arXiv subjects

Mikołaj J. Kasprzak

Publications and source records attributed to Mikołaj J. Kasprzak.

5 recordsLinked to original sources

arXiv2026

Asymptotics of cut distributions and robust modular inference using Posterior Bootstrap

Bayesian inference provides a framework to combine various model components with shared parameters, allowing joint uncertainty estimation and the use of all available data sources. Unfortunately, misspecification of any part of the model might propagate to all other parts and can lead to unsatisfactory results. Cut distributions have been proposed as a remedy, where the information is prevented from flowing along certain directions. We study cut distributions from an asymptotic perspective and obtain a Bernstein-von Mises theorem, as well as a Laplace approximation with quantitative bounds. We then propose an algorithm based on the Posterior Bootstrap that delivers credible regions with the nominal frequentist asymptotic coverage. The proposed methods are illustrated with numerical experiments in a variety of examples, including causal inference with propensity scores.

stat.ME↗

arXiv2025

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 constructing a Gaussian approximation to the Bayesian posterior and thereby approximating the posterior mean and variance. But approximation quality is a concern. One might consider using rate-of-convergence bounds from certain versions of the Bayesian Central Limit Theorem (BCLT) to provide quality guarantees. But existing bounds require assumptions that are unrealistic even for relatively simple real-life Bayesian analyses; more specifically, existing bounds either (1) require knowing the true data-generating parameter, (2) are asymptotic in the number of samples, (3) do not control the Bayesian posterior mean, or (4) require strongly log concave models to compute. In this work, we provide the first computable bounds on quality that simultaneously (1) do not require knowing the true parameter, (2) apply to finite samples, (3) control posterior means and variances, and (4) apply generally to models that satisfy the conditions of the asymptotic BCLT. Moreover, we substantially improve the dimension dependence of existing bounds; in fact, we achieve the lowest-order dimension dependence possible in the general case. 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.

math.ST↗

arXiv2022

Vector-valued statistics of binomial processes: Berry-Esseen bounds in the convex distance

We study the discrepancy between the distribution of a vector-valued functional of i.i.d. random elements and that of a Gaussian vector. Our main contribution is an explicit bound on the convex distance between the two distributions, holding in every dimension. Such a finding constitutes a substantial extension of the one-dimensional bounds deduced in Chatterjee (2007) and Lachièze-Rey and Peccati (2017), as well as of the multidimensional bounds for smooth test functions and indicators of rectangles derived, respectively, in Dung (2019), and Fang and Koike (2021). Our techniques involve the use of Stein's method, combined with a suitable adaptation of the recursive approach inaugurated by Schulte and Yukich (2017): this yields rates of converge that have a presumably optimal dependence on the sample size. We develop several applications of a geometric nature, among which is a new collection of multidimensional quantitative limit theorems for the intrinsic volumes associated with coverage processes in Euclidean spaces.

math.PR↗

arXiv2021

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 Erdős-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.

math.PR↗

arXiv2020

Stein's method for multivariate Brownian approximations of sums under dependence

We use Stein's method to obtain a bound on the distance between scaled $p$-dimensional random walks and a $p$-dimensional (correlated) Brownian Motion. We consider dependence schemes including those in which the summands in scaled sums are weakly dependent and their $p$ components are strongly correlated. As an example application, we prove a functional limit theorem for exceedances in an $m$-scans process, together with a bound on the rate of convergence. We also find a bound on the rate of convergence of scaled U-statistics to Brownian Motion, representing an example of a sum of strongly dependent terms.

math.PR↗