arXiv:1412.6370v1 [stat.ME] 19 Dec 2014 Adaptive Monte Carlo Maximum Likelihood B la˙ zej Miasojedow 1 , Wojciech Niemiro 1,3 , Jan Palczewski 2 , and Wojciech Rejchel 3 1 Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland W.Miasojedow@mimuw.edu.pl W.Niemiro@mimuw.edu.pl 2 School of Mathematics, University of Leeds, Leeds, UK J.Palczewski@leeds.ac.uk 3 Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toru´ n, Poland wrejchel@gmail.com Abstract. We consider Monte Carlo approximations to the maximum likelihood estimator in models with intractable norming constants. This paper deals with adaptive Monte Carlo algorithms, which adjust control parameters in the course of simulation. We examine asymptotics of adap- tive importance sampling and a new algorithm, which uses resampling and MCMC. This algorithm is designed to reduce problems with degen- eracy of importance weights. Our analysis is based on martingale limit theorems. We also describe how adaptive maximization algorithms of Newton-Raphson type can be combined with the resampling techniques. The paper includes results of a small scale simulation study in which we compare the performance of adaptive and non-adaptive Monte Carlo maximum likelihood algorithms. Keywords: maximum likelihood, importance sampling, adaptation, MCMC, resampling 1 Introduction Maximum likelihood (ML) is a well-known and often used method in estimation of parameters in statistical models. However, for many complex models exact calculation of such estimators is very difficult or impossible. Such problems arise if considered densities are known only up to intractable norming constants, for instance in Markov random fields or spatial statistics. The wide range of applica- tions of models with unknown norming constants is discussed e.g. in [10]. Meth- ods proposed to overcome the problems with computing ML estimates in such models include, among others, maximum pseudolikelihood [2], “coding method” [9] and Monte Carlo maximum likelihood (MCML) [4], [15], [9], [17]. In our paper we focus on MCML.