We derive the first explicit bounds for the spectral gap of a random walk Metropolis algorithm on $R^d$ for any value of the proposal variance, which when scaled appropriately recovers the correct $d^{-1}$ dependence on dimension for suitably regular invariant distributions. We also obtain explicit bounds on the ${\rm L}^2$-mixing time for a broad class of models. In obtaining these results, we refine the use of isoperimetric profile inequalities to obtain conductance profile bounds, which also enable the derivation of explicit bounds in a much broader class of models. We also obtain similar results for the preconditioned Crank--Nicolson Markov chain, obtaining dimension-independent bounds under suitable assumptions.
翻译:我们从随机步行大都会算法的光谱差距中得出第一个明确界限,该算法以美元计算,不论提案差异的价值为何,如果按比例调整,就能够适当恢复对适当正常差异分布的维度的依赖度的正确值。我们还获得了对广泛类型模型的2美元混合时间的清晰界限。在获得这些结果时,我们改进了对等光谱不平等的利用,以获得行为剖面的界限,这也使得在范围更广的模型中可以得出明确的界限。我们还获得了对Crank-Nicolson Markov 链的前提条件的类似结果,在适当的假设下获得了维度独立的界限。