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Antonio Lijoi

Publications and source records attributed to Antonio Lijoi.

At least 19 recordsLinked to original sources
arXiv2026

Ferguson's Dirichlet Process Breakthrough: A Lasting Legacy

Ferguson's 1973 introduction of the Dirichlet process marked a breakthrough in Bayesian nonparametric statistics. For the first time, a prior on the space of probability measures fulfilled two key desiderata: large support and analytical tractability. In this paper, we review three complementary constructions of the Dirichlet process, whose roots can be traced back to Ferguson: through finite-dimensional distributions, via normalization of a gamma process, and through predictive distributions. Each perspective not only deepens the understanding of the Dirichlet process but also provides a template for generalizations, from normalized random measures with independent increments to Gibbs--type priors and beyond. Over the past fifty years, the Dirichlet process has become the cornerstone of Bayesian nonparametric methodology and applications, while simultaneously inspiring the expansion of the landscape of nonparametric priors. Since de Finetti laid out the Bayesian nonparametric framework in the 1930s, the key obstacle had been the absence of a tractable nonparametric prior. Ferguson's contribution overcame this challenge, providing a solution to a decades-long open problem. In recognition of this decisive advance, it seems appropriate to refer to the Dirichlet process as the Ferguson--Dirichlet process.

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arXiv2026

Principled Estimation and Prediction with Competing Risks: a Bayesian Nonparametric Approach

Competing risks occur in survival analysis when multiple causes of death are present. They play a prominent role in several domains extending beyond biostatistics to encompass epidemiology, actuarial sciences, and reliability theory. This paper adopts a multi-state modeling framework to competing risks. We introduce a class of flexible nonparametric priors, defined through hierarchical completely random measures, to model the transition probabilities, and identify the specific (conditionally) conjugate member of this general class. Furthermore, we determine the joint marginal distribution of the data and of a latent random partition, and characterize the posterior distribution of the model. Leveraging these distributional results, we evaluate the predictive probability that a future event is of a specific type (e.g. death from a particular cause), as a function of the time at which the event occurs. The resulting function, derived on sound principles, is termed the prediction curve, and represents a major innovation in the literature. In addition, we provide posterior estimates for the survival function, and for the cause-specific incidence and subdistribution functions. Suitable simulation algorithms for posterior inference are also devised. The model's performance, as well as the algorithms' effectiveness, is evaluated through simulation studies. Finally, we illustrate our approach on clinical datasets.

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arXiv2026

Multivariate Species Sampling Models

Species sampling processes have long served as the fundamental framework for modeling random discrete distributions and exchangeable sequences. However, data arising from distinct but related sources require a broader notion of probabilistic invariance, making partial exchangeability a natural choice. Countless models for partially exchangeable data, collectively known as dependent nonparametric priors, have been proposed. These include hierarchical, nested and additive processes, widely used in statistics and machine learning. Still, a unifying framework is lacking and key questions about their underlying learning mechanisms remain unanswered. We fill this gap by introducing multivariate species sampling models, a new general class of nonparametric priors that encompasses most existing finite- and infinite-dimensional dependent processes. They are characterized by the induced partially exchangeable partition probability function encoding their multivariate clustering structure. We establish their core distributional properties and analyze their dependence structure, demonstrating that borrowing of information across groups is entirely determined by shared ties. This provides new insights into the underlying learning mechanisms, offering, for instance, a principled rationale for the previously unexplained correlation structure observed in existing models. Beyond providing a cohesive theoretical foundation, our approach serves as a constructive tool for developing new models and opens novel research directions for capturing richer dependence structures beyond the framework of multivariate species sampling processes.

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arXiv2026

Markov Stick-breaking Processes

Stick-breaking has a long history and is one of the most popular procedures for constructing random discrete distributions in Statistics and Machine Learning. In particular, due to their intuitive construction and computational tractability they are ubiquitous in modern Bayesian nonparametric inference. Most widely used models, such as the Dirichlet and the Pitman-Yor processes, rely on iid or independent length variables. Here we pursue a completely unexplored research direction by considering Markov length variables and investigate the corresponding general class of stick-breaking processes, which we term Markov stick-breaking processes. We establish conditions under which the associated species sampling process is proper and the distribution of a Markov stick-breaking process has full topological support, two fundamental desiderata for Bayesian nonparametric models. We also analyze the stochastic ordering of the weights and provide a new characterization of the Pitman-Yor process as the only stick-breaking process invariant under size-biased permutations, under mild conditions. Moreover, we identify two notable subclasses of Markov stick-breaking processes that enjoy appealing properties and include Dirichlet, Pitman-Yor and Geometric priors as special cases. Our findings include distributional results enabling posterior inference algorithms and methodological insights.

