arXiv subjects

Emanuele Dolera

Publications and source records attributed to Emanuele Dolera.

At least 19 recordsLinked to original sources

arXiv2026

The variance of the Pitman--Yor process: Cifarelli--Regazzini-type identities and inversion formulas

The Cifarelli--Regazzini identity lies at the foundation of the distributional theory of the Dirichlet process and has played a central role in Bayesian nonparametrics. By providing the generalized Cauchy--Stieltjes transform of linear functionals of the Dirichlet process, together with an analytic inversion, it yields, in particular, explicit representations for the distribution function and density of the Dirichlet mean. This transform identity was subsequently extended to the Pitman--Yor process, a generalization of the Dirichlet process, leading to an analogous distributional theory for its linear functionals. Nonlinear functionals, by contrast, remain substantially less understood, with the available distributional literature limited to a small number of specific examples and confined to the Dirichlet process. In this paper, we move beyond linear functionals by considering the variance of the Pitman--Yor process, a genuinely quadratic functional. In particular, we establish a Cifarelli--Regazzini-type identity providing the generalized Cauchy--Stieltjes transform of the Pitman--Yor variance for general base probability measures under suitable integrability assumptions. For the Uniform base probability measure on $[0,1]$, analytic inversion yields explicit representations for both the distribution function and density. Corresponding formulas are also obtained for the Dirichlet variance, which provide more explicit representations than those currently available in the literature.

math.PR↗

arXiv2026

Posterior contraction rates in Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families

We study posterior contraction in positive-order Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families. We embed the natural parameter in a Hilbert scale and model it via a standard Gaussian series prior expanded in the eigenbasis generating the scale. Under a two-sided link condition on the Fisher information and suitable local regularity assumptions, we show that smoothness-matching priors achieve minimax-optimal posterior contraction rates in any Hilbert scale norm up to the regularity of the ground truth. Our analysis builds on the novel approach to posterior contraction based on the Wasserstein distance recently introduced by Dolera et al. (2024). It combines refined Laplace-type estimates for infinite-dimensional integrals associated to the posterior kernels with a mixed-geometry estimate controlling their stability under fluctuations in the data, itself resting on a tailored Poincaré inequality for posterior distributions conditioned on neighbourhoods of the truth. We apply the general theory to density estimation with a logistic parametrisation, Poisson intensity estimation with an exponential link, and the Gaussian white-noise model, yielding minimax contraction rates in Sobolev norms across all three settings. In particular, these yield optimal recovery of density score functions and derivatives of Poisson intensities.

math.ST↗

arXiv2026

AGILE detection of transient γ-ray emission from the region of the supergiant fast X-ray transient source IGR J17354-3255

Context. On April 14, 2009, the AGILE satellite detected a γ-ray flare from an unknown transient source. Subsequent X-ray follow-up observations with Swift and INTEGRAL identified the supergiant fast X-ray transient (SFXT) IGR J17354-3255 as the best candidate counterpart, based on positional coincidence and a similar temporal behaviour. Aside from this hint, no SFXT has been firmly detected at high energies to date. Overall, SFXTs comprise a subclass of high-mass X-ray binaries (HMXBs) that host a massive OB supergiant star as a companion donor. They tend to display the most extreme X-ray variability among HMXBs. These systems might be able to emit photons at MeV-TeV energies in the form of fast flares lasting from hours to a few days, with a low-duty cycle. Aims. In this work, we analyse archival AGILE data to search for γ-ray flares consistent with IGR J17354-3255 and evaluate their possible physical origin. Methods. We identified a transient source, AGL J1736-3250, which emitted 19 γ-ray flares and was seen to be positionally consistent with IGR J17354-3255. Most flares, detected on a 1 d timescale, concentrate most of their emission on two, four, and six hour timescales, resembling those observed in the X-ray band from IGR J17354-3255. Results. An orbital phase analysis revealed that approximately half of the γ-ray activity occurs around the apastron passage of the compact object hosted in the binary system. We also incorporated archival Swift and INTEGRAL observations to provide phase-folded light curves at lower energies. Our collected results strongly support a physical association between IGR J17354-3255 and AGL J1736-3250, offering evidence that SFXTs could constitute a new class of Galactic high-energy transient emitters.

