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Species-sampling problems (SSPs) refer to a vast class of statistical problems calling for the estimation of (discrete) functionals of the unknown species composition of an unobservable population. A common feature of SSPs is their invariance with respect to species labeling, which is at the core of the Bayesian nonparametric (BNP) approach to SSPs under the popular Pitman-Yor process (PYP) prior. In this paper, we develop a BNP approach to SSPs that are not “invariant” to species labeling, in the sense that an ordering or ranking is assigned to species’ labels. Inspired by the population genetics literature on age-ordered alleles’ compositions, we study the following SSP with ordering: given an observable sample from an unknown population of individuals belonging to species (alleles), with species’ labels being ordered according to weights (ages), estimate the frequencies of the first r order species’ labels in an enlarged sample obtained by including additional unobservable samples. By relying on an ordered PYP prior, we obtain an explicit posterior distribution of the first r order frequencies, with estimates being of easy implementation and computationally efficient. We apply our approach to the analysis of genetic variation, showing its effectiveness in estimating the frequency of the oldest allele, and then we discuss other potential applications.

Gibbs-type priors are combinatorial processes widely used as key components in several Bayesian nonparametric models. By virtue of their flexibility and mathematical tractability, they turn out to be predominant priors in species sampling problems and mixture modeling. We introduce a new family of processes which extends the Gibbs-type one, by including a contaminant component in the model to account for an excess of observations with frequency one. We first investigate the induced random partition, the associated predictive distribution, the asymptotic behavior of the total number of blocks and the number of blocks with a given frequency: all the results we obtain are in closed form and easily interpretable. A remarkable aspect of contaminated Gibbs-type priors relies on their predictive structure, compared to the one of the standard Gibbs-type family: it depends on the additional sampling information on the number of observations with frequency one out of the observed sample. As a noteworthy example we focus on the contaminated version of the Pitman-Yor process, which turns out to be analytically tractable and computationally feasible. Finally we pinpoint the advantage of our construction in different applications: we show how it helps to improve predictive inference in a species-related dataset exhibiting a high number of species with frequency one; we also discuss the use of the proposed construction in mixture models to perform density estimation and outlier detection.

There is a growing interest in the estimation of the number of unseen features, mostly driven by biological applications. A recent work brought out a peculiar property of the popular completely random measures (CRMs) as prior models in Bayesian nonparametric (BNP) inference for the unseen-features problem: for fixed prior’s parameters, they all lead to a Poisson posterior distribution for the number of unseen features, which depends on the sampling information only through the sample size. CRMs are thus not a flexible prior model for the unseen-features problem and, while the Poisson posterior distribution may be appealing for analytical tractability and ease of interpretability, its independence from the sampling information makes the BNP approach a questionable oversimplification, with posterior inferences being completely determined by the estimation of unknown prior’s parameters. In this article, we introduce the stable-Beta scaled process (SB-SP) prior, and we show that it allows to enrich the posterior distribution of the number of unseen features arising under CRM priors, while maintaining its analytical tractability and interpretability. That is, the SB-SP prior leads to a negative Binomial posterior distribution, which depends on the sampling information through the sample size and the number of distinct features, with corresponding estimates being simple, linear in the sampling information and computationally efficient. We apply our BNP approach to synthetic data and to real cancer genomic data, showing that: (i) it outperforms the most popular parametric and nonparametric competitors in terms of estimation accuracy; (ii) it provides improved coverage for the estimation with respect to a BNP approach under CRM priors. Supplementary materials for this article are available online.

The use of large datasets for targeted therapeutic interventions requires new ways to characterize the heterogeneity observed across subgroups of a specific population. In particular, models for partially exchangeable data are needed for inference on nested datasets, where the observations are assumed to be organized in different units and some sharing of information is required to learn distinctive features of the units. In this manuscript, we propose a nested common atoms model (CAM) that is particularly suited for the analysis of nested datasets where the distributions of the units are expected to differ only over a small fraction of the observations sampled from each unit. The proposed CAM allows a two-layered clustering at the distributional and observational level and is amenable to scalable posterior inference through the use of a computationally efficient nested slice sampler algorithm. We further discuss how to extend the proposed modeling framework to handle discrete measurements, and we conduct posterior inference on a real microbiome dataset from a diet swap study to investigate how the alterations in intestinal microbiota composition are associated with different eating habits. We further investigate the performance of our model in capturing true distributional structures in the population by means of a simulation study.

