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Ramsés H. Mena

Publications and source records attributed to Ramsés H. Mena.

17 recordsLinked to original sources
arXiv2026

Exact two-stage finite-mixture representations for species sampling processes

Discrete random probability measures are central to Bayesian inference, particularly as priors for mixture modeling and clustering. A broad and unifying class is that of proper species sampling processes (SSPs), encompassing many Bayesian nonparametric priors. We show that any proper SSP admits an exact two-stage finite-mixture representation built from a latent truncation index and a simple reweighting of the atoms. For each realized truncation index, the representation has finitely many atoms, and averaging over the induced law of that index recovers the original SSP setwise. This yields at least two consequences: (i) an exact two-stage finite construction for arbitrary SSPs, without user-chosen truncation levels; and (ii) posterior inference in SSP mixture models via standard finite-mixture machinery, leading to tractable MCMC algorithms without ad hoc truncations. We explore these consequences by deriving explicit total-variation bounds for the approximation error when the truncation level is fixed, and by studying practical performance in mixture modeling, with emphasis on Dirichlet and geometric SSPs.

stat.ME↗
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.

math.ST↗
arXiv2025

Weighted Support Points from Random Measures: An Interpretable Alternative for Generative Modeling

Support points summarize a large dataset through a smaller set of representative points that can be used for data operations, such as Monte Carlo integration, without requiring access to the full dataset. In this sense, support points offer a compact yet informative representation of the original data. We build on this idea to introduce a generative modeling framework based on random weighted support points, where the randomness arises from a weighting scheme inspired by the Dirichlet process and the Bayesian bootstrap. The proposed method generates diverse and interpretable sample sets from a fixed dataset, without relying on probabilistic modeling assumptions or neural network architectures. We present the theoretical formulation of the method and develop an efficient optimization algorithm based on the Convex--Concave Procedure (CCP). Empirical results on the MNIST and CelebA-HQ datasets show that our approach produces high-quality and diverse outputs at a fraction of the computational cost of black-box alternatives such as Generative Adversarial Networks (GANs) or Denoising Diffusion Probabilistic Models (DDPMs). These results suggest that random weighted support points offer a principled, scalable, and interpretable alternative for generative modeling. A key feature is their ability to produce genuinely interpolative samples that preserve underlying data structure.

stat.ML↗
arXiv2025

Reconstruction of the Probability Measure and the Coupling Parameters in a Curie-Weiss Model

The Curie-Weiss model is used to study phase transitions in statistical mechanics and has been the object of rigorous analysis in mathematical physics. We analyse the problem of reconstructing the probability measure of a multi-group Curie-Weiss model from a sample of data by employing the maximum likelihood estimator for the coupling parameters of the model, under the assumption that there is interaction within each group but not across group boundaries. The estimator has a number of positive properties, such as consistency, asymptotic normality, and exponentially decaying probabilities of large deviations of the estimator with respect to the true parameter value. A shortcoming in practice is the necessity to calculate the partition function of the Curie-Weiss model, which scales exponentially with respect to the population size. There are a number of applications of the estimator in political science, sociology, and automated voting, centred on the idea of identifying the degree of social cohesion in a population. In these applications, the coupling parameter is a natural way to quantify social cohesion. We treat the estimation of the optimal weights in a two-tier voting system, which requires the estimation of the coupling parameter.

math.PR↗
arXiv2025

On a divergence-based prior analysis of stick-breaking processes

The nonparametric view of Bayesian inference has transformed statistics and many of its applications. The canonical Dirichlet process and other more general families of nonparametric priors have served as a gateway to solve frontier uncertainty quantification problems of large, or infinite, nature. This success has been greatly due to available constructions and representations of such distributions, the two most useful constructions are the one based on normalization of homogeneous completely random measures and that based on stick-breaking processes. Hence, understanding their distributional features and how different random probability measures compare among themselves is a key ingredient for their proper application. In this paper, we analyse the discrepancy among some nonparametric priors employed in the literature. Initially, we compute the mean and variance of the random Kullback-Leibler divergence between the Dirichlet process and the geometric process. Subsequently, we extend our analysis to encompass a broader class of exchangeable stick-breaking processes, which includes the Dirichlet and geometric processes as extreme cases. Our results establish quantitative conditions where all the aforementioned priors are close in total variation distance. In such instances, adhering to Occam's razor principle advocates for the preference of the simpler process.