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arXiv2025

Partially exchangeable stochastic block models for (node-colored) multilayer networks

Multilayer networks generalize single-layered connectivity data in several directions. These generalizations include, among others, settings where multiple types of edges are observed among the same set of nodes (edge-colored networks) or where a single notion of connectivity is measured between nodes belonging to different pre-specified layers (node-colored networks). While progress has been made in statistical modeling of edge-colored networks, principled approaches that flexibly account for both within and across layer block-connectivity structures while incorporating layer information through a rigorous probabilistic construction are still lacking for node-colored multilayer networks. We fill this gap by introducing a novel class of partially exchangeable stochastic block models specified in terms of a hierarchical random partition prior for the allocation of nodes to groups, whose number is learned by the model. This goal is achieved without jeopardizing probabilistic coherence, uncertainty quantification and derivation of closed-form predictive within- and across-layer co-clustering probabilities. Our approach facilitates prior elicitation, the understanding of theoretical properties and the development of yet-unexplored predictive strategies for both the connections and the allocations of future incoming nodes. Posterior inference proceeds via a tractable collapsed Gibbs sampler, while performance is illustrated in simulations and in a real-world criminal network application. The notable gains achieved over competitors clarify the importance of developing general stochastic block models based on suitable node-exchangeability structures coherent with the type of multilayer network being analyzed.

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arXiv2024

Restricted mean survival times for comparing grouped survival data: a Bayesian nonparametric approach

Comparing survival experiences of different groups of data is an important issue in several applied problems. A typical example is where one wishes to investigate treatment effects. Here we propose a new Bayesian approach based on restricted mean survival times (RMST). A nonparametric prior is specified for the underlying survival functions: this extends the standard univariate neutral to the right processes to a multivariate setting and induces a prior for the RMST's. We rely on a representation as exponential functionals of compound subordinators to determine closed form expressions of prior and posterior mixed moments of RMST's. These results are used to approximate functionals of the posterior distribution of RMST's and are essential for comparing time--to--event data arising from different samples.

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arXiv2023

Nonparametric priors with full-range borrowing of information

Modeling of the dependence structure across heterogeneous data is crucial for Bayesian inference since it directly impacts the borrowing of information. Despite the extensive advances over the last two decades, most available proposals allow only for non-negative correlations. We derive a new class of dependent nonparametric priors that can induce correlations of any sign, thus introducing a new and more flexible idea of borrowing of information. This is achieved thanks to a novel concept, which we term hyper-tie, and represents a direct and simple measure of dependence. We investigate prior and posterior distributional properties of the model and develop algorithms to perform posterior inference. Illustrative examples on simulated and real data show that our proposal outperforms alternatives in terms of prediction and clustering.

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arXiv2023

A Wasserstein index of dependence for random measures

Optimal transport and Wasserstein distances are flourishing in many scientific fields as a means for comparing and connecting random structures. Here we pioneer the use of an optimal transport distance between Lévy measures to solve a statistical problem. Dependent Bayesian nonparametric models provide flexible inference on distinct, yet related, groups of observations. Each component of a vector of random measures models a group of exchangeable observations, while their dependence regulates the borrowing of information across groups. We derive the first statistical index of dependence in $[0,1]$ for (completely) random measures that accounts for their whole infinite-dimensional distribution, which is assumed to be equal across different groups. This is accomplished by using the geometric properties of the Wasserstein distance to solve a max-min problem at the level of the underlying Lévy measures. The Wasserstein index of dependence sheds light on the models' deep structure and has desirable properties: (i) it is $0$ if and only if the random measures are independent; (ii) it is $1$ if and only if the random measures are completely dependent; (iii) it simultaneously quantifies the dependence of $d \ge 2$ random measures, avoiding the need for pairwise comparisons; (iv) it can be evaluated numerically. Moreover, the index allows for informed prior specifications and fair model comparisons for Bayesian nonparametric models.