In the 1920s, the English philosopher W.E. Johnson introduced a characterization of the symmetric Dirichlet prior distribution in terms of its predictive distribution. This is typically referred to as Johnson’s “sufficientness” postulate, and it has been the subject of many contributions in Bayesian statistics, leading to predictive characterization for infinite-dimensional generalizations of the Dirichlet distribution, i.e., species-sampling models. In this paper, we review “sufficientness” postulates for species-sampling models, and then investigate analogous predictive characterizations for the more general feature-sampling models. In particular, we present a “sufficientness” postulate for a class of feature-sampling models referred to as Scaled Processes (SPs), and then discuss analogous characterizations in the general setup of feature-sampling models.

Partially exchangeable datasets are characterized by observations grouped into known, heterogeneous units. The recently developed Common Atoms Model (CAM) is a Bayesian nonparametric technique suited for analyzing this type of data. CAM induces a two-layered clustering structure: one across observations and another across units. In particular, the units are clustered according to their distributional similarities. In this article, we illustrate the versatility of CAM with an application to an openly available Spotify dataset. The dataset contains quantitative audio features for a large number of songs grouped by artists. After describing the data preprocessing steps, we employ CAM to group the Spotify artists according to the distributions of the energy of their songs.

While the cost of sequencing genomes has decreased dramatically in recent years, this expense often remains nontrivial. Under a fixed budget, scientists face a natural trade-off between quantity and quality: spending resources to sequence a greater number of genomes or spending resources to sequence genomes with increased accuracy. Our goal is to find the optimal allocation of resources between quantity and quality. Optimizing resource allocation promises to reveal as many new variations in the genome as possible. We introduce a Bayesian nonparametric methodology to predict the number of new variants in a follow-up study based on a pilot study. When experimental conditions are kept constant between the pilot and follow-up, we find that our prediction is competitive with the best existing methods. Unlike current methods, though, our new method allows practitioners to change experimental conditions between the pilot and the follow-up. We demonstrate how this distinction allows our method to be used for more realistic predictions and for optimal allocation of a fixed budget between quality and quantity. We validate our method on cancer and human genomics data.

The problem of maximizing cell type discovery under budget constraints is a fundamental challenge for the collection and analysis of single-cell RNAsequencing (scRNA-seq) data. In this paper we introduce a simple, computationally efficient and scalable Bayesian nonparametric sequential approach to optimize the budget allocation when designing a large-scale experiment for the collection of scRNA-seq data for the purpose of, but not limited to, creating cell atlases. Our approach relies on the following tools: (i) a hierarchical Pitman–Yor prior that recapitulates biological assumptions regarding cellular differentiation, and (ii) a Thompson sampling multiarmed bandit strategy that balances exploitation and exploration to prioritize experiments across a sequence of trials. Posterior inference is performed by using a sequential Monte Carlo approach which allows us to fully exploit the sequential nature of our species sampling problem. We empirically show that our approach outperforms state-of-the-art methods and achieves near-Oracle performance on simulated and scRNA-seq data alike.

Post randomization methods are among the most popular disclosure limitation techniques for both categorical and continuous data. In the categorical case, given a stochastic matrix M and a specified variable, an individual belonging to category i is changed to category j with probability Mi,j. Every approach to choose the randomization matrix M has to balance between two desiderata: (1) preserving as much statistical information from the raw data as possible; (2) guaranteeing the privacy of individuals in the dataset. This trade-off has generally been shown to be very challenging to solve. In this work, we use recent tools from the computer science literature and propose to choose M as the solution of a constrained maximization problems. Specifically, M is chosen as the solution of a constrained maximization problem, where we maximize the mutual information between raw and transformed data, given the constraint that the transformation satisfies the notion of differential privacy. For the general categorical model, it is shown how this maximization problem reduces to a convex linear programming and can be therefore solved with known optimization algorithms.