math.ST↗
arXiv2023

Detection of an Arbitrary Number of Communities in a Block Spin Ising Model

We study the problem of community detection in a general version of the block spin Ising model featuring M groups, a model inspired by the Curie-Weiss model of ferromagnetism in statistical mechanics. We solve the general problem of identifying any number of groups with any possible coupling constants. Up to now, the problem was only solved for the specific situation with two groups of identical size and identical interactions. Our results can be applied to the most realistic situations, in which there are many groups of different sizes and different interactions. In addition, we give an explicit algorithm that permits the reconstruction of the structure of the model from a sample of observations based on the comparison of empirical correlations of the spin variables, thus unveiling easy applications of the model to real-world voting data and communities in biology.

math.PR↗
arXiv2022

Clustering constrained on linear networks

An unsupervised classification method for point events occurring on a network of lines is proposed. The idea relies on the distributional flexibility and practicality of random partition models to discover the clustering structure featuring observations from a particular phenomenon taking place on a given set of edges. By incorporating the spatial effect in the random partition distribution, induced by a Dirichlet process, one is able to control the distance between edges and events, thus leading to an appealing clustering method. A Gibbs sampler algorithm is proposed and evaluated with a sensitivity analysis. The proposal is motivated and illustrated by the analysis of crime and violence patterns in Mexico City.

stat.ME↗
arXiv2021

A Copula-based Fully Bayesian Nonparametric Evaluation of Cardiovascular Risk Markers in the Mexico City Diabetes Study

Cardiovascular disease lead the cause of death world wide and several studies have been carried out to understand and explore cardiovascular risk markers in normoglycemic and diabetic populations. In this work, we explore the association structure between hyperglycemic markers and cardiovascular risk markers controlled by triglycerides, body mass index, age and gender, for the normoglycemic population in The Mexico City Diabetes Study. Understanding the association structure could contribute to the assessment of additional cardiovascular risk markers in this low income urban population with a high prevalence of classic cardiovascular risk biomarkers. The association structure is measured by conditional Kendall's tau, defined through conditional copula functions. The latter are in turn modeled under a fully Bayesian nonparametric approach, which allows the complete shape of the copula function to vary for different values of the controlled covariates.

stat.AP↗
arXiv2021

Stick-breaking processes with exchangeable length variables

Our object of study is the general class of stick-breaking processes with exchangeable length variables. These generalize well-known Bayesian non-parametric priors in an unexplored direction. We give conditions to assure the respective species sampling process is proper and the corresponding prior has full support. For a rich sub-class we explain how, by tuning a single $[0,1]$-valued parameter, the stochastic ordering of the weights can be modulated, and Dirichlet and Geometric priors can be recovered. A general formula for the distribution of the latent allocation variables is derived and an MCMC algorithm is proposed for density estimation purposes.

math.ST↗
arXiv2021

COVID-19 Clinical footprint to infer about mortality

Information of 1.6 million patients identified as SARS-CoV-2 positive in Mexico is used to understand the relationship between comorbidities, symptoms, hospitalizations and deaths due to the COVID-19 disease. Using the presence or absence of these latter variables a clinical footprint for each patient is created. The risk, expected mortality and the prediction of death outcomes, among other relevant quantities, are obtained and analyzed by means of a multivariate Bernoulli distribution. The proposal considers all possible footprint combinations resulting in a robust model suitable for Bayesian inference.

stat.AP↗
arXiv2021

Asymptotic behavior of the number of distinct values in a sample from the geometric stick-breaking process

