Adaptive Monte Carlo Maximum Likelihood

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.

Appendix A: Martingale Limit Theorems

For completeness, we cite the following martingale central limit theorem (CLT):

Theorem A.1

([8, Theorem 2.5]) Let $X_n = \xi _1 + \cdots + \xi _n$ be a mean-zero (vector valued) martingale. If there exists a symmetric positive definite matrix V such that

$$\begin{aligned} \frac{1}{n} \sum _{j=1}^n \mathbb {E}\Big ( \xi _j \xi _j^T | \mathcal {F}_{j-1} \Big ) \xrightarrow {\mathrm{p}}V, \end{aligned}$$

(A.1)

$$\begin{aligned} \frac{1}{n} \sum _{j=1}^n \mathbb {E}\Big ( \xi _j \xi _j^T \mathbf {1}_{|\xi _j| > \varepsilon \sqrt{n}}\, | \mathcal {F}_{j-1} \Big ) \xrightarrow {\mathrm{p}}0 \quad \text { for each } \varepsilon > 0, \end{aligned}$$

(A.2)

then

$$\begin{aligned} \frac{X_n}{\sqrt{n}} \xrightarrow {\mathrm{d}}\mathcal {N} (0, V). \end{aligned}$$

The Lindeberg condition (A.2) can be replaced by a stronger Lyapunov condition

$$\begin{aligned} \frac{1}{n} \sum _{j=1}^n \mathbb {E}\Big ( |\xi _j|^{2+\alpha } | \mathcal {F}_{j-1} \Big )\le M \quad \text { for some } \alpha > 0 \text { and }M<\infty . \end{aligned}$$

(A.3)

A simple consequence of [6, Theorem 2.18] (see also [3]) is the following strong law of large numbers (SLLN).

Theorem A.2

Let $X_n = \xi _1 + \cdots + \xi _n$ be a mean-zero martingale. If

$$\begin{aligned} \sup _j \mathbb {E}\Big (|\xi _j|^{1+\alpha } | \mathcal {F}_{j-1} \Big )\le M \quad \text { for some } \alpha > 0 \text { and }M<\infty \end{aligned}$$

then

$$\begin{aligned} \frac{X_n}{n} \xrightarrow {\mathrm{a.s.}}0. \end{aligned}$$