Output Analysis Of Monte Carlo Methods With Applications To Networks And Functional Approximation

Abstract

The overall objective of the Monte Carlo method is to use data simulated in a computer to learn about complex systems. This is a highly flexible approach and can be applied in a variety of settings. For instance, Monte Carlo methods are used to estimate network properties or to approximate functions. Although the use of these methods in such cases is common, little to no work exists on assessing the reliability of the estimation procedure. Thus, the contribution of this work lies in further developing methods to better address the reliability of Monte Carlo estimation, particularly with respect to estimating network properties and approximating functions. In network analysis, there exist many networks which can only be studied via sampling methods due to the scale or complexity of the network, access limitations, or the population of interest is hard to reach. In such cases, the application of random walk-based Markov chain Monte Carlo (MCMC) methods to estimate multiple network features is common. However, the reliability of these estimates has been largely ignored. We consider and further develop multivariate MCMC output analysis methods in the context of network sampling to directly address the reliability of the multivariate estimation. This approach yields principled, computationally efficient, and broadly applicable methods for assessing the Monte Carlo estimation procedure. We also study the Monte Carlo estimation reliability in approximating functions using Importance Sampling. Although we focus on approximating difficult to compute density and log-likelihood functions, we develop a general framework for constructing simultaneous confidence bands that could be applied in other contexts. In addition, we propose a correction to improve the reliability of the log-likelihood function estimation using the Monte Carlo Likelihood Approximation approach.