We study Bayesian histograms for distribution estimation on $[0,1]^d$ under the Wasserstein $W_v, 1 \leq v < \infty$ distance in the i.i.d sampling regime. We newly show that when $d < 2v$, histograms possess a special \textit{memory efficiency} property, whereby in reference to the sample size $n$, order $n^{d/2v}$ bins are needed to obtain minimax rate optimality. This result holds for the posterior mean histogram and with respect to posterior contraction: under the class of Borel probability measures and some classes of smooth densities. The attained memory footprint overcomes existing minimax optimal procedures by a polynomial factor in $n$; for example an $n^{1 - d/2v}$ factor reduction in the footprint when compared to the empirical measure, a minimax estimator in the Borel probability measure class. Additionally constructing both the posterior mean histogram and the posterior itself can be done super--linearly in $n$. Due to the popularity of the $W_1,W_2$ metrics and the coverage provided by the $d < 2v$ case, our results are of most practical interest in the $(d=1,v =1,2), (d=2,v=2), (d=3,v=2)$ settings and we provide simulations demonstrating the theory in several of these instances.

翻译：我们研究独立同分布采样机制下，在$[0,1]^d$空间中以Wasserstein $W_v$距离（$1 \leq v < \infty$）为度量的分布估计贝叶斯直方图方法。我们首次证明：当$d < 2v$时，直方图具备特殊的\textit{内存效率}特性——仅需$n^{d/2v}$量级的箱数即可达到极小极大最优收敛速率（$n$为样本量）。该结论对后验均值直方图及后验收缩均成立，适用于Borel概率测度类与若干光滑密度类。相较现有极小极大最优方法，本节内存需求在$n$的多项式因子上实现突破：例如与Borel概率测度类中的极小极大估计量——经验测度相比，内存消耗降低$n^{1-d/2v}$倍。此外，后验均值直方图与后验本身的构建均可实现超线性时间复杂度。鉴于$d < 2v$情形覆盖了广泛使用的$W_1, W_2$度量，且$(d=1, v=1,2)$、$(d=2, v=2)$、$(d=3, v=2)$等设置具有最高实用性，我们针对多个典型场景提供了验证理论的模拟实验。