Stein's method of exchangeable pairs in multivariate functional approximations

In this paper we develop a framework for multivariate functional approximation by a suitable Gaussian process via an exchangeable pairs coupling that satisfies a suitable approximate linear regression property, thereby building on work by Barbour (1990) and Kasprzak (2020). We demonstrate the applicability of our results by applying it to joint subgraph counts in an Erd\H{o}s-Renyi random graph model on the one hand and to vectors of weighted, degenerate $U$-processes on the other hand. As a concrete instance of the latter class of examples, we provide a bound for the functional approximation of a vector of success runs of different lengths by a suitable Gaussian process which, even in the situation of just a single run, would be outside the scope of the existing theory

Practical bounds on the error of Bayesian posterior approximations: A nonasymptotic approach

Bayesian inference typically requires the computation of an approximation to the posterior distribution. An important requirement for an approximate Bayesian inference algorithm is to output high-accuracy posterior mean and uncertainty estimates. Classical Monte Carlo methods, particularly Markov Chain Monte Carlo, remain the gold standard for approximate Bayesian inference because they have a robust finite-sample theory and reliable convergence diagnostics. However, alternative methods, which are more scalable or apply to problems where Markov Chain Monte Carlo cannot be used, lack the same finite-data approximation theory and tools for evaluating their accuracy. In this work, we develop a flexible new approach to bounding the error of mean and uncertainty estimates of scalable inference algorithms. Our strategy is to control the estimation errors in terms of Wasserstein distance, then bound the Wasserstein distance via a generalized notion of Fisher distance. Unlike computing the Wasserstein distance, which requires access to the normalized posterior distribution, the Fisher distance is tractable to compute because it requires access only to the gradient of the log posterior density. We demonstrate the usefulness of our Fisher distance approach by deriving bounds on the Wasserstein error of the Laplace approximation and Hilbert coresets. We anticipate that our approach will be applicable to many other approximate inference methods such as the integrated Laplace approximation, variational inference, and approximate Bayesian computationComment: 22 pages, 2 figure

A Fourier representation of kernel Stein discrepancy with application to Goodness-of-Fit tests for measures on infinite dimensional Hilbert spaces

Kernel Stein discrepancy (KSD) is a widely used kernel-based measure of discrepancy between probability measures. It is often employed in the scenario where a user has a collection of samples from a candidate probability measure and wishes to compare them against a specified target probability measure. KSD has been employed in a range of settings including goodness-of-fit testing, parametric inference, MCMC output assessment and generative modelling. However, so far the method has been restricted to finite-dimensional data. We provide the first analysis of KSD in the generality of data lying in a separable Hilbert space, for example functional data. The main result is a novel Fourier representation of KSD obtained by combining the theory of measure equations with kernel methods. This allows us to prove that KSD can separate measures and thus is valid to use in practice. Additionally, our results improve the interpretability of KSD by decoupling the effect of the kernel and Stein operator. We demonstrate the efficacy of the proposed methodology by performing goodness-of-fit tests for various Gaussian and non-Gaussian functional models in a number of synthetic data experiments.Comment: To appear in Bernoull

How good is your Laplace approximation of the Bayesian posterior? Finite-sample computable error bounds for a variety of useful divergences

The Laplace approximation is a popular method for providing posterior mean and variance estimates. But can we trust these estimates for practical use? One might consider using rate-of-convergence bounds for the Bayesian Central Limit Theorem (BCLT) to provide quality guarantees for the Laplace approximation. But the bounds in existing versions of the BCLT either: require knowing the true data-generating parameter, are asymptotic in the number of samples, do not control the Bayesian posterior mean, or apply only to narrow classes of models. Our work provides the first closed-form, finite-sample quality bounds for the Laplace approximation that simultaneously (1) do not require knowing the true parameter, (2) control posterior means and variances, and (3) apply generally to models that satisfy the conditions of the asymptotic BCLT. In fact, our bounds work even in the presence of misspecification. We compute exact constants in our bounds for a variety of standard models, including logistic regression, and numerically demonstrate their utility. We provide a framework for analysis of more complex models.Comment: Major update to the structure of the paper and discussion of the main result