Consider an (observable) random sample of size n from an infinite population of individuals, each individual being endowed with a finite set of "features" from a collection of features (F-j)(j>1) with unknown probabilities (p(j))(j>1), i.e., p(j) is the probability that an individual displays feature F-j. Under this feature sampling framework, in recent years there has been a growing interest in estimating the sum of the probability masses p(j)'s of features observed with frequency r >= 0 in the sample, here denoted by M-n,M-r. This is the natural feature sampling counterpart of the classical problem of estimating small probabilities in the species sampling framework, where each individual is endowed with only one feature (or "species"). In this paper we study the problem of consistent estimation of the small mass M-n,M-r. We first show that there do not exist universally consistent estimators, in the multiplicative sense, of the missing mass M-n,M-0. Then, we introduce an estimator of M-n,M-r and identify sufficient conditions under which the estimator is consistent. In particular, we propose a nonparametric estimator (M) over cap (n,r) of M-n,M-r which has the same analytic form of the celebrated Good-Turing estimator for small probabilities, with the sole difference that the two estimators have different ranges (supports). Then, we show that (M) over cap (n,r) is strongly consistent, in the multiplicative sense, under the assumption that (p(j))(j >= 1) has regularly varying heavy tails.

Hierarchical nonparametric processes are popular tools for defining priors on collections of probability distributions, which induce dependence across multiple samples. In survival analysis problems, one is typically interested in modeling the hazard rates, rather than the probability distributions themselves, and the currently available methodologies are not applicable. Here, we fill this gap by introducing a novel, and analytically tractable, class of multivariate mixtures whose distribution acts as a prior for the vector of sample-specific baseline hazard rates. The dependence is induced through a hierarchical specification of the mixing random measures that ultimately corresponds to a composition of random discrete combinatorial structures. Our theoretical results allow to develop a full Bayesian analysis for this class of models, which can also account for right-censored survival data and covariates, and we also show posterior consistency. In particular, we emphasize that the posterior characterization we achieve is the key for devising both marginal and conditional algorithms for evaluating Bayesian inferences of interest. The effectiveness of our proposal is illustrated through some synthetic and real data examples.

In the present paper, we address the problem of prediction within the setting of species sampling models. We consider d populations composed of different species with unknown proportions. Our goal is to predict specific features of additional and unobserved samples from the d populations by adopting a Bayesian nonparametric model. We focus on a broad class of hierarchical priors. These were introduced and investigated in[1], where also an algorithm for drawing predictions is devised, however, without any specific numerical illustration. The aim of this paper is twofold: on the one hand, we provide an illustration with an actual implementation of the algorithm of[1] and, on the other hand, we discuss its relevance with respect to complex prediction problems with species sampling data.

Hierarchies of discrete probability measures are remarkably popular as nonparametric priors in applications, arguably due to two key properties: (i) they naturally represent multiple heterogeneous populations; (ii) they produce ties across populations, resulting in a shrinkage property often described as “sharing of information.” In this paper, we establish a distribution theory for hierarchical random measures that are generated via normalization, thus encompassing both the hierarchical Dirichlet and hierarchical Pitman–Yor processes. These results provide a probabilistic characterization of the induced (partially exchangeable) partition structure, including the distribution and the asymptotics of the number of partition sets, and a complete posterior characterization. They are obtained by representing hierarchical processes in terms of completely random measures, and by applying a novel technique for deriving the associated distributions. Moreover, they also serve as building blocks for new simulation algorithms, and we derive marginal and conditional algorithms for Bayesian inference.

