Adaptive Monte Carlo Maximum Likelihood

Abstract

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 adaptive importance sampling and a new algorithm, which uses resampling and MCMC. This algorithm is designed to reduce problems with degeneracy 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.

Asymptotics of maximum likelihood estimators based on Markov chain Monte Carlo methods

“…However, to the best of our knowledge, there is no full theoretical justification of it in the literature. Our paper fills this gap and can be viewed as a generalization of the results contained in [12,16]. The methods used in these papers are extended and developed to work in the case where the MC sample is a Markov chain.…”

“…(Note that behaviour of these terms would be much easier to examine if we considered an unrealistic scenario of i.i.d. Monte Carlo, as in [12]. )…”

“…In influential papers [7,8] the Authors prove consistency and asymptotic normality of MCML estimators under the assumption that the initial sample is fixed, and only the Monte Carlo sample size tends to infinity. Both sources of randomness (one due to the initial sample and the other due to Monte Carlo simulations) are considered in [2,12,16,22]. The authors of the first mentioned paper apply the general importance sampling recipe.…”

“…The log-likelihood is first decomposed into independent summands and then importance sampling is applied. Papers [2,16] describe asymptotic properties of MCML estimators only for models with missing data while [12] investigates models with intractable norming constants and explanatory variables. However, in [2,12,16] it is assumed that one can efficiently generate independent samples from a given distribution.…”

“…Papers [2,16] describe asymptotic properties of MCML estimators only for models with missing data while [12] investigates models with intractable norming constants and explanatory variables. However, in [2,12,16] it is assumed that one can efficiently generate independent samples from a given distribution. In many practical problems it is impossible and instead of independent sampling one has to use Markov chain simulation.…”