论文标题
局部近似MCMC的最佳优化策略
Rate-optimal refinement strategies for local approximation MCMC
论文作者
论文摘要
许多贝叶斯推论问题涉及目标分布，其密度函数在计算上的评估昂贵。基于少量精心选择的密度评估，用局部近似替换目标密度可以显着减少马尔可夫链蒙特卡洛（MCMC）采样的计算费用。此外，局部近似的持续细化可以保证渐近精确的采样。由于与MCMC差异的近似值，我们设计了一种平衡偏差衰减率的新策略。我们证明，所得局部近似MCMC（LA-MCMC）算法的误差大约是预期的$ 1/\ sqrt {t} $ rate，我们以数值来证明此速率。我们还引入了一个算法参数，该参数可以保证鉴于非常弱的尾部界限，可以显着增强先前的收敛结果。最后，我们将LA-MCMC应用于地下水水文学中产生的计算密集贝叶斯逆问题。
Many Bayesian inference problems involve target distributions whose density functions are computationally expensive to evaluate. Replacing the target density with a local approximation based on a small number of carefully chosen density evaluations can significantly reduce the computational expense of Markov chain Monte Carlo (MCMC) sampling. Moreover, continual refinement of the local approximation can guarantee asymptotically exact sampling. We devise a new strategy for balancing the decay rate of the bias due to the approximation with that of the MCMC variance. We prove that the error of the resulting local approximation MCMC (LA-MCMC) algorithm decays at roughly the expected $1/\sqrt{T}$ rate, and we demonstrate this rate numerically. We also introduce an algorithmic parameter that guarantees convergence given very weak tail bounds, significantly strengthening previous convergence results. Finally, we apply LA-MCMC to a computationally intensive Bayesian inverse problem arising in groundwater hydrology.
