arXiv subjects

Sergios Agapiou

Publications and source records attributed to Sergios Agapiou.

14 recordsLinked to original sources

arXiv2026

Leveraging tails for adaptation

We consider contraction of Bayesian posterior distributions in nonparametric settings where coefficients of a function over a basis or dictionary are given priors with $p$--exponential tails, including Laplace tails $(p=1)$ and heavier tails $(p<1)$. It is shown that contraction rates improve as $p$ decreases and that full adaptation to smoothness, up to logarithmic factors, is obtained in an appropriate $p\to 0$ regime. As applications, we consider both series priors in white noise regression and shallow ReLU neural networks in random design regression. In particular, we show that overparametrised shallow ReLU networks can adapt to any regularity $0\le \beta\le 2$. Through a simulation study, we show strong empirical agreement with the behavior predicted by our theory.

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arXiv2025

Heavy-tailed and Horseshoe priors for regression and sparse Besov rates

The large variety of functions encountered in nonparametric statistics, calls for methods that are flexible enough to achieve optimal or near-optimal performance over a wide variety of functional classes, such as Besov balls, as well as over a large array of loss functions. In this work, we show that a class of heavy-tailed prior distributions on basis function coefficients introduced in \cite{AC} and called Oversmoothed heavy-Tailed (OT) priors, leads to Bayesian posterior distributions that satisfy these requirements; the case of horseshoe distributions is also investigated, for the first time in the context of nonparametrics, and we show that they fit into this framework. Posterior contraction rates are derived in two settings. The case of Sobolev--smooth signals and $L_2$--risk is considered first, along with a lower bound result showing that the imposed form of the scalings on prior coefficients by the OT prior is necessary to get full adaptation to smoothness. Second, the broader case of Besov-smooth signals with $L_{p'}$--risks, $p' \geq 1$, is considered, and minimax posterior contraction rates, adaptive to the underlying smoothness, and including rates in the so-called {\em sparse} zone, are derived. We provide an implementation of the proposed method and illustrate our results through a simulation study.

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arXiv2023

Heavy-tailed Bayesian nonparametric adaptation

We propose a new Bayesian strategy for adaptation to smoothness in nonparametric models based on heavy tailed series priors. We illustrate it in a variety of settings, showing in particular that the corresponding Bayesian posterior distributions achieve adaptive rates of contraction in the minimax sense (up to logarithmic factors) without the need to sample hyperparameters. Unlike many existing procedures, where a form of direct model (or estimator) selection is performed, the method can be seen as performing a soft selection through the prior tail. In Gaussian regression, such heavy tailed priors are shown to lead to (near-)optimal simultaneous adaptation both in the $L^2$- and $L^\infty$-sense. Results are also derived for linear inverse problems, for anisotropic Besov classes, and for certain losses in more general models through the use of tempered posterior distributions. We present numerical simulations corroborating the theory.

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arXiv2022

Adaptive inference over Besov spaces in the white noise model using $p$-exponential priors

In many scientific applications the aim is to infer a function which is smooth in some areas, but rough or even discontinuous in other areas of its domain. Such spatially inhomogeneous functions can be modelled in Besov spaces with suitable integrability parameters. In this work we study adaptive Bayesian inference over Besov spaces, in the white noise model from the point of view of rates of contraction, using $p$-exponential priors, which range between Laplace and Gaussian and possess regularity and scaling hyper-parameters. To achieve adaptation, we employ empirical and hierarchical Bayes approaches for tuning these hyper-parameters. Our results show that, while it is known that Gaussian priors can attain the minimax rate only in Besov spaces of spatially homogeneous functions, Laplace priors attain the minimax or nearly the minimax rate in both Besov spaces of spatially homogeneous functions and Besov spaces permitting spatial inhomogeneities.

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arXiv2021

Laplace priors and spatial inhomogeneity in Bayesian inverse problems

Spatially inhomogeneous functions, which may be smooth in some regions and rough in other regions, are modelled naturally in a Bayesian manner using so-called Besov priors which are given by random wavelet expansions with Laplace-distributed coefficients. This paper studies theoretical guarantees for such prior measures - specifically, we examine their frequentist posterior contraction rates in the setting of non-linear inverse problems with Gaussian white noise. Our results are first derived under a general local Lipschitz assumption on the forward map. We then verify the assumption for two non-linear inverse problems arising from elliptic partial differential equations, the Darcy flow model from geophysics as well as a model for the Schr\"odinger equation appearing in tomography. In the course of the proofs, we also obtain novel concentration inequalities for penalized least squares estimators with $\ell^1$ wavelet penalty, which have a natural interpretation as maximum a posteriori (MAP) estimators. The true parameter is assumed to belong to some spatially inhomogeneous Besov class $B^{\alpha}_{11}$, $\alpha>0$. In a setting with direct observations, we complement these upper bounds with a lower bound on the rate of contraction for arbitrary Gaussian priors. An immediate consequence of our results is that while Laplace priors can achieve minimax-optimal rates over $B^{\alpha}_{11}$-classes, Gaussian priors are limited to a (by a polynomial factor) slower contraction rate. This gives information-theoretical justification for the intuition that Laplace priors are more compatible with $\ell^1$ regularity structure in the underlying parameter.