Discrete random structures are important tools in Bayesian nonparametrics and the resulting models have proven effective in density estimation, clustering, topic modeling and prediction, among others. In this paper, we consider nested processes and study the dependence structures they induce. Dependence ranges between homogeneity, corresponding to full exchangeability, and maximum heterogeneity, corresponding to (unconditional) independence across samples. The popular nested Dirichlet process is shown to degenerate to the fully exchangeable case when there are ties across samples at the observed or latent level. To overcome this drawback, inherent to nesting general discrete random measures, we introduce a novel class of latent nested processes. These are obtained by adding common and group-specific completely random measures and, then, normalizing to yield dependent random probability measures. We provide results on the partition distributions induced by latent nested processes, and develop a Markov Chain Monte Carlo sampler for Bayesian inferences. A test for distributional homogeneity across groups is obtained as a by-product. The results and their inferential implications are showcased on synthetic and real data.

A very active line of research in Bayesian statistics has aimed at defining and investigating general classes of nonparametric priors. A notable example, which includes the Dirichlet process, is obtained through normalization or transformation of completely random measures. These have been extensively studied for the exchangeable setting. However in a large variety of applied problems data are heterogeneous, being generated by different, though related, experiments; in such situations partial exchangeability is a more appropriate assumption. In this spirit we propose a nonparametric hierarchical model based on transformations of completely random measures, which extends the hierarchical Dirichlet process. The model allows us to handle related groups of observations, creating a borrowing of strength between them. From the theoretical viewpoint, we analyze the induced partition structure, which plays a pivotal role in a very large number of inferential problems. The resulting partition probability function has a feasible expression, suitable to address predictionin its generality, as suggested by de Finetti. Finally we propose a set of applications which include inference on genomic and survival data.

In Bayesian Nonparametrics partial exchangeability is a useful assumption tailored for heterogeneous, though related, groups of observations. Recent contributions in Bayesian literature have focused on the construction of dependent nonparametric priors to accommodate for partially exchangeable sequences of observations. In the present paper we concentrate on vectors of hierarchical Pitman-Yor processes, in which the dependence is created by choosing a common random base measure for each group of observations. These hierarchical processes are then used to define dependent hierarchical mixtures. We finally apply the model to estimate densities arising from multiple groups of observations by performing a suitable Gibbs sampling algorithm.

The definition and investigation of general classes of nonparametric priors has recently been an active research line in Bayesian statistics. Among the various proposals, the Gibbs-type family, which includes the Dirichlet process as a special case, stands out as the most tractable class of nonparametric priors for exchangeable sequences of observations. This is the consequence of a key simplifying assumption on the learning mechanism, which, however, has justification except that of ensuring mathematical tractability. In this paper, we remove such an assumption and investigate a general class of random probability measures going beyond the Gibbs-type framework. More specifically, we present a nonparametric hierarchical structure based on transformations of completely random measures, which extends the popular hierarchical Dirichlet process. This class of priors preserves a good degree of tractability, given that we are able to determine the fundamental quantities for Bayesian inference. In particular, we derive the induced partition structure and the prediction rules and characterize the posterior distribution. These theoretical results are also crucial to devise both a marginal and a conditional algorithm for posterior inference. An illustration concerning prediction in genomic sequencing is also provided.

The mean density of a random closed set with integer Hausdorff dimension is a crucial notion in stochastic geometry, in fact it is a fundamental tool in a large variety of applied problems, such as image analysis, medicine, computer vision, etc. Hence the estimation of the mean density is a problem of interest both from a theoretical and computational standpoint. Nowadays different kinds of estimators are available in the literature, in particular here we focus on a kernel–type estimator, which may be considered as a generalization of the traditional kernel density estimator of random variables to the case of random closed sets. The aim of the present paper is to provide asymptotic properties of such an estimator in the context of Boolean models, which are a broad class of random closed sets. More precisely we are able to prove large and moderate deviation principles, which allow us to derive the strong consistency of the estimator of the mean density as well as asymptotic confidence intervals. Finally we underline the connection of our theoretical findings with classical literature concerning density estimation of random variables

Vectors of hierarchical random probability measures are popular tools in Bayesian nonparametrics. They may be used as priors whenever partial exchangeability is assumed at the level of either the observations or of some latent variables involved in the model. The first contribution in this direction can be found in Teh et al. (2006), who introduced the hierarchical Dirichlet process. Recently, Camerlenghi et al. (2018) have developed a general distribution theory for hierarchical processes, which includes the derivation of the partition structure, the posterior distribution and the prediction rules. The present paper is a review of these theoretical findings for vectors of hierarchies of Pitman--Yor processes.

