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Source author record

Julyan Arbel

Julyan Arbel appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

ResearcherUnclaimed source record
MethodologyMachine Learningmath.STStatistics TheoryComputationArtificial Intelligencemath.HOstat.OT

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Published work

13 published item(s)

preprint2026arXiv

Bayesian inference with sources of uncertainty: from confidence modelling to sparse estimation

We introduce a general framework that extends Bayesian inference by allowing the researcher to explicitly encode confidence in each source of uncertainty within the model. This mechanism provides a new handle for model design and regularisation control. Building on this framework, we develop a general approach for inducing sparsity in statistical models and illustrate its use in linear and logistic regression, as well as in Bayesian neural networks.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Error analysis of a compositional score-based algorithm for simulation-based inference

Simulation-based inference (SBI) has become a widely used framework in applied sciences for estimating the parameters of stochastic models that best explain experimental observations. A central question in this setting is how to effectively combine multiple observations in order to improve parameter inference and obtain sharper posterior distributions. Recent advances in score-based diffusion methods address this problem by constructing a compositional score, obtained by aggregating individual posterior scores within the diffusion process. While it is natural to suspect that the accumulation of individual errors may significantly degrade sampling quality as the number of observations grows, this important theoretical issue has so far remained unexplored. In this paper, we study the compositional score produced by the GAUSS algorithm of Linhart et al. (2024) and establish an upper bound on its mean squared error in terms of both the individual score errors and the number of observations. We illustrate our theoretical findings on a Gaussian example, where all analytical expressions can be derived in a closed form.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Cold Posteriors through PAC-Bayes

We investigate the cold posterior effect through the lens of PAC-Bayes generalization bounds. We argue that in the non-asymptotic setting, when the number of training samples is (relatively) small, discussions of the cold posterior effect should take into account that approximate Bayesian inference does not readily provide guarantees of performance on out-of-sample data. Instead, out-of-sample error is better described through a generalization bound. In this context, we explore the connections between the ELBO objective from variational inference and the PAC-Bayes objectives. We note that, while the ELBO and PAC-Bayes objectives are similar, the latter objectives naturally contain a temperature parameter $λ$ which is not restricted to be $λ=1$. For both regression and classification tasks, in the case of isotropic Laplace approximations to the posterior, we show how this PAC-Bayesian interpretation of the temperature parameter captures the cold posterior effect.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Approximate Bayesian computation via the energy statistic

Approximate Bayesian computation (ABC) has become an essential part of the Bayesian toolbox for addressing problems in which the likelihood is prohibitively expensive or entirely unknown, making it intractable. ABC defines a pseudo-posterior by comparing observed data with simulated data, traditionally based on some summary statistics, the elicitation of which is regarded as a key difficulty. Recently, using data discrepancy measures has been proposed in order to bypass the construction of summary statistics. Here we propose to use the importance-sampling ABC (IS-ABC) algorithm relying on the so-called two-sample energy statistic. We establish a new asymptotic result for the case where both the observed sample size and the simulated data sample size increase to infinity, which highlights to what extent the data discrepancy measure impacts the asymptotic pseudo-posterior. The result holds in the broad setting of IS-ABC methodologies, thus generalizing previous results that have been established only for rejection ABC algorithms. Furthermore, we propose a consistent V-statistic estimator of the energy statistic, under which we show that the large sample result holds, and prove that the rejection ABC algorithm, based on the energy statistic, generates pseudo-posterior distributions that achieves convergence to the correct limits, when implemented with rejection thresholds that converge to zero, in the finite sample setting. Our proposed energy statistic based ABC algorithm is demonstrated on a variety of models, including a Gaussian mixture, a moving-average model of order two, a bivariate beta and a multivariate $g$-and-$k$ distribution. We find that our proposed method compares well with alternative discrepancy measures.

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preprint2020arXiv

Approximating predictive probabilities of Gibbs-type priors

Gibbs-type random probability measures, or Gibbs-type priors, are arguably the most "natural" generalization of the celebrated Dirichlet prior. Among them the two parameter Poisson-Dirichlet prior certainly stands out for the mathematical tractability and interpretability of its predictive probabilities, which made it the natural candidate in several applications. Given a sample of size $n$, in this paper we show that the predictive probabilities of any Gibbs-type prior admit a large $n$ approximation, with an error term vanishing as $o(1/n)$, which maintains the same desirable features as the predictive probabilities of the two parameter Poisson-Dirichlet prior.

