Strong posterior contraction rates via Wasserstein dynamics

In this paper, we develop a novel approach to posterior contractions rates (PCRs), for both finite-dimensional (parametric) and infinite-dimensional (nonparametric) Bayesian models. Critical to our approach is the combination of an assumption of local Lipschitz-continuity for the posterior distribution with a dynamic formulation of the Wasserstein distance, here referred to as Wasserstein dynamics, which allows to set forth a connection between the problem of establishing PCRs and some classical problems in mathematical analysis, probability theory and mathematical statistics: Laplace methods for approximating integrals, Sanov's large deviation principle under the Wasserstein distance, rates of convergence of mean Glivenko-Cantelli theorems, and estimates of weighted Poincar\'e-Wirtinger constants. Under dominated Bayesian models, we present two main results: i) a theorem on PCRs for the regular infinite-dimensional exponential family of statistical models; ii) a theorem on PCRs for a general dominated statistical models. Some applications of our results are presented for the regular parametric model, the multinomial model, the finite-dimensional and the infinite-dimensional logistic-Gaussian model and the infinite-dimensional linear regression. It turns out that our approach leads to optimal PCRs in finite dimension, whereas in infinite dimension it is shown explicitly how prior distributions affect the corresponding PCRs. In general, with regards to infinite-dimensional Bayesian models for density estimation, our approach to PCRs is the first to consider strong norm distances on parameter spaces of functions, such as Sobolev-like norms, as most of the literature deals with spaces of density functions endowed with $\mathrm{L}^p$ norms or the Hellinger distance.

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. Under the popular framework of $(\varepsilon,\delta)$-differential privacy for perturbed data, we introduce a private likelihood-ratio (LR) test for goodness-of-fit and we study its large sample properties, showing the importance of taking the perturbation into account to avoid a loss in the statistical significance of the test. Our main contribution provides a quantitative characterization of the trade-off between confidentiality, measured via differential privacy parameters $\varepsilon$ and $\delta$, and utility, measured via the power of the test. In particular, we establish a precise Bahadur-Rao type large deviation expansion for the power of the private LR test, which leads to: i) identify a critical quantity, as a function of the sample size and $(\varepsilon,\delta)$, which determines a loss in the power of the private LR test; ii) quantify the sample cost of $(\varepsilon,\delta)$-differential privacy in the private LR test, namely the additional sample size that is required to recover the power of the LR test in the absence of perturbation. Such a result relies on a novel multidimensional large deviation principle for sum of i.i.d. random vectors, which is of independent interest. Our work presents the first rigorous treatment of privacy-protecting LR tests for goodness-of-fit in frequency tables, making use of the power of the test to quantify the trade-off between confidentiality and utility.

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.

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, i.e. point queries, based on random hashed data. Learning-augmented CMSs improve the CMS by learning models that allow to better exploit data properties. In this paper, we focus on the learning-augmented CMS of Cai, Mitzenmacher and Adams (\textit{NeurIPS} 2018), which relies on Bayesian nonparametric (BNP) modeling of a data stream via Dirichlet process (DP) priors. This is referred to as the CMS-DP, and it leads to BNP estimates of a point query as posterior means of the point query given the hashed data. While BNPs is proved to be a powerful tool for developing robust learning-augmented CMSs, ideas and methods behind the CMS-DP are tailored to point queries under DP priors, and they can not be used for other priors or more general queries. In this paper, we present an alternative, and more flexible, derivation of the CMS-DP such that: i) it allows to make use of the Pitman-Yor process (PYP) prior, which is arguably the most popular generalization of the DP prior; ii) it can be readily applied to the more general problem of estimating range queries. This leads to develop a novel learning-augmented CMS under power-law data streams, referred to as the CMS-PYP, which relies on BNP modeling of the stream via PYP priors. Applications to synthetic and real data show that the CMS-PYP outperforms the CMS and the CMS-DP in the estimation of low-frequency tokens; this known to be a critical feature in natural language processing, where it is indeed common to encounter power-law data streams.