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arXiv2022

Smoothing distributions for conditional Fleming-Viot and Dawson-Watanabe diffusions

We study the distribution of the unobserved states of two measure-valued diffusions of Fleming-Viot and Dawson-Watanabe type, conditional on observations from the underlying populations collected at past, present and future times. If seen as nonparametric hidden Markov models, this amounts to finding the smoothing distributions of these processes, which we show can be explicitly described in recursive form as finite mixtures of laws of Dirichlet and gamma random measures respectively. We characterize the time-dependent weights of these mixtures, accounting for potentially different time intervals between data collection times, and fully describe the implications of assuming a discrete or a nonatomic distribution for the underlying process that drives mutations. In particular, we show that with a nonatomic mutation offspring distribution, the inference automatically upweights mixture components that carry, as atoms, observed types shared at different collection times. The predictive distributions for further samples from the population conditional on the data are also identified and shown to be mixtures of generalized Polya urns, conditionally on a latent variable in the Dawson-Watanabe case.

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arXiv2022

Model Selection for Maternal Hypertensive Disorders with Symmetric Hierarchical Dirichlet Processes

Hypertensive disorders of pregnancy occur in about 10% of pregnant women around the world. Though there is evidence that hypertension impacts maternal cardiac functions, the relation between hypertension and cardiac dysfunctions is only partially understood. The study of this relationship can be framed as a joint inferential problem on multiple populations, each corresponding to a different hypertensive disorder diagnosis, that combines multivariate information provided by a collection of cardiac function indexes. A Bayesian nonparametric approach seems particularly suited for this setup and we demonstrate it on a dataset consisting of transthoracic echocardiography results of a cohort of Indian pregnant women. We are able to perform model selection, provide density estimates of cardiac function indexes and a latent clustering of patients: these readily interpretable inferential outputs allow to single out modified cardiac functions in hypertensive patients compared to healthy subjects and progressively increased alterations with the severity of the disorder. The analysis is based on a Bayesian nonparametric model that relies on a novel hierarchical structure, called symmetric hierarchical Dirichlet process. This is suitably designed so that the mean parameters are identified and used for model selection across populations, a penalization for multiplicity is enforced, and the presence of unobserved relevant factors is investigated through a latent clustering of subjects. Posterior inference relies on a suitable Markov Chain Monte Carlo algorithm and the model behaviour is also showcased on simulated data.

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arXiv2022

Clustering consistency with Dirichlet process mixtures

Dirichlet process mixtures are flexible non-parametric models, particularly suited to density estimation and probabilistic clustering. In this work we study the posterior distribution induced by Dirichlet process mixtures as the sample size increases, and more specifically focus on consistency for the unknown number of clusters when the observed data are generated from a finite mixture. Crucially, we consider the situation where a prior is placed on the concentration parameter of the underlying Dirichlet process. Previous findings in the literature suggest that Dirichlet process mixtures are typically not consistent for the number of clusters if the concentration parameter is held fixed and data come from a finite mixture. Here we show that consistency for the number of clusters can be achieved if the concentration parameter is adapted in a fully Bayesian way, as commonly done in practice. Our results are derived for data coming from a class of finite mixtures, with mild assumptions on the prior for the concentration parameter and for a variety of choices of likelihood kernels for the mixture.

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arXiv2022

Flexible clustering via hidden hierarchical Dirichlet priors

The Bayesian approach to inference stands out for naturally allowing borrowing information across heterogeneous populations, with different samples possibly sharing the same distribution. A popular Bayesian nonparametric model for clustering probability distributions is the nested Dirichlet process, which however has the drawback of grouping distributions in a single cluster when ties are observed across samples. With the goal of achieving a flexible and effective clustering method for both samples and observations, we investigate a nonparametric prior that arises as the composition of two different discrete random structures and derive a closed-form expression for the induced distribution of the random partition, the fundamental tool regulating the clustering behavior of the model. On the one hand, this allows to gain a deeper insight into the theoretical properties of the model and, on the other hand, it yields an MCMC algorithm for evaluating Bayesian inferences of interest. Moreover, we single out limitations of this algorithm when working with more than two populations and, consequently, devise an alternative more efficient sampling scheme, which as a by-product, allows testing homogeneity between different populations. Finally, we perform a comparison with the nested Dirichlet process and provide illustrative examples of both synthetic and real data.