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preprint2020arXiv

Dirichlet process mixtures under affine transformations of the data

Location-scale Dirichlet process mixtures of Gaussians (DPM-G) have proved extremely useful in dealing with density estimation and clustering problems in a wide range of domains. Motivated by an astronomical application, in this work we address the robustness of DPM-G models to affine transformations of the data, a natural requirement for any sensible statistical method for density estimation and clustering. First, we devise a coherent prior specification of the model which makes posterior inference invariant with respect to affine transformations of the data. Second, we formalise the notion of asymptotic robustness under data transformation and show that mild assumptions on the true data generating process are sufficient to ensure that DPM-G models feature such a property. Our investigation is supported by an extensive simulation study and illustrated by the analysis of an astronomical dataset consisting of physical measurements of stars in the field of the globular cluster NGC 2419.

0 citations0 reviewsNo code linkedOpen access
preprint2018arXiv

A simple proof of Pitman-Yor's Chinese restaurant process from its stick-breaking representation

For a long time, the Dirichlet process has been the gold standard discrete random measure in Bayesian nonparametrics. The Pitman--Yor process provides a simple and mathematically tractable generalization, allowing for a very flexible control of the clustering behaviour. Two commonly used representations of the Pitman--Yor process are the stick-breaking process and the Chinese restaurant process. The former is a constructive representation of the process which turns out very handy for practical implementation, while the latter describes the partition distribution induced. However, the usual proof of the connection between them is indirect and involves measure theory. We provide here an elementary proof of Pitman--Yor's Chinese Restaurant process from its stick-breaking representation.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Bayesian Survival Model based on Moment Characterization

Bayesian nonparametric marginal methods are very popular since they lead to fairly easy implementation due to the formal marginalization of the infinite-dimensional parameter of the model. However, the straightforwardness of these methods also entails some limitations: they typically yield point estimates in the form of posterior expectations, but cannot be used to estimate non-linear functionals of the posterior distribution, such as median, mode or credible intervals. This is particularly relevant in survival analysis where non-linear functionals such as e.g. the median survival time, play a central role for clinicians and practitioners. The main goal of this paper is to summarize the methodology introduced in [Arbel et al., Comput. Stat. Data. An., 2015] for hazard mixture models in order to draw approximate Bayesian inference on survival functions that is not limited to the posterior mean. In addition, we propose a practical implementation of an R package called momentify designed for moment-based density approximation, and, by means of an extensive simulation study, we thoroughly compare the introduced methodology with standard marginal methods and empirical estimation.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Full Bayesian inference with hazard mixture models

Bayesian nonparametric inferential procedures based on Markov chain Monte Carlo marginal methods typically yield point estimates in the form of posterior expectations. Though very useful and easy to implement in a variety of statistical problems, these methods may suffer from some limitations if used to estimate non-linear functionals of the posterior distribution. The main goal of the present paper is to develop a novel methodology that extends a well-established marginal procedure designed for hazard mixture models, in order to draw approximate inference on survival functions that is not limited to the posterior mean but includes, as remarkable examples, credible intervals and median survival time. Our approach relies on a characterization of the posterior moments that, in turn, is used to approximate the posterior distribution by means of a technique based on Jacobi polynomials. The inferential performance of our methodology is analyzed by means of an extensive study of simulated data and real data consisting of leukemia remission times. Although tailored to the survival analysis context, the procedure we introduce can be adapted to a range of other models for which moments of the posterior distribution can be estimated.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Bayesian optimal adaptive estimation using a sieve prior

We derive rates of contraction of posterior distributions on nonparametric models resulting from sieve priors. The aim of the paper is to provide general conditions to get posterior rates when the parameter space has a general structure, and rate adaptation when the parameter space is, e.g., a Sobolev class. The conditions employed, although standard in the literature, are combined in a different way. The results are applied to density, regression, nonlinear autoregression and Gaussian white noise models. In the latter we have also considered a loss function which is different from the usual l2 norm, namely the pointwise loss. In this case it is possible to prove that the adaptive Bayesian approach for the l2 loss is strongly suboptimal and we provide a lower bound on the rate.

0 citations0 reviewsNo code linkedOpen access