In this paper we consider a new probability sampling methods based on Langevin diffusion dynamics to resolve the problem of existing Monte Carlo algorithms when draw samples from high dimensional target densities. We extent Metropolis-Adjusted Langevin Diffusion algorithm by modelling the stochasticity of precondition matrix as a random matrix. An advantage compared to other proposal method is that it only requires the gradient of log-posterior. The proposed method provides fully adaptation mechanisms to tune proposal densities to exploits and adapts the geometry of local structures of statistical models. We clarify the benefits of the new proposal by modelling a Quantum Probability Density Functions of a free particle in a plane (energy Eigen-functions). The proposed model represents a remarkable improvement in terms of performance accuracy and computational time over standard MCMC method.

翻译：本文提出了一种基于朗之万扩散动力学的新型概率抽样方法，以解决现有蒙特卡洛算法从高维目标密度中抽取样本时存在的问题。我们通过将预条件矩阵的随机性建模为随机矩阵，扩展了Metropolis-Adjusted Langevin Diffusion算法。相较于其他提案方法，其优势在于仅需对数后验的梯度信息。所提出的方法提供了完整的自适应机制，通过调整提案密度以利用并适应统计模型局部结构的几何特性。我们通过建模平面自由粒子（能量本征函数）的量子概率密度函数，阐明了新提案方法的优势。实验表明，相较于标准MCMC方法，该模型在计算精度与运行效率方面均表现出显著提升。