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arXiv2021

Designing truncated priors for direct and inverse Bayesian problems

The Bayesian approach to inverse problems with functional unknowns, has received significant attention in recent years. An important component of the developing theory is the study of the asymptotic performance of the posterior distribution in the frequentist setting. The present paper contributes to the area of Bayesian inverse problems by formulating a posterior contraction theory for linear inverse problems, with truncated Gaussian series priors, and under general smoothness assumptions. Emphasis is on the intrinsic role of the truncation point both for the direct as well as for the inverse problem, which are related through the modulus of continuity as this was recently highlighted by Knapik and Salomond (2018).

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arXiv2020

Modeling of Covid-19 Pandemic in Cyprus

The Republic of Cyprus is a small island in the southeast of Europe and member of the European Union. The first wave of COVID-19 in Cyprus started in early March, 2020 (imported cases) and peaked in late March-early April. The health authorities responded rapidly and rigorously to the COVID-19 pandemic by scaling-up testing, increasing efforts to trace and isolate contacts of cases, and implementing measures such as closures of educational institutions, and travel and movement restrictions. The pandemic was also a unique opportunity that brought together experts from various disciplines including epidemiologists, clinicians, mathematicians, and statisticians. The aim of this paper is to present the efforts of this new, multidisciplinary research team in modelling the COVID-19 pandemic in the Republic of Cyprus.

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arXiv2018

Rates of contraction of posterior distributions based on $p$-exponential priors

We consider a family of infinite dimensional product measures with tails between Gaussian and exponential, which we call $p$-exponential measures. We study their measure-theoretic properties and in particular their concentration. Our findings are used to develop a general contraction theory of posterior distributions on nonparametric models with $p$-exponential priors in separable Banach parameter spaces. Our approach builds on the general contraction theory for Gaussian process priors in van der Vaart and van Zanten 2008, namely we use prior concentration to verify prior mass and entropy conditions sufficient for posterior contraction. However, the specific concentration properties of $p$-exponential priors lead to a more complex entropy bound which can influence negatively the obtained rate of contraction, depending on the topology of the parameter space. Subject to the more complex entropy bound, we show that the rate of contraction depends on the position of the true parameter relative to a certain Banach space associated to $p$-exponential measures and on the small ball probabilities of these measures. For example, we apply our theory in the white noise model under Besov regularity of the truth and obtain minimax rates of contraction using (rescaled) $\alpha$-regular $p$-exponential priors. In particular, our results suggest that when interested in spatially inhomogeneous unknown functions, in terms of posterior contraction, it is preferable to use Laplace rather than Gaussian priors.

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arXiv2017

Sparsity-promoting and edge-preserving maximum a posteriori estimators in non-parametric Bayesian inverse problems

We consider the inverse problem of recovering an unknown functional parameter $u$ in a separable Banach space, from a noisy observation $y$ of its image through a known possibly non-linear ill-posed map ${\mathcal G}$. The data $y$ is finite-dimensional and the noise is Gaussian. We adopt a Bayesian approach to the problem and consider Besov space priors (see Lassas et al. 2009), which are well-known for their edge-preserving and sparsity-promoting properties and have recently attracted wide attention especially in the medical imaging community. Our key result is to show that in this non-parametric setup the maximum a posteriori (MAP) estimates are characterized by the minimizers of a generalized Onsager--Machlup functional of the posterior. This is done independently for the so-called weak and strong MAP estimates, which as we show coincide in our context. In addition, we prove a form of weak consistency for the MAP estimators in the infinitely informative data limit. Our results are remarkable for two reasons: first, the prior distribution is non-Gaussian and does not meet the smoothness conditions required in previous research on non-parametric MAP estimates. Second, the result analytically justifies existing uses of the MAP estimate in finite but high dimensional discretizations of Bayesian inverse problems with the considered Besov priors.

