论文检索

McKean–Vlasov 模型的贝叶斯非参数推断AI 解读

We consider nonparametric statistical inference on a periodic interaction potential W from noisy discrete space-time measurements of solutions ρ=ρW of the nonlinear McKean–Vlasov equation, describing the probability density of the mean field limit of an interacting particle system. We show how Gaussian process priors assigned to W give rise to posterior mean estimators that exhibit fast convergence rates for the implied estimated densities ρ¯ towards ρW. We further show that…

Kolyan Ray 与 Botond Szabo 对“鞅后验分布”的讨论AI 解读

We congratulate the authors for their thought-provoking article, which includes proposing a copula-based update for the predictive density and establishing its frequentist consistency under relatively mild assumptions. In this contribution, we further explore the frequentist properties of (Bayesian) predictive densities and illustrate through a simple example that, similarly to the posterior distribution, the predictive distribution can also be inconsistent. One therefore…

可逆多维扩散的非参数贝叶斯推断AI 解读

We study nonparametric Bayesian models for reversible multidimensional diffusions with periodic drift. For continuous observation paths, reversibility is exploited to prove a general posterior contraction rate theorem for the drift gradient vector field under approximation-theoretic conditions on the induced prior for the invariant measure. The general theorem is applied to Gaussian priors and p-exponential priors, which are shown to converge to the truth at the optimal…

稀疏先验下高维线性回归的变分贝叶斯方法AI 解读

We study a mean-field spike and slab variational Bayes (VB) approximation to Bayesian model selection priors in sparse high-dimensional linear regression. Under compatibility conditions on the design matrix, oracle inequalities are derived for the mean-field VB approximation, implying that it converges to the sparse truth at the optimal rate and gives optimal prediction of the response vector. The empirical performance of our algorithm is studied, showing that it works…

半参数贝叶斯因果推断AI 解读

We develop a semiparametric Bayesian approach for estimating the mean response in a missing data model with binary outcomes and a nonparametrically modelled propensity score. Equivalently, we estimate the causal effect of a treatment, correcting nonparametrically for confounding. We show that standard Gaussian process priors satisfy a semiparametric Bernstein–von Mises theorem under smoothness conditions. We further propose a novel propensity score-dependent prior that…

多维扩散过程漂移向量场的非参数统计推断AI 解读

The problem of determining a periodic Lipschitz vector field $b=(b_{1},\ldots ,b_{d})$ from an observed trajectory of the solution $(X_{t}:0\le t\le T)$ of the multi-dimensional stochastic differential equation \begin{equation*}dX_{t}=b(X_{t})\,dt+dW_{t},\quad t\geq 0,\end{equation*} where $W_{t}$ is a standard $d$-dimensional Brownian motion, is considered. Convergence rates of a penalised least squares estimator, which equals the maximum a posteriori (MAP) estimate…

高斯白噪声模型中的自适应Bernstein–von Mises定理AI 解读

We investigate Bernstein–von Mises theorems for adaptive nonparametric Bayesian procedures in the canonical Gaussian white noise model. We consider both a Hilbert space and multiscale setting with applications in $L^{2}$ and $L^{\infty}$, respectively. This provides a theoretical justification for plug-in procedures, for example the use of certain credible sets for sufficiently smooth linear functionals. We use this general approach to construct optimal frequentist…