Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers
Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers
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DOI：
10.3150/15-bej785
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发表时间：
2018-05-01
期刊：
影响因子：
1.5
通讯作者：
Vihola, Matti
中科院分区：
文献类型：
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作者：
Andrieu, Christophe;Lee, Anthony;Vihola, Matti
我们建立了迭代条件序列蒙特卡罗（i-cSMC）马尔可夫链和相关粒子吉布斯采样器的收敛速度和渐近方差的定量界[J. R。Stat. Soc. Ser B. Stat.美沙酮72（2010）269-342]。我们的主要研究结果是，与i-cSMC算法相关的潜在功能的本质有界性提供了必要和充分条件的一致遍历的i-cSMC马尔可夫链，以及其（一致几何）收敛速度的定量界。此外，我们表明，i-cSMC马尔可夫链甚至不能是几何遍历的，如果这个基本的有界性不持有在许多感兴趣的应用。我们的充分性和定量界依赖于一个新的非渐近分析的期望标准归一化常数估计相对于一个“双条件”SMC算法。此外，我们对i-cSMC的研究结果表明，收敛速度可以通过增加算法中的粒子数N来任意提高，并且在存在混合假设的情况下，收敛速度可以通过随时间范围线性增加N来保持恒定。我们将i-cSMC的有界性条件的充分性转化为粒子Gibbs Markov链几何遍历的充分条件和几何收敛速度的定量界，这意味着粒子Gibbs Markov链的性质收敛于相应的Gibbs采样器的性质.这些结果补充了最近发现的，和相关的，颗粒边际大都会黑斯廷斯（PMMH）马尔可夫链的条件。
We establish quantitative bounds for rates of convergence and asymptotic variances for iterated conditional sequential Monte Carlo (i-cSMC) Markov chains and associated particle Gibbs samplers [J. R. Stat. Soc. Ser B. Stat. Methodol. 72 (2010) 269-342]. Our main findings are that the essential boundedness of potential functions associated with the i-cSMC algorithm provide necessary and sufficient conditions for the uniform ergodicity of the i-cSMC Markov chain, as well as quantitative bounds on its (uniformly geometric) rate of convergence. Furthermore, we show that the i-cSMC Markov chain cannot even be geometrically ergodic if this essential boundedness does not hold in many applications of interest. Our sufficiency and quantitative bounds rely on a novel non-asymptotic analysis of the expectation of a standard normalizing constant estimate with respect to a "doubly conditional" SMC algorithm. In addition, our results for i-cSMC imply that the rate of convergence can be improved arbitrarily by increasing N, the number of particles in the algorithm, and that in the presence of mixing assumptions, the rate of convergence can be kept constant by increasing N linearly with the time horizon. We translate the sufficiency of the boundedness condition for i-cSMC into sufficient conditions for the particle Gibbs Markov chain to be geometrically ergodic and quantitative bounds on its geometric rate of convergence, which imply convergence of properties of the particle Gibbs Markov chain to those of its corresponding Gibbs sampler. These results complement recently discovered, and related, conditions for the particle marginal Metropolis-Hastings (PMMH) Markov chain.