The prediction of future outcomes of a random phenomenon is typically based on a certain number of “analogous” observations from the past. When observations are generated by multiple samples, a natural notion of analogy is partial exchangeability and the problem of prediction can be effectively addressed in a Bayesian nonparametric setting. Instead of confining ourselves to the prediction of a single future experimental outcome, as in most treatments of the subject, we aim at predicting features of an unobserved additional sample of any size. We first provide a structural property of prediction rules induced by partially exchangeable arrays, without assuming any specific nonparametric prior. Then we focus on a general class of hierarchical random probability measures and devise a simulation algorithm to forecast the outcome of m future observations, for any m≥1. The theoretical result and the algorithm are illustrated by means of a real dataset, which also highlights the “borrowing strength” behavior across samples induced by the hierarchical specification.

The problem of the evaluation and estimation of the mean density of random closed sets in Rd with integer Hausdorff dimension 0 < n < d, is of great interest in many different scientific and technological fields. Among the estimators of the mean density available in literature, the so-called "Minkowski content"-based estimator reveals its benefits in applications in the non-stationary cases. We introduce here a multivariate version of such estimator, and we study its asymptotical properties by means of large and moderate deviation results. In particular we prove that the estimator is strongly consistent and asymptotically Normal. Furthermore we also provide confidence regions for the mean density of the involved random closed set in m ≥ 1 distinct points x1, …, xm ∈ Rd.

We review some recent constructions of time-dependent randomprobabilitymeasures of Gibbs type which exhibit diffusive behaviour. The characterization of the dynamics of the type frequencies, and those of the type heterogeneity in the underlying population, also known as alpha diversity, allows for a qualitative classification of these models according to whether the heterogeneity is driven by state-dependent

The estimation of the mean density of random closed sets in $$\mathbb {R}^d$$Rd with integer Hausdorff dimension $$n

Many real phenomena may be modelled as random closed sets in ℝd, of different Hausdorff dimensions. The problem of the estimation of pointwise mean densities of absolutely continuous, and spatially inhomogeneous, random sets with Hausdorff dimension n < d; has been the subject of extended mathematical analysis by the authors. In particular, two different kinds of estimators have been recently proposed, the first one is based on the notion of Minkowski content, the second one is a kernel-type estimator generalizing the well-known kernel density estimator for random variables. The specific aim of the present paper is to validate the theoretical results on statistical properties of those estimators by numerical experiments. We provide a set of simulations which illustrates their valuable properties via typical examples of lower dimensional random sets

Many real phenomena may be modeled as random closed sets in Rd, of different Hausdorff dimensions. Of particular interest are cases in which their Hausdorff dimension, say n, is strictly less than d, such as fiber processes, boundaries of germ-grain models, and n-facets of random tessellations. A crucial problem is the estimation of pointwise mean densities of absolutely continuous, and spatially inhomogeneous random sets, as defined by the authors in a series of recent papers. While the case n= 0 (random vectors, point processes, etc.) has been, and still is, the subject of extensive literature, in this paper we face the general case of any n<. d; pointwise density estimators which extend the notion of kernel density estimators for random vectors are analyzed, together with a previously proposed estimator based on the notion of Minkowski content. In a series of papers, the authors have established the mathematical framework for obtaining suitable approximations of such mean densities. Here we study the unbiasedness and consistency properties, and identify optimal bandwidths for all proposed estimators, under sufficient regularity conditions. We show how some known results in the literature follow as particular cases. A series of examples throughout the paper, both non-stationary, and stationary, are provided to illustrate various relevant situations. © 2013 Elsevier Inc