Publications
Convex Order Comparisons for Sub-Gamma Random Variables
Submitted
arXiv | BibTeX | Google Scholar
Recent work shows that a sub-Gaussian variable is dominated in convex order by a sharp multiple of a Gaussian. We ask the same question for the sub-Gamma class underlying Bernstein’s inequality, with the Laplace law as majorant, \[X \preceq_{\mathrm{cx}} c\,L, \qquad L \sim \mathrm{Laplace},\] where the moment generating function is controlled only over a finite range of frequencies. We give a variational formula for the optimal multiple \(c\), show that it is strictly larger than the natural scale \(\sigma \vee \alpha\), and prove sharpness: the value is attained by an asymmetric two-point distribution. The sub-exponential class behaves differently. There the optimal multiple is exactly \(\sigma \vee \alpha\), but it is attained only when \(\alpha \leq \sigma\); for \(\alpha > \sigma\) the constant stays sharp while equality fails for every non-affine convex function. Both results rest on a sharp finite-range variant of the Kearns-Saul inequality, of independent interest.
Robust Bayesian Inference via Variational Approximations of Generalized Rho-Posteriors
Submitted
arXiv | BibTeX | Google Scholar
We introduce the \(\tilde{\rho}\)-posterior, a modified version of the \(\rho\)-posterior of Baraud & Birgé (2020), obtained by replacing the supremum over competitor parameters with a softmax aggregation. This modification allows a PAC-Bayesian analysis of the \(\tilde{\rho}\)-posterior. This yields finite-sample oracle inequalities with explicit convergence rates that inherit the key robustness properties of the original framework, in particular graceful degradation under model misspecification and data contamination. Crucially, the PAC-Bayesian oracle inequalities extend to variational approximations of the \(\tilde{\rho}\)-posterior, providing theoretical guarantees for tractable inference. Numerical experiments on exponential families, regression, and real-world datasets confirm that the resulting variational procedures achieve robustness competitive with theoretical predictions at computational cost comparable to standard variational Bayes.
On importance sampling and independent Metropolis–Hastings with an unbounded weight function
Accepted, The Annals of Statistics
arXiv | BibTeX | Google Scholar
We study importance sampling (IS) and the particle independent Metropolis–Hastings (PIMH) algorithm when the weight function is unbounded but has finite moments of order \(p\). For PIMH with \(N\) particles, we establish the convergence rate \[\left|\bar{q}P^t - \bar{\pi}\right|_{\mathrm{TV}} \leq \frac{C}{\sqrt{N}}\,\frac{1}{(1+t)^{p-1}}.\] For the single-chain IMH, we prove that the common random numbers (CRN) coupling is maximal, yielding the exact identity \[\left|P^t(x,\cdot) - P^t(y,\cdot)\right|_{\mathrm{TV}} = P_{x,y}(\tau > t).\] This allows a formal comparison of the finite-time biases of IS and IMH, showing IMH to have strictly smaller bias.
Convergence of Statistical Estimators via Mutual Information Bounds
Submitted
arXiv | BibTeX | Google Scholar
We introduce a unified mutual information bound for general statistical models, bridging PAC-Bayesian theory, Bayesian nonparametrics, and classical estimation. The bound yields sharper contraction rates for fractional posteriors and applies to a wide family of estimators including variational inference and MLE. The central inequality is \[\mathbb{E}_{\theta\sim\hat{\rho}}\!\left[D_\alpha(P_\theta\|P_{\theta_0})\right] - \frac{\alpha}{n(1-\alpha)}\,\mathbb{E}_{\theta\sim\hat{\rho}}\!\left[r_n(\theta,\theta_0)\right] \leq \frac{\mathcal{I}(\theta,\mathcal{S})}{n(1-\alpha)},\] where \(D_\alpha\) is the Rényi divergence of order \(\alpha\), \(r_n\) is the log-likelihood ratio evaluated on the sample \(\mathcal{S}\), and \(\mathcal{I}(\theta,\mathcal{S})\) is the mutual information between the estimator and the data.