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arXiv2021

BNPdensity: Bayesian nonparametric mixture modeling in R

Robust statistical data modelling under potential model mis-specification often requires leaving the parametric world for the nonparametric. In the latter, parameters are infinite dimensional objects such as functions, probability distributions or infinite vectors. In the Bayesian nonparametric approach, prior distributions are designed for these parameters, which provide a handle to manage the complexity of nonparametric models in practice. However, most modern Bayesian nonparametric models seem often out of reach to practitioners, as inference algorithms need careful design to deal with the infinite number of parameters. The aim of this work is to facilitate the journey by providing computational tools for Bayesian nonparametric inference. The article describes a set of functions available in the \R package BNPdensity in order to carry out density estimation with an infinite mixture model, including all types of censored data. The package provides access to a large class of such models based on normalized random measures, which represent a generalization of the popular Dirichlet process mixture. One striking advantage of this generalization is that it offers much more robust priors on the number of clusters than the Dirichlet. Another crucial advantage is the complete flexibility in specifying the prior for the scale and location parameters of the clusters, because conjugacy is not required. Inference is performed using a theoretically grounded approximate sampling methodology known as the Ferguson & Klass algorithm. The package also offers several goodness of fit diagnostics such as QQ-plots, including a cross-validation criterion, the conditional predictive ordinate. The proposed methodology is illustrated on a classical ecological risk assessment method called the Species Sensitivity Distribution (SSD) problem, showcasing the benefits of the Bayesian nonparametric framework.

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arXiv2021

Inner spike and slab Bayesian nonparametric models

Discrete Bayesian nonparametric models whose expectation is a convex linear combination of a point mass at some point of the support and a diffuse probability distribution allow to incorporate strong prior information, while still being extremely flexible. Recent contributions in the statistical literature have successfully implemented such a modelling strategy in a variety of applications, including density estimation, nonparametric regression and model-based clustering. We provide a thorough study of a large class of nonparametric models we call inner spike and slab hNRMI models, which are obtained by considering homogeneous normalized random measures with independent increments (hNRMI) with base measure given by a convex linear combination of a point mass and a diffuse probability distribution. In this paper we investigate the distributional properties of these models and our results include: i) the exchangeable partition probability function they induce, ii) the distribution of the number of distinct values in an exchangeable sample, iii) the posterior predictive distribution, and iv) the distribution of the number of elements that coincide with the only point of the support with positive probability. Our findings are the main building block for an actual implementation of Bayesian inner spike and slab hNRMI models by means of a generalized Pólya urn scheme.

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arXiv2020

Predictive inference with Fleming--Viot-driven dependent Dirichlet processes

We consider predictive inference using a class of temporally dependent Dirichlet processes driven by Fleming--Viot diffusions, which have a natural bearing in Bayesian nonparametrics and lend the resulting family of random probability measures to analytical posterior analysis. Formulating the implied statistical model as a hidden Markov model, we fully describe the predictive distribution induced by these Fleming--Viot-driven dependent Dirichlet processes, for a sequence of observations collected at a certain time given another set of draws collected at several previous times. This is identified as a mixture of Pólya urns, whereby the observations can be values from the baseline distribution or copies of previous draws collected at the same time as in the usual Pòlya urn, or can be sampled from a random subset of the data collected at previous times. We characterise the time-dependent weights of the mixture which select such subsets and discuss the asymptotic regimes. We describe the induced partition by means of a Chinese restaurant process metaphor with a conveyor belt, whereby new customers who do not sit at an occupied table open a new table by picking a dish either from the baseline distribution or from a time-varying offer available on the conveyor belt. We lay out explicit algorithms for exact and approximate posterior sampling of both observations and partitions, and illustrate our results on predictive problems with synthetic and real data.

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arXiv2018

Limiting behaviour of the stationary search cost distribution driven by a generalized gamma process

Consider a list of labeled objects that are organized in a heap. At each time, object $j$ is selected with probability $p_j$ and moved to the top of the heap. This procedure defines a Markov chain on the set of permutations which is referred to in the literature as Move-to-Front rule. The present contribution focuses on the stationary search cost, namely the position of the requested item in the heap when the Markov chain is in equilibrium. We consider the scenario where the number of objects is infinite and the probabilities $p_j$'s are defined as the normalization of the increments of a subordinator. In this setting, we provide an exact formula for the moments of any order of the stationary search cost distribution. We illustrate the new findings in the case of a generalized gamma subordinator and deal with an extension to the two--parameter Poisson--Dirichlet process, also known as Pitman--Yor process.

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arXiv2018

Latent nested nonparametric priors

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, normalising to yield dependent random probability measures. We provide results on the partition distributions induced by latent nested processes, and develop an 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.

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