astro-ph.HE↗

arXiv2026

A Central Limit Theorem for the Ewens-Pitman random partition in the large-$θ$ regime via a martingale approach

The Ewens-Pitman model defines a distribution on random partitions of $\{1,\ldots,n\}$, with parameters $α\in [0,1)$ and $θ> -α$; the case $α=0$ reduces to the classical Ewens model from population genetics. We investigate the large-$n$ asymptotic behaviour of the Ewens-Pitman random partition in the nonstandard regime $θ=λn$ with $λ>0$, establishing joint fluctuation results for the total number of blocks $K_n^{\{n\}}$ and the counts $K_{r,n}^{\{n\}}$ of blocks of sizes $r=1,\dots,d$, for fixed $d\in\mathbb{N}$. In particular, for $α\in[0,1)$ and $θ=λn$, our main result provides a strong law of large numbers and a central limit theorem for the $(d+1)$-dimensional vector $\mathbf{K}_{d,n}^{\{n\}} = \bigl(K_n^{\{n\}}, K_{1,n}^{\{n\}}, \dots, K_{d,n}^{\{n\}}\bigr)^T$ as $n \to \infty$. The proof exploits the Chinese restaurant sequential construction under $θ=λn$ and a central limit theorem for triangular arrays of martingales, extending techniques previously developed for the classical regime with fixed $θ$. As corollaries of our results, we recover known asymptotics for $K_n^{\{n\}}$ and derive new strong laws and central limit theorems for each fixed $K_{r,n}^{\{n\}}$, thereby completing earlier weak-law results and providing a comprehensive asymptotic description of the Ewens-Pitman partition structure in the large-$θ$ setting.

math.PR↗

arXiv2025

Gaussian credible intervals in Bayesian nonparametric estimation of the unseen

The unseen-species problem assumes $n\geq1$ samples from a population of individuals belonging to different species, possibly infinite, and calls for estimating the number $K_{n,m}$ of hitherto unseen species that would be observed if $m\geq1$ new samples were collected from the same population. This is a long-standing problem in statistics, which has gained renewed relevance in biological and physical sciences, particularly in settings with large values of $n$ and $m$. In this paper, we adopt a Bayesian nonparametric approach to the unseen-species problem under the Pitman-Yor prior, and propose a novel methodology to derive large $m$ asymptotic credible intervals for $K_{n,m}$, for any $n\geq1$. By leveraging a Gaussian central limit theorem for the posterior distribution of $K_{n,m}$, our method improves upon competitors in two key aspects: firstly, it enables the full parameterization of the Pitman-Yor prior, including the Dirichlet prior; secondly, it avoids the need of Monte Carlo sampling, enhancing computational efficiency. We validate the proposed method on synthetic and real data, demonstrating that it improves the empirical performance of competitors by significantly narrowing the gap between asymptotic and exact credible intervals for any $m\geq1$.

stat.ME↗

arXiv2024

Laws of large numbers and central limit theorem for Ewens-Pitman model

The Ewens-Pitman model is a distribution for random partitions of the set $\{1,\ldots,n\}$, with $n\in\mathbb{N}$, indexed by parameters $α\in [0,1)$ and $θ>-α$, such that $α=0$ is the Ewens model in population genetics. The large $n$ asymptotic behaviour of the number $K_{n}$ of blocks in the Ewens-Pitman random partition has been extensively investigated in terms of almost-sure and Gaussian fluctuations, which show that $K_{n}$ scales as $\log n$ and $n^α$ depending on whether $α=0$ or $α\in(0,1)$, providing non-random and random limiting behaviours, respectively. In this paper, we study the large $n$ asymptotic behaviour of $K_{n}$ when the parameter $θ$ is allowed to depend linearly on $n\in\mathbb{N}$, a non-standard asymptotic regime first considered for $α=0$ in Feng (\textit{The Annals of Applied Probability}, \textbf{17}, 2007). In particular, for $α\in[0,1)$ and $θ=λn$, with $λ>0$, we establish a law of large numbers (LLN) and a central limit theorem (CLT) for $K_{n}$, which show that $K_{n}$ scales as $n$, providing non-random limiting behaviours. Depending on whether $α=0$ or $α\in(0,1)$, our results rely on different arguments. For $α=0$ we rely on the representation of $K_{n}$ as a sum of independent, but not identically distributed, Bernoulli random variables, which leads to a refinement of the CLT in terms of a Berry-Esseen theorem. Instead, for $α\in(0,1)$, we rely on a compound Poisson construction of $K_{n}$, leading to prove LLNs, CLTs and Berry-Esseen theorems for the number of blocks of the negative-Binomial compound Poisson random partition, which are of independent interest.

