Bayesian multidimensional scaling (BMDS) embeds $n$ objects in a low-dimensional space to approximately preserve an observed dissimilarity matrix. Compared to classic MDS, BMDS is more robust to model misspecification and supports posterior uncertainty quantification and joint estimation within hierarchical models. However, standard BMDS inference is computationally prohibitive, requiring $O(n^2)$ operations per MCMC iteration to evaluate the likelihood. We propose Barnes--Hut BMDS (BH-BMDS), which uses a tree-based approximation to the likelihood and a Gibbs sampler that leverages this structure, remaining compatible with hierarchical extensions. BH-BMDS reduces computational complexity to $O(n \log n)$ while preserving the geometric fidelity of the embedding. We further establish consistency for the stationary measure of BH-BMDS, proving that it concentrates around the true latent configuration even as the total error of the surrogate likelihood diverges. Notably, this consistency holds in the infinite-dimensional limit. We evaluate the approximation on datasets with diverse structure, including air traffic networks, arXiv abstracts, MNIST images and neural activity recordings from mouse models of tau pathology. Across all settings, BH-BMDS closely matches BMDS while achieving substantial computational gains, with approximately 10-fold speedups at $n=1{,}000$ and 70-fold speedups at $n=10{,}000$. These gains increase with $n$, demonstrating strong empirical scalability.

翻译：贝叶斯多维尺度分析（BMDS）通过在低维空间中嵌入n个对象，以近似保留观测到的相异性矩阵。与经典MDS相比，BMDS对模型误设更具鲁棒性，支持后验不确定性量化，并可在层次模型内进行联合估计。然而，标准BMDS推理在计算上代价高昂，每次MCMC迭代需O(n²)次操作来评估似然函数。我们提出Barnes-Hut BMDS（BH-BMDS），该方法采用基于树的似然近似以及利用该结构的Gibbs采样器，且与层次化扩展兼容。BH-BMDS将计算复杂度降至O(n log n)，同时保留嵌入的几何保真度。我们进一步证明了BH-BMDS平稳测度的一致性，证明即使在替代似然的总误差发散的情况下，其仍集中围绕真实潜在配置。值得注意的是，该一致性在无穷维极限下成立。我们在具有不同结构的数据集上评估了该近似，包括空中交通网络、arXiv摘要、MNIST图像以及tau病理小鼠模型的神经活动记录。在所有设置下，BH-BMDS均与BMDS高度吻合，同时实现显著的计算增益：在n=1,000时加速约10倍，在n=10,000时加速约70倍。这些增益随n增加而增大，展现了强大的经验可扩展性。