标题： 一种贝叶斯对集合卡尔曼更新的描述 显示英文标题

标题： A Bayesian Characterization of Ensemble Kalman Updates

摘要： 集合卡尔曼滤波（EnKF）的更新，称为集合卡尔曼更新（EnKU），广泛用于反问题和数据同化的贝叶斯推断。在每个过滤步骤中，它通过从联合测度$\pi$中采样的粒子集合$(X_i,Y_i)\sim\pi$和观测$y_\ast\in\mathbb{R}^m$来近似无似然函数的贝叶斯反演解。后验分布${\pi}_{X|Y=y_\ast}$通过由卡尔曼增益确定的仿射映射$L^{\mathrm{EnKU}}_{y_\ast}(x,y)$将$(X_i,Y_i)$进行传输来近似。 虽然EnKU在平均场极限下对于高斯联合分布$\pi$是精确的，但仅精确性本身并不能确定更新：无限多的仿射映射$L_{y_\ast}$将一个高斯$\pi$推至$\pi_{X|Y=y_\ast}$。 这引发了一个问题：应该使用哪个仿射映射来估计后验？ 我们给出了所有此类映射中EnKU的特征描述。 首先，我们描述了EnKU产生精确条件化的分布集合$\mathrm{E}^{\mathrm{EnKU}}$，表明它比高斯族更大。 接下来，我们证明了除了$\mathrm{E}^{\mathrm{EnKU}}$中一小类高度对称分布（包括高斯分布）外，EnKU是唯一的精确仿射条件化映射。 最后，我们要求出最大的可能集合$\mathrm{F}$，其中任何依赖于测度的仿射传输都可以是精确的；在表征$\mathrm{F}$后，我们证明 EnKU 的精确性集合几乎是最大的：$\mathrm{F}=\mathrm{E}^{\mathrm{EnKU}}\cup\mathrm{S}_{\mathrm{nl-dec}}$，其中$\mathrm{S}_{\mathrm{nl-dec}}$是一个小的对称类。因此，在仿射传输中，EnKU 在高斯分布之外的精确条件方面是近最优的，并且是唯一一种对于$\mathrm{F}$中任何测度（除了强对称定律的一个子类）实现精确性的仿射更新。 显示英文摘要

摘要： The update in the Ensemble Kalman Filter (EnKF), called the Ensemble Kalman Update (EnKU), is widely used for Bayesian inference in inverse problems and data assimilation. At each filtering step, it approximates the solution to a likelihood-free Bayesian inversion from an ensemble of particles $(X_i,Y_i)\sim\pi$ sampled from a joint measure $\pi$ and an observation $y_\ast\in\mathbb{R}^m$. The posterior ${\pi}_{X|Y=y_\ast}$ is approximated by transporting $(X_i,Y_i)$ through an affine map $L^{\mathrm{EnKU}}_{y_\ast}(x,y)$ determined by the Kalman gain. While the EnKU is exact for Gaussian joints $\pi$ in the mean-field limit, exactness alone does not fix the update: infinitely many affine maps $L_{y_\ast}$ push a Gaussian $\pi$ to $\pi_{X|Y=y_\ast}$. This raises a question: which affine map should estimate the posterior? We provide a characterization of the EnKU among all such maps. First, we describe the set $\mathrm{E}^{\mathrm{EnKU}}$ of laws where the EnKU yields exact conditioning, showing it is larger than the Gaussian family. Next, we prove that, except for a small class of highly symmetric distributions in $\mathrm{E}^{\mathrm{EnKU}}$ (including Gaussians), the EnKU is the unique exact affine conditioning map. Finally, we ask for the largest possible set $\mathrm{F}$ where any measure-dependent affine transport could be exact; after characterizing $\mathrm{F}$, we show the EnKU's exactness set is almost maximal: $\mathrm{F}=\mathrm{E}^{\mathrm{EnKU}}\cup\mathrm{S}_{\mathrm{nl-dec}}$, where $\mathrm{S}_{\mathrm{nl-dec}}$ is a small symmetry class. Thus, among affine transports, the EnKU is near-optimal for exact conditioning beyond Gaussians and is the unique affine update achieving exactness for any measure in $\mathrm{F}$ except a subclass of strongly symmetric laws.