Stein's method for functional approximations

We extend the ideas of Barbour's paper from 1990 and adapt Stein's method for distributional approximation to infinite-dimensional distributions. Hence, we obtain theoretical results bounding the rate of functional convergence of certain classes of stochastic processes to diffusions. Those are applied to examples coming from queuing theory, random-graph theory, statistics and combinatorics. We firstly look at the motivation for this thesis and an overview of Stein's method. Then we present our original work, contained in four articles. The first one is a collaboration with Andrew Duncan and Sebastian Vollmer, published in the Electronic Communications in Probability and the other ones, for which I am the sole author, are currently under consideration for publication. The first paper corrects a mistake in Barbour's seminal work from 1990. The second paper considers the approximation of a time-changed Poisson process by a time-changed Brownian motion for time changes independent of the processes they are applied to. As an application, we study the M/M/1 queue and a time-changed Brownian Motion and bound a distance between the two. The third paper studies the asymptotic behaviour of scaled sums of random vectors having different dependence structures. As an application, a bound on the distance between scaled non-degenerate U-statistics and Brownian Motion is proved. Moreover, we prove a quantitative functional limit theorem for exceedances in the m-scans process. In the fourth paper, we adapt the exchangeable-pair approach to Stein's method to approximations by infinite-dimensional laws. It is used to provide the rate of convergence in a functional combinatorial central limit theorem, extending the result of Barbour and Janson from 2009. We further apply this approach to study the asymptotics of edge and two-star counts in a certain graph-valued process. The final part of the thesis presents the conclusions and suggestions for future work.</p

The multivariate functional de Jong CLT

Abstract We prove a multivariate functional version of de Jong’s CLT (J Multivar Anal 34(2):275–289, 1990) yielding that, given a sequence of vectors of Hoeffding-degenerate U-statistics, the corresponding empirical processes on [0, 1] weakly converge in the Skorohod space as soon as their fourth cumulants in $$t=1$$ t = 1 vanish asymptotically and a certain strengthening of the Lindeberg-type condition is verified. As an application, we lift to the functional level the ‘universality of Wiener chaos’ phenomenon first observed in Nourdin et al. (Ann Probab 38(5):1947–1985, 2010)

Clinical Characteristics and Predictors of In-Hospital Mortality of Patients Hospitalized with COVID-19 Infection

Background: The identification of parameters that would serve as predictors of prognosis in COVID-19 patients is very important. In this study, we assessed independent factors of in-hospital mortality of COVID-19 patients during the second wave of the pandemic. Material and methods: The study group consisted of patients admitted to two hospitals and diagnosed with COVID-19 between October 2020 and May 2021. Clinical and demographic features, the presence of comorbidities, laboratory parameters, and radiological findings at admission were recorded. The relationship of these parameters with in-hospital mortality was evaluated. Results: A total of 1040 COVID-19 patients (553 men and 487 women) qualified for the study. The in-hospital mortality rate was 26% across all patients. In multiple logistic regression analysis, age ≥ 70 years with OR = 7.8 (95% CI 3.17–19.32), p p = 0.004, the presence of typical COVID-19-related lung abnormalities visualized in chest computed tomography ≥40% with OR = 2.5 (95% CI 1.05–6.23), p = 0.037, and a concomitant diagnosis of coronary artery disease with OR = 3.5 (95% CI 1.38–9.10), p = 0.009 were evaluated as independent risk factors for in-hospital mortality. Conclusion: The relationship between clinical and laboratory markers, as well as the advancement of lung involvement by typical COVID-19-related abnormalities in computed tomography of the chest, and mortality is very important for the prognosis of these patients and the determination of treatment strategies during the COVID-19 pandemic