Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions
Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions
复制标题
多维扩散漂移矢量场的非参数统计推断
作者：
Richard Nickl;Kolyan Ray
从观察到的解决方案的轨迹$（x_t：0 \ le t \ le t \ le t）$中确定周期性Lipschitz vector Field $ b =（b_1，\ dots，b_d）$的问题开始{等式*} dx_t = b（x_t）dt + dw_t，\ quad t \ geq 0， \ end {equation*}，其中$ w_t $是标准的$ d $ - 二维棕色运动。得出了与高维高斯产品相对应的最大后验（MAP）估计值的最大最小二乘估计量的收敛速率。这些结果是从相关后验分布的相应收缩率中得出的。获得的费率是最佳的，以$ l^2 $ -loss在任何维度中，也是$ d \ le 4 $时的最高规范损失。此外，当$ d \ le 3 $时，事实证明，非参数伯恩斯坦 - 冯·米塞斯定理被证明是$ b $的后验分布。由此，我们将功能性中心限制定理用于不变度度量的隐含估计器$ \ mu_b $。从信息理论的角度来看，限制的高斯过程分布具有协方差结构。
The problem of determining a periodic Lipschitz vector field $b=(b_1, \dots, b_d)$ 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 corresponding to a high-dimensional Gaussian product prior, are derived. These results are deduced from corresponding contraction rates for the associated posterior distributions. The rates obtained are optimal up to log-factors in $L^2$-loss in any dimension, and also for supremum norm loss when $d \le 4$. Further, when $d \le 3$, nonparametric Bernstein-von Mises theorems are proved for the posterior distributions of $b$. From this we deduce functional central limit theorems for the implied estimators of the invariant measure $\mu_b$. The limiting Gaussian process distributions have a covariance structure that is asymptotically optimal from an information-theoretic point of view.
$X$ 射线变换的高效非参数贝叶斯推理
DOI：
10.1214/18-aos1708
发表时间：
2019
期刊：
The Annals of Statistics
影响因子：
--
作者：
Monard, François;Nickl, Richard;Paternain, Gabriel P.
通讯作者：
Paternain, Gabriel P.