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arXiv2014

Unbiased Monte Carlo: posterior estimation for intractable/infinite-dimensional models

We provide a general methodology for unbiased estimation for intractable stochastic models. We consider situations where the target distribution can be written as an appropriate limit of distributions, and where conventional approaches require truncation of such a representation leading to a systematic bias. For example, the target distribution might be representable as the $L^2$-limit of a basis expansion in a suitable Hilbert space; or alternatively the distribution of interest might be representable as the weak limit of a sequence of random variables, as in MCMC. Our main motivation comes from infinite-dimensional models which can be parame- terised in terms of a series expansion of basis functions (such as that given by a Karhunen-Loeve expansion). We consider schemes for direct unbiased estimation along such an expansion, as well as those based on MCMC schemes which, due to their dimensionality, cannot be directly imple- mented, but which can be effectively estimated unbiasedly. For all our methods we give theory to justify the numerical stability for robust Monte Carlo implementation, and in some cases we illustrate using simulations. Interestingly the computational efficiency of our methods is usually comparable to simpler methods which are biased. Crucial to the effectiveness of our proposed methodology is the construction of appropriate couplings, many of which resonate strongly with the Monte Carlo constructions used in the coupling from the past algorithm and its variants.

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arXiv2014

Preconditioning the prior to overcome saturation in Bayesian inverse problems

We study Bayesian inference in statistical linear inverse problems with Gaussian noise and priors in Hilbert space. We focus our interest on the posterior contraction rate in the small noise limit. Existing results suffer from a certain saturation phenomenon, when the data generating element is too smooth compared to the smoothness inherent in the prior. We show how to overcome this saturation in an empirical Bayesian framework by using a non-centered data-dependent prior. The center is obtained from a preconditioning regularization step, which provides us with additional information to be used in the Bayesian framework. We use general techniques known from regularization theory. To highlight the significance of the findings we provide several examples. In particular, our approach allows to obtain and, using preconditioning improve after saturation, minimax rates of contraction established in previous studies. We also establish minimax contraction rates in cases which have not been considered so far.

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arXiv2014

Analysis of the Gibbs sampler for hierarchical inverse problems

Many inverse problems arising in applications come from continuum models where the unknown parameter is a field. In practice the unknown field is discretized resulting in a problem in $\mathbb{R}^N$, with an understanding that refining the discretization, that is increasing $N$, will often be desirable. In the context of Bayesian inversion this situation suggests the importance of two issues: (i) defining hyper-parameters in such a way that they are interpretable in the continuum limit $N \to \infty$ and so that their values may be compared between different discretization levels; (ii) understanding the efficiency of algorithms for probing the posterior distribution, as a function of large $N.$ Here we address these two issues in the context of linear inverse problems subject to additive Gaussian noise within a hierarchical modelling framework based on a Gaussian prior for the unknown field and an inverse-gamma prior for a hyper-parameter, namely the amplitude of the prior variance. The structure of the model is such that the Gibbs sampler can be easily implemented for probing the posterior distribution. Subscribing to the dogma that one should think infinite-dimensionally before implementing in finite dimensions, we present function space intuition and provide rigorous theory showing that as $N$ increases, the component of the Gibbs sampler for sampling the amplitude of the prior variance becomes increasingly slower. We discuss a reparametrization of the prior variance that is robust with respect to the increase in dimension; we give numerical experiments which exhibit that our reparametrization prevents the slowing down. Our intuition on the behaviour of the prior hyper-parameter, with and without reparametrization, is sufficiently general to include a broad class of nonlinear inverse problems as well as other families of hyper-priors.

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arXiv2013

Bayesian Posterior Contraction Rates for Linear Severely Ill-posed Inverse Problems

We consider a class of linear ill-posed inverse problems arising from inversion of a compact operator with singular values which decay exponentially to zero. We adopt a Bayesian approach, assuming a Gaussian prior on the unknown function. If the observational noise is assumed to be Gaussian then this prior is conjugate to the likelihood so that the posterior distribution is also Gaussian. We study Bayesian posterior consistency in the small observational noise limit. We assume that the forward operator and the prior and noise covariance operators commute with one another. We show how, for given smoothness assumptions on the truth, the scale parameter of the prior can be adjusted to optimize the rate of posterior contraction to the truth, and we explicitly compute the logarithmic rate.

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arXiv2013

Posterior Contraction Rates for the Bayesian Approach to Linear Ill-Posed Inverse Problems

We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting with Gaussian noise. We assume Gaussian priors, which are conjugate to the model, and present a method of identifying the posterior using its precision operator. Working with the unbounded precision operator enables us to use partial differential equations (PDE) methodology to obtain rates of contraction of the posterior distribution to a Dirac measure centered on the true solution. Our methods assume a relatively weak relation between the prior covariance, noise covariance and forward operator, allowing for a wide range of applications.

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