Variance reduction for Markov chains with application to MCMC
In this paper we propose a novel variance reduction approach for additive functionals of Markov chains based on minimization of an estimate for the asymptotic variance of these functionals over suitable classes of control variates. A distinctive feature of the proposed approach is its ability to significantly reduce the overall finite sample variance. This feature is theoretically demonstrated by means of a deep non asymptotic analysis of a variance reduced functional as well as by a thorough simulation study. In particular we apply our method to various MCMC Bayesian estimation problems where it favourably compares to the existing variance reduction approaches.
In this paper we propose a novel and practical variance reduction approach for additive functionals of dependent sequences. Our approach combines the use of control variates with the minimisation of an empirical variance estimate. We analyse finite sample properties of the proposed method and derive finite-time bounds of the excess asymptotic variance to zero. We apply our methodology to Stochastic Gradient MCMC (SGMCMC) methods for Bayesian inference on large data sets and combine it with existing variance reduction methods for SGMCMC. We present empirical results carried out on a number of benchmark examples showing that our variance reduction method achieves significant improvement as compared to state-of-the-art methods at the expense of a moderate increase of computational overhead.
In his book, Rowan Wilken, lecturer at the University of Swinburne, Australia, makes an attempt at providing a theoretical frame for a three-dimensional problem: the relation between new technologies, communities and places. His main goal is to sculpt an understanding of the relationship between place and community, both of which are transcended by what he calls 'teletechnologies' such as mobile phones, internet and their eventual derivatives. Looking for ‘productive theoretical possibilities to make sense of the complex interactions and interconnections between teletechnologies, place, and community’ appears to be a very difficult task.
What factors best explain the low incidence of skills training in a late industrial society like Russia? This research undertakes a multilevel analysis of the role of occupational structure against the probability of training. The explanatory power of occupation-specific determinants and skills polarisation are evaluated, using a representative 2012 sample from the Russian Longitudinal Monitoring Survey. Applying a two-level Bayesian logistic regression model, we show that the incidence of training in Russia is significantly contextualised within the structure of occupations and the inequalities between them. The study shows that extremely high wage gaps within managerial class jobs can discourage training, an unusual finding. Markets accumulating interchangeable and disposable labour best explain the low incidence of training; workers within generic labour are less likely to develop their skills formally, except in urban markets. Although we did not find strong evidence of skills polarisation, Russians are yet to live in a knowledge economy.
We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.
We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.
We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.