arXiv subjects

Francesco Pozza

Publications and source records attributed to Francesco Pozza.

5 recordsLinked to original sources

arXiv2026

Laplace and skew-Laplace approximations for Dirichlet process mixture posterior density

Posterior inference for Dirichlet process mixture models is analytically intractable and typically relies on Markov chain Monte Carlo methods, which can become computationally prohibitive at moderate to large sample sizes. In this work, we investigate the performance of Laplace and skew-Laplace posterior approximations for density estimation in this setting. Through an extensive numerical study covering four simulation scenarios with sample sizes ranging from n = 20 to n = 2,000 and four standard real datasets, we compare the standard Laplace approximation, its skew-corrected extension, and a slice sampling benchmark, assessing accuracy through total variation distance and computational efficiency through runtime. Our results show that the Gaussian Laplace approximation is more effective in this setting than might be anticipated, and that the skew-Laplace approximation consistently improves posterior recovery while remaining substantially faster than state-of-the-art Markov chain Monte Carlo samplers across all settings considered. In particular, the use of skew-Laplace in place of the standard Laplace approximation is especially beneficial in more complex density structures, where we observe error reductions typically on the order of 30%.

stat.CO↗

arXiv2026

Zeroth-order parallel sampling

Finding effective ways to exploit parallel computing to accelerate Markov chain Monte Carlo methods is an important problem in Bayesian computation and related disciplines. In this paper, we consider the zeroth-order setting where the unnormalized target distribution can be evaluated but its gradient is unavailable for theoretical, practical, or computational reasons. We also assume access to $m$ parallel processors to accelerate convergence. The proposed approach is inspired by modern zeroth-order optimization methods, which mimic gradient-based schemes by replacing the gradient with a zeroth-order stochastic gradient estimator. Our contribution is twofold. First, we show that a naive application of popular zeroth-order stochastic gradient estimators within Markov chain Monte Carlo methods leads to algorithms with poor dependence on $m$, both for unadjusted and Metropolis-adjusted schemes. We then propose a simple remedy to this problem, based on a random-slice perspective, as opposed to a stochastic gradient one, obtaining a new class of zeroth-order samplers that provably achieve a polynomial speed-up in $m$. Theoretical findings are supported by numerical studies.

stat.CO↗

arXiv2024

On the fundamental limitations of multiproposal Markov chain Monte Carlo algorithms

We study multiproposal Markov chain Monte Carlo algorithms, such as Multiple-try or generalised Metropolis-Hastings schemes, which have recently received renewed attention due to their amenability to parallel computing. First, we prove that no multiproposal scheme can speed-up convergence relative to the corresponding single proposal scheme by more than a factor of $K$, where $K$ denotes the number of proposals at each iteration. This result applies to arbitrary target distributions and it implies that serial multiproposal implementations are always less efficient than single proposal ones. Secondly, we consider log-concave distributions over Euclidean spaces, proving that, in this case, the speed-up is at most logarithmic in $K$, which implies that even parallel multiproposal implementations are fundamentally limited in the computational gain they can offer. Crucially, our results apply to arbitrary multiproposal schemes and purely rely on the two-step structure of the associated kernels (i.e. first generate $K$ candidate points, then select one among those). Our theoretical findings are validated through numerical simulations.

stat.CO↗

arXiv2024

Skew-symmetric approximations of posterior distributions

Popular deterministic approximations of posterior distributions from, e.g. the Laplace method, variational Bayes and expectation-propagation, generally rely on symmetric approximating families, often taken to be Gaussian. This choice facilitates optimization and inference, but typically affects the quality of the overall approximation. In fact, even in basic parametric models, the posterior distribution often displays asymmetries that yield bias and a reduced accuracy when considering symmetric approximations. Recent research has moved towards more flexible approximating families which incorporate skewness. However, current solutions are often model specific, lack a general supporting theory, increase the computational complexity of the optimization problem, and do not provide a broadly applicable solution to incorporate skewness in any symmetric approximation. This article addresses such a gap by introducing a general and provably optimal strategy to perturb any off-the-shelf symmetric approximation of a generic posterior distribution. This novel perturbation scheme is derived without additional optimization steps, and yields a similarly tractable approximation within the class of skew-symmetric densities that provably enhances the finite sample accuracy of the original symmetric counterpart. Furthermore, under suitable assumptions, it improves the convergence rate to the exact posterior by at least a $\sqrt{n}$ factor, in asymptotic regimes. These advancements are illustrated in numerical studies focusing on skewed perturbations of state-of-the-art Gaussian approximations.

stat.ME↗

arXiv2023

Skewed Bernstein-von Mises theorem and skew-modal approximations

Gaussian approximations are routinely employed in Bayesian statistics to ease inference when the target posterior is intractable. Although these approximations are asymptotically justified by Bernstein-von Mises type results, in practice the expected Gaussian behavior may poorly represent the shape of the posterior, thus affecting approximation accuracy. Motivated by these considerations, we derive an improved class of closed-form approximations of posterior distributions which arise from a new treatment of a third-order version of the Laplace method yielding approximations in a tractable family of skew-symmetric distributions. Under general assumptions which account for misspecified models and non-i.i.d. settings, this family of approximations is shown to have a total variation distance from the target posterior whose rate of convergence improves by at least one order of magnitude the one established by the classical Bernstein-von Mises theorem. Specializing this result to the case of regular parametric models shows that the same improvement in approximation accuracy can be also derived for polynomially bounded posterior functionals. Unlike other higher-order approximations, our results prove that it is possible to derive closed-form and valid densities which are expected to provide, in practice, a more accurate, yet similarly-tractable, alternative to Gaussian approximations of the target posterior, while inheriting its limiting frequentist properties. We strengthen such arguments by developing a practical skew-modal approximation for both joint and marginal posteriors that achieves the same theoretical guarantees of its theoretical counterpart by replacing the unknown model parameters with the corresponding MAP estimate. Empirical studies confirm that our theoretical results closely match the remarkable performance observed in practice, even in finite, possibly small, sample regimes.

math.ST↗