math.PR↗

arXiv2024

On strong posterior contraction rates for Besov-Laplace priors in the white noise model

In this article, we investigate the problem of estimating a spatially inhomogeneous function and its derivatives in the white noise model using Besov-Laplace priors. We show that smoothness-matching priors attains minimax optimal posterior contraction rates, in strong Sobolev metrics, over the Besov spaces $B^β_{11}$, $β> d/2$, closing a gap in the existing literature. Our strong posterior contraction rates also imply that the posterior distributions arising from Besov-Laplace priors with matching regularity enjoy a desirable plug-in property for derivative estimation, entailing that the push-forward measures under differential operators optimally recover the derivatives of the unknown regression function. The proof of our results relies on the novel approach to posterior contraction rates, based on Wasserstein distance, recently developed by Dolera, Favaro and Mainini (Probability Theory and Related Fields, 2024). We show how this approach allows to overcome some technical challenges that emerge in the frequentist analysis of smoothness-matching Besov-Laplace priors.

math.ST↗

arXiv2023

Strong posterior contraction rates via Wasserstein dynamics

In Bayesian statistics, posterior contraction rates (PCRs) quantify the speed at which the posterior distribution concentrates on arbitrarily small neighborhoods of a true model, in a suitable way, as the sample size goes to infinity. In this paper, we develop a new approach to PCRs, with respect to strong norm distances on parameter spaces of functions. Critical to our approach is the combination of a local Lipschitz-continuity for the posterior distribution with a dynamic formulation of the Wasserstein distance, which allows to set forth an interesting connection between PCRs and some classical problems arising in mathematical analysis, probability and statistics, e.g., Laplace methods for approximating integrals, Sanov's large deviation principles in the Wasserstein distance, rates of convergence of mean Glivenko-Cantelli theorems, and estimates of weighted Poincaré-Wirtinger constants. We first present a theorem on PCRs for a model in the regular infinite-dimensional exponential family, which exploits sufficient statistics of the model, and then extend such a theorem to a general dominated model. These results rely on the development of novel techniques to evaluate Laplace integrals and weighted Poincaré-Wirtinger constants in infinite-dimension, which are of independent interest. The proposed approach is applied to the regular parametric model, the multinomial model, the finite-dimensional and the infinite-dimensional logistic-Gaussian model and the infinite-dimensional linear regression. In general, our approach leads to optimal PCRs in finite-dimensional models, whereas for infinite-dimensional models it is shown explicitly how the prior distribution affect PCRs.

math.ST↗

arXiv2023

Lipschitz continuity of probability kernels in the optimal transport framework

In Bayesian statistics, a continuity property of the posterior distribution with respect to the observable variable is crucial as it expresses well-posedness, i.e., stability with respect to errors in the measurement of data. Essentially, this requires to analyze the continuity of a probability kernel or, equivalently, of a conditional probability distribution with respect to the conditioning variable. Here, we give general conditions for the Lipschitz continuity of probability kernels with respect to metric structures arising within the optimal transport framework, such as the Wasserstein metric. For dominated probability kernels over finite-dimensional spaces, we show Lipschitz continuity results with a Lipschitz constant enjoying explicit bounds in terms of Fisher-information functionals and weighted Poincaré constants. We also provide results for kernels with moving support, for infinite-dimensional spaces and for non dominated kernels. We show applications to several problems in Bayesian statistics, such as approximation of posterior distributions by mixtures and posterior consistency.

