Conference Agenda

By characterizing the posterior distribution as the minimizer of a divergence functional, variational inference allows to identify the path of steepest descent toward the posterior as the gradient flow with respect to the divergence, and to calculate rates of convergence determined by the convexity of the functionals. While this approach has been successful for parametric families of distributions, we present an extension to the nonparametric setting, which is based on the construction of a divergence that generalizes the Mahalanobis divergence; an instance of this divergence has been constructed by Hallin, using multivariate quantiles based on optimal transport. We use an extension of this idea deemed Sinkhorn divergence, which is based on a regularized version of optimal transport and provides better estimates for the rates of convergence, given its stronger convexity as a functional. We present the estimation and inference result for this procedure as well as various approximation schemes to different types of distributions.

The present talk is based on the joint research with Denis Belomestny and Vladimir Panov, available as a preprint [1], which is devoted to the problem of statistical inference for random sums \begin{equation}\label{main} X=\sum\limits_{k=1} ^N \xi_k, \end{equation} where $\xi_1,\xi_2,\dots$ is a sequence of i.i.d. random variables, and $N$ is a positive integer-valued random variable independent of $\xi_1,\xi_2,\dots$. A natural problem arising in this setting is that of recovering the distribution of either $N$ or $\xi_1$ based on a sample from $X$, assuming that the law of the other variable ($\xi_1$ or $N$, respectively) is known. The current research considers the second task. In the most popular case of Poisson random sums, which appears when $N$ has the Poisson distribution, this problem is well-known and is typically referred to as decompounding [3]. Nowadays, there exists a great variety of methods for recovering the distribution of summands in the compound Poisson model, including, but not limited to, kernel-type estimators [7], spectral methods [4] and Bayesian approach [5]. Some results are also available for the case when the law of $N$ is geometric [6]. However, very few research has been devoted to the general case, when $N$ has an arbitrary distribution supported on non-negative integers.

In this research, we propose an estimator for the distribution of $\xi_1$ without imposing any parametric assumptions on the law of $N$. While the problem of nonparametric inference for the distribution of $\xi_1$ in the random sum model has already been considered by Bøgsted and Pitts [2], it should be noted that their estimation scheme requires the summands to be positive, which is not necessary for our estimation procedure. In addition, the rates of convergence are out of the scope of paper [2], while we put a special emphasis on proving the bounds for the error of the proposed estimator. We demonstrate that for several large classes of distributions the rates of convergence are of polynomial order, and moreover, we show that in certain cases the upper and lower bounds coincide, implying that the proposed estimator is minimax optimal.

References:

1. Belomestny, D., Morozova, E. and Panov, V. (2024). Decompounding under general mixing distributions. Preprint, arXiv:2405.05419, series "math.ST".

2. Bøgsted, M. and Pitts, S.M. (2010). Decompounding random sums: a nonparametric approach. Annals of the Institute of Statistical Mathematics, 62:855–872.

3. Buchmann, B. and Grübel, R. (2003). Decompounding: an estimation problem for Poisson random sums. The Annals of Statistics, 31(4):1054–1074.

4. Coca, A.J. (2018). Efficient nonparametric inference for discretely observed compound Poisson processes. Probability Theory and Related Fields, 170(1-2):475–523.

5. Gugushvili, S., Mariucci, E. and van der Meulen, F. (2020). Decompounding discrete distributions: A nonparametric Bayesian approach. Scandinavian journal of statistics, 47(2):464–492.

6. Hansen, M. B. and Pitts, S. M. (2006). Nonparametric inference from the M/G/1 workload. Bernoulli, 12(4):737–759.

7. Van Es, B., Gugushvili, S. and Spreij, P. (2007). A kernel type nonparametric density estimator for decompounding. Bernoulli, 13(3):672–694.

In recent years, statistical methodology based on optimal transport (OT) witnessed a considerable increase in practical and theoretical interest. A central reason for this trend is the ability of optimal transport to efficiently compare data in a geometrically meaningful way. This development was further amplified by computational advances spurred by the introduction of entropy regularized optimal transport (EOT). In applications, the OT or EOT cost are often estimated through an empirical plug-in approach, raising statistical questions about the performance and uncertainty of these estimators. The convergence behavior of the empirical OT cost for increasing sample size is dictated by various aspects. Remarkably, under distinct population measures with different intrinsic dimensions, we show that the convergence rate for the empirical OT cost is determined by the population measure with lower intrinsic dimension -- a novel phenomenon we term "lower complexity adaptation“. For the empirical EOT cost, we establish a similar phenomenon and show that the dependency on the entropy regularization parameter in the convergence rate is determined by the minimum intrinsic dimension of the population measures. Concerning the fluctuation of the empirical OT cost around its population counterpart, we show for settings where one measure is sufficiently low dimensional we show that the asymptotic fluctuation is given by the supremum over a Gaussian process. This is in strict contrast to the entropy regularized setting, where we establish a central limit theorem with a centered normal asymptotic law. Altogether this talk will highlight key similarities and differences between empirical OT and EOT, offering comprehensive insights into strengths and limitations of transport-based methodologies in statistical contexts.

This talk is based on joint work with Thomas Staudt, Marcel Klatt, Michel Groppe, Alberto-Gonzáles-Sanz, and Axel Munk.

The statistical properties of entropic optimal transport recently became of great interest as this quantity has been shown to be useful for complex data analysis, among others due to its fast computability. Prominent applications meanwhile include tasks in machine learning, computer vision and various subject matters, such as econometrics or particle physics. In cell biology, colocalization analysis based on entropic optimal transport (EOT) has been used as a measure for quantification of spatial proximity of different protein assemblies. Using properties of the entropic optimal transport plans, we derive asymptotic weak convergence result for a large class of functionals of the EOT plan, in which the colocalization process is included. The proof is based on Hadamard differentiability and the extended delta method. As applications, we obtain uniform confidence bands for colocalization curves, bootstrap consistency and a notion of conditional colocalization. Our theory is supported by simulation studies and is illustrated by real world data analysis from mitochondrial protein colocalization.