norms are scant, non-performance goes unpunished, and proposals on any fundamental reforms (including some suggested by the previous review committees) have so far proved difficult to implement. The Review Committee noted that despite many challenges, there are pockets of excellence in the Institute that are world class, and ISI can rise to its potential, which is huge. The Review Committee realizes that many challenges that are holding the Institute back from attaining its true potential are internal, and could and should have been addressed by the internal leadership. That this did not happen points to some fundamental flaws in the system. The Review Committee took a deep dive into the institutional entities, inter-relationships and mechanisms and concluded that ISI has dated governance structure that needs urgent overhaul and modernization. The Council of the Institute is bulky, with an unusually large proportion of it comprising Institute employees, almost all of whom, barring the Director and the Centre Heads, are elected. The Heads of the Scientific Divisions are elected, the representatives of the scientific and non-scientific workers to the Council are elected, the General Body members, who will be members of the council are elected, and even the Dean of Studies is elected. Elections test one’s popularity, which does not necessarily translate into a measure of merit and excellence in terms of capability and academic accomplishment, the cornerstones of all great institutions. An institution as great as the ISI can ill-afford to waste time in conducting institute wide elections every two years, and all the paraphernalia that is associated with electioneering, including the almost regular lobbying that frequent elections usually bring in their wake. Keeping these in mind, the Review Committee has made a number of recommendations that are targeted towards addressing the fundamental issues in the governance and administrative structure of the Institute. Besides these, some of the recommendations aim to rationalize the structure and focus of the scientific divisions, and distribution of scientific personnel across divisions, in order to make the scientific activities more aligned to the grand overarching objectives of the Institute. Related recommendations about creation, continuation and discontinuation of focused Centres/Units/Cells have been made to enhance productivity and to encourage inter- and cross-disciplinary research. Any academic institution advances mainly on the strength of its faculty, scientific staff, and students and scholars. Several of the recommendations of the Review Committee aim to increase accountability of the scientific members, set tangible targets of teaching, research supervision, fund generation, etc., provide a broad mechanism of feedback, link performance (in terms of research, teaching/training, administration and revenue generation) to future career prospects/institutional support, and strengthen and create a deep connect with the alumni, who are a valuable resource for any organization. The scientific staff of an academic institute can perform to their full potential and also outshine themselves only if they receive outstanding support from the administration. Several recommendations of the Review Committee are targeted towards rationalization of administrative staff, creation of structures for smooth conduct of research projects and engagement with the external world including the industry, efficient management of finances, adoption of digital technology and the like. Infrastructure including buildings, laboratories and computing systems need to be modernized urgently. The Institute has to significantly expand its outreach activities with a target to become a natural go-to destination for students, industry and the government. Significant effort and resources need to be spent on brand building that has tremendous real-life value in attracting the best talents and the best projects to the Institute. Finally, the Review Committee argues for greater autonomy to the Institute on its internal administrative matters, recruitments and the rules/procedures of recruitment, and handling the internal revenue that it generates, as long as they are within the broad guidelines of the Government of India. In terms of attracting outstanding talent to the Institute, particularly on short-term basis, the Institute should be given a special consideration to fix the remuneration and facilities at internationally competitive values. The Council, consisting of eminent and responsible persons as well as representatives of the government, should be empowered to take such decisions, without the need to thereafter seek government approval once again, as long as the total financial burden remains within reasonable limits of the sanctioned budget. The Review Committee recognizes that it is the power of ideas together with the power of execution that matters. Therefore, the recommendations have been broken down into categories in terms of ease of implementation as well as urgency into short, medium and long term. The Review Committee strongly feels that as ISI reaches its centenary year in 2031, it should aim to create a new ISI@100, transitioning from the current good to the very best, to an extent that it makes an international impact and becomes one of the foremost institutes globally.
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