math.PR↗

arXiv2022

A counterexample to $L^{\infty}$-gradient type estimates for Ornstein-Uhlenbeck operators

Let $(λ_k)$ be a strictly increasing sequence of positive numbers such that $\sum_{k=1}^{\infty} \frac{1}{λ_k} < \infty.$ Let $f $ be a bounded smooth function and denote by $u= u^f$ the bounded classical solution to $u(x) - \frac{1}{2}\sum_{k=1}^m D^2_{kk} u(x) + \sum_{k =1}^m λ_k x_k D_k u(x) = f(x), $ $ x \in \R^m$. It is known that the following dimension-free estimate holds: $$ \displaystyle \int_{\R^m} \Big (\sum_{k=1}^m λ_k \, (D_k u (y))^2 \Big)^{p/2} μ_m (dy) \le (c_p)^p \, \int_{\R^m} |f( y)|^p μ_m (dy),\;\;\; 1 < p < \infty; $$ here $μ_m$ is the "diagonal" Gaussian measure determined by $λ_1, \ldots, λ_m$ and $c_p > 0$ is independent of $f$ and $m$. This is a consequence of generalized Meyer's inequalities [Chojnowska-Michalik, Goldys, J. Funct. Anal. 182 (2001)]. We show that, if $λ_k \sim k^2$, then such estimate does not hold when $p= \infty$. Indeed we prove $$ \sup_{\substack{f \in C^{ 2}_b(\R^m),\;\; \|f\|_{\infty} \leq 1}} \Big \{ \sum_{k=1}^m λ_k \, (D_k u^f (0))^2 \Big \} \to \infty \;\; \text {as} \; m \to \infty. $$ This is in contrast to the case of $λ_k = λ>0$, $k \ge 1$, where a dimension-free bound holds for $p =\infty$.

math.PR↗

arXiv2022

Learning-augmented count-min sketches via Bayesian nonparametrics

The count-min sketch (CMS) is a time and memory efficient randomized data structure that provides estimates of tokens' frequencies in a data stream of tokens, i.e. point queries, based on random hashed data. A learning-augmented version of the CMS, referred to as CMS-DP, has been proposed by Cai, Mitzenmacher and Adams (\textit{NeurIPS} 2018), and it relies on Bayesian nonparametric (BNP) modeling of the data stream of tokens via a Dirichlet process (DP) prior, with estimates of a point query being obtained as suitable mean functionals of the posterior distribution of the point query, given the hashed data. While the CMS-DP has proved to improve on some aspects of CMS, it has the major drawback of arising from a ``constructive" proof that builds upon arguments tailored to the DP prior, namely arguments that are not usable for other nonparametric priors. In this paper, we present a ``Bayesian" proof of the CMS-DP that has the main advantage of building upon arguments that are usable, in principle, within a broad class of nonparametric priors arising from normalized completely random measures. This result leads to develop a novel learning-augmented CMS under power-law data streams, referred to as CMS-PYP, which relies on BNP modeling of the data stream of tokens via a Pitman-Yor process (PYP) prior. Under this more general framework, we apply the arguments of the ``Bayesian" proof of the CMS-DP, suitably adapted to the PYP prior, in order to compute the posterior distribution of a point query, given the hashed data. Applications to synthetic data and real textual data show that the CMS-PYP outperforms the CMS and the CMS-DP in estimating low-frequency tokens, which are known to be of critical interest in textual data, and it is competitive with respect to a variation of the CMS designed for low-frequency tokens. An extension of our BNP approach to more general queries is also discussed.

stat.ML↗

arXiv2022

The power of private likelihood-ratio tests for goodness-of-fit in frequency tables