Discrete random probability measures are a key ingredient of Bayesian nonparametric inferential procedures. A sample generates ties with positive probability and a fundamental object of both theoretical and applied interest is the corresponding random number of distinct values. The growth rate can be determined from the rate of decay of the small frequencies implying that, when the decreasingly ordered frequencies admit a tractable form, the asymptotics of the number of distinct values can be conveniently assessed. We focus on the geometric stick-breaking process and we investigate the effect of the choice of the distribution for the success probability on the asymptotic behavior of the number of distinct values. We show that a whole range of logarithmic behaviors are obtained by appropriately tuning the prior. We also derive a two-term expansion and illustrate its use in a comparison with a larger family of discrete random probability measures having an additional parameter given by the scale of the negative binomial distribution.

math.ST↗
arXiv2020

Beta-Binomial stick-breaking non-parametric prior

A new class of nonparametric prior distributions, termed Beta-Binomial stick-breaking process, is proposed. By allowing the underlying length random variables to be dependent through a Beta marginals Markov chain, an appealing discrete random probability measure arises. The chain's dependence parameter controls the ordering of the stick-breaking weights, and thus tunes the model's label-switching ability. Also, by tuning this parameter, the resulting class contains the Dirichlet process and the Geometric process priors as particular cases, which is of interest for fast convergence of MCMC implementations. Some properties of the model are discussed and a density estimation algorithm is proposed and tested with simulated datasets.

math.ST↗
arXiv2020

Using posterior predictive distributions to analyse epidemic models: COVID-19 in Mexico City

Epidemiological models contain a set of parameters that must be adjusted based on available observations. Once a model has been calibrated, it can be used as a forecasting tool to make predictions and to evaluate contingency plans. It is customary to employ only point estimators for such predictions. However, some models may fit the same data reasonably well for a broad range of parameter values, and this flexibility means that predictions stemming from such models will vary widely, depending on the particular parameter values employed within the range that give a good fit. When data are poor or incomplete, model uncertainty widens further. A way to circumvent this problem is to use Bayesian statistics to incorporate observations and use the full range of parameter estimates contained in the posterior distribution to adjust for uncertainties in model predictions. Specifically, given the epidemiological model and a probability distribution for observations, we use the posterior distribution of model parameters to generate all possible epidemiological curves via the posterior predictive distribution. From the envelope of all curves one can extract the worst-case scenario and study the impact of implementing contingency plans according to this assessment. We apply this approach to the potential evolution of COVID-19 in Mexico City and assess whether contingency plans are being successful and whether the epidemiological curve has flattened.

q-bio.PE↗
arXiv2019

Modeling failures times with dependent renewal type models via exchangeability

Failure times of a machinery cannot always be assumed independent and identically distributed, e.g. if after reparations the machinery is not restored to a same-as-new condition. Framed within the renewal processes approach, a generalization that considers exchangeable inter-arrival times is presented. The resulting model provides a more realistic approach to capture the dependence among events occurring at random times, while retaining much of the tractability of the classical renewal process. Extensions of some classical results and special cases of renewal functions are analyzed, in particular the one corresponding to an exchangeable sequence driven by a Dirichlet process. The proposal is tested through an estimation procedure using simulated data sets and with an application to the reliability of hydraulic subsystems in load-haul-dump machines.

stat.AP↗
arXiv2019

On a flexible construction of a negative binomial model

This work presents a construction of stationary Markov models with negative-binomial marginal distributions. A simple closed form expression for the corresponding transition probabilities is given, linking the proposal to well-known classes of birth and death processes and thus revealing interesting characterizations. The advantage of having such closed form expressions is tested on simulated and real data.

stat.ME↗
arXiv2016

Dynamic density estimation with diffusive Dirichlet mixtures

We introduce a new class of nonparametric prior distributions on the space of continuously varying densities, induced by Dirichlet process mixtures which diffuse in time. These select time-indexed random functions without jumps, whose sections are continuous or discrete distributions depending on the choice of kernel. The construction exploits the widely used stick-breaking representation of the Dirichlet process and induces the time dependence by replacing the stick-breaking components with one-dimensional Wright-Fisher diffusions. These features combine appealing properties of the model, inherited from the Wright-Fisher diffusions and the Dirichlet mixture structure, with great flexibility and tractability for posterior computation. The construction can be easily extended to multi-parameter GEM marginal states, which include, for example, the Pitman--Yor process. A full inferential strategy is detailed and illustrated on simulated and real data.

stat.ME↗