Privacy-protecting data analysis investigates statistical methods under privacy constraints. This is a rising challenge in modern statistics, as the achievement of confidentiality guarantees, which typically occurs through suitable perturbations of the data, may determine a loss in the statistical utility of the data. In this paper, we consider privacy-protecting tests for goodness-of-fit in frequency tables, this being arguably the most common form of releasing data, and present a rigorous analysis of the large sample behaviour of a private likelihood-ratio (LR) test. Under the framework of $(\varepsilon,δ)$-differential privacy for perturbed data, our main contribution is the power analysis of the private LR test, which characterizes the trade-off between confidentiality, measured via the differential privacy parameters $(\varepsilon,δ)$, and statistical utility, measured via the power of the test. This is obtained through a Bahadur-Rao large deviation expansion for the power of the private LR test, bringing out a critical quantity, as a function of the sample size, the dimension of the table and $(\varepsilon,δ)$, that determines a loss in the power of the test. Such a result is then applied to characterize the impact of the sample size and the dimension of the table, in connection with the parameters $(\varepsilon,δ)$, on the loss of the power of the private LR test. In particular, we determine the (sample) cost of $(\varepsilon,δ)$-differential privacy in the private LR test, namely the additional sample size that is required to recover the power of the Multinomial LR test in the absence of perturbation. Our power analysis rely on a non-standard large deviation analysis for the LR, as well as the development of a novel (sharp) large deviation principle for sum of i.i.d. random vectors, which is of independent interest.

math.ST↗

arXiv2022

A new approach to posterior contraction rates via Wasserstein dynamics

This paper presents a new approach to the classical problem of quantifying posterior contraction rates (PCRs) in Bayesian statistics. Our approach relies on Wasserstein distance, and it leads to two main contributions which improve on the existing literature of PCRs. The first contribution exploits the dynamic formulation of Wasserstein distance, for short referred to as Wasserstein dynamics, in order to establish PCRs under dominated Bayesian statistical models. As a novelty with respect to existing approaches to PCRs, Wasserstein dynamics allows us to circumvent the use of sieves in both stating and proving PCRs, and it sets forth a natural connection between PCRs and three well-known classical problems in statistics and probability theory: the speed of mean Glivenko-Cantelli convergence, the estimation of weighted Poincaré-Wirtinger constants and Sanov large deviation principle for Wasserstein distance. The second contribution combines the use of Wasserstein distance with a suitable sieve construction to establish PCRs under full Bayesian nonparametric models. As a novelty with respect to existing literature of PCRs, our second result provides with the first treatment of PCRs under non-dominated Bayesian models. Applications of our results are presented for some classical Bayesian statistical models, e.g., regular parametric models, infinite-dimensional exponential families, linear regression in infinite dimension and nonparametric models under Dirichlet process priors.

math.ST↗

arXiv2022

Wasserstein posterior contraction rates in non-dominated Bayesian nonparametric models

Posterior contractions rates (PCRs) strengthen the notion of Bayesian consistency, quantifying the speed at which the posterior distribution concentrates on arbitrarily small neighborhoods of the true model, with probability tending to 1 or almost surely, as the sample size goes to infinity. Under the Bayesian nonparametric framework, a common assumption in the study of PCRs is that the model is dominated for the observations; that is, it is assumed that the posterior can be written through the Bayes formula. In this paper, we consider the problem of establishing PCRs in Bayesian nonparametric models where the posterior distribution is not available through the Bayes formula, and hence models that are non-dominated for the observations. By means of the Wasserstein distance and a suitable sieve construction, our main result establishes PCRs in Bayesian nonparametric models where the posterior is available through a more general disintegration than the Bayes formula. To the best of our knowledge, this is the first general approach to provide PCRs in non-dominated Bayesian nonparametric models, and it relies on minimal modeling assumptions and on a suitable continuity assumption for the posterior distribution. Some refinements of our result are presented under additional assumptions on the prior distribution, and applications are given with respect to the Dirichlet process prior and the normalized extended Gamma process prior.

math.ST↗

arXiv2021

A compound Poisson perspective of Ewens-Pitman sampling model

The Ewens-Pitman sampling model (EP-SM) is a distribution for random partitions of the set $\{1,\ldots,n\}$, with $n\in\mathbb{N}$, which is index by real parameters $α$ and $θ$ such that either $α\in[0,1)$ and $θ>-α$, or $α<0$ and $θ=-mα$ for some $m\in\mathbb{N}$. For $α=0$ the EP-SM reduces to the celebrated Ewens sampling model (E-SM), which admits a well-known compound Poisson perspective in terms of the log-series compound Poisson sampling model (LS-CPSM). In this paper, we consider a generalization of the LS-CPSM, which is referred to as the negative Binomial compound Poisson sampling model (NB-CPSM), and we show that it leads to extend the compound Poisson perspective of the E-SM to the more general EP-SM for either $α\in(0,1)$, or $α<0$. The interplay between the NB-CPSM and the EP-SM is then applied to the study of the large $n$ asymptotic behaviour of the number of blocks in the corresponding random partitions, leading to a new proof of Pitman's $α$ diversity. We discuss the proposed results, and conjecture that analogous compound Poisson representations may hold for the class of $α$-stable Poisson-Kingman sampling models, of which the EP-SM is a noteworthy special case.

math.PR↗

arXiv2021

A Bayesian nonparametric approach to count-min sketch under power-law data streams

The count-min sketch (CMS) is a randomized data structure that provides estimates of tokens' frequencies in a large data stream using a compressed representation of the data by random hashing. In this paper, we rely on a recent Bayesian nonparametric (BNP) view on the CMS to develop a novel learning-augmented CMS under power-law data streams. We assume that tokens in the stream are drawn from an unknown discrete distribution, which is endowed with a normalized inverse Gaussian process (NIGP) prior. Then, using distributional properties of the NIGP, we compute the posterior distribution of a token's frequency in the stream, given the hashed data, and in turn corresponding BNP estimates. Applications to synthetic and real data show that our approach achieves a remarkable performance in the estimation of low-frequency tokens. This is known to be a desirable feature in the context of natural language processing, where it is indeed common in the context of the power-law behaviour of the data.

stat.ML↗

arXiv2020

A Berry-Esseen theorem for Pitman's $α$-diversity

This paper is concerned with the study of the random variable $K_n$ denoting the number of distinct elements in a random sample $(X_1, \dots, X_n)$ of exchangeable random variables driven by the two parameter Poisson-Dirichlet distribution, $PD(α,θ)$. For $α\in(0,1)$, Theorem 3.8 in \cite{Pit(06)} shows that $\frac{K_n}{n^α}\stackrel{\text{a.s.}}{\longrightarrow} S_{α,θ}$ as $n\rightarrow+\infty$. Here, $S_{α,θ}$ is a random variable distributed according to the so-called scaled Mittag-Leffler distribution. Our main result states that $$ \sup_{x \geq 0} \Big| \ppsf\Big[\frac{K_n}{n^α} \leq x \Big] - \ppsf[S_{α,θ} \leq x] \Big| \leq \frac{C(α, θ)}{n^α} $$ holds with an explicit constant $C(α, θ)$. The key ingredients of the proof are a novel probabilistic representation of $K_n$ as compound distribution and new, refined versions of certain quantitative bounds for the Poisson approximation and the compound Poisson distribution.

math.PR↗

arXiv2020

De Finetti's theorem: rate of convergence in Kolmogorov distance

This paper provides a quantitative version of de Finetti law of large numbers. Given an infinite sequence $\{X_n\}_{n \geq 1}$ of exchangeable Bernoulli variables, it is well-known that $\frac{1}{n} \sum_{i = 1}^n X_i \stackrel{a.s.}{\longrightarrow} Y$, for a suitable random variable $Y$ taking values in $[0,1]$. Here, we consider the rate of convergence in law of $\frac{1}{n} \sum_{i = 1}^n X_i$ towards $Y$, with respect to the Kolmogorov distance. After showing that any rate of the type of $1/n^α$ can be obtained for any $α\in (0,1]$, we find a sufficient condition on the probability distribution of $Y$ for the achievement of the optimal rate of convergence, that is $1/n$. Our main result improve on existing literature: in particular, with respect to \cite{MPS}, we study a stronger metric while, with respect to \cite{Mna}, we weaken the regularity hypothesis on the probability distribution of $Y$.

math.PR↗