[Paper Review] Maximum Likelihood Estimation for Stochastic Differential Equations Using Sequential Kriging-Based Optimization

Grant Schneider, Craigmile, Peter F.+1 more|arXiv (Cornell University)|Aug 11, 2014

Advanced Multi-Objective Optimization Algorithms15 references3 citations

TL;DR

This paper proposes a sequential kriging-based optimization (SKBO) method for maximum likelihood estimation (MLE) in stochastic differential equations (SDEs), where likelihoods are estimated via sequential Monte Carlo (SMC). By modeling the log-likelihood as a Gaussian process and using expected improvement to sequentially select promising parameter values, the method efficiently locates the MLE without requiring gradient information or grid searches, demonstrating significant computational gains over existing approaches.

ABSTRACT

Stochastic Differential Equations (SDEs) are used as statistical models in many disciplines. However, intractable likelihood functions for SDEs make inference challenging, and we need to resort to simulation-based techniques to estimate and maximize the likelihood function. While sequential Monte Carlo methods have allowed for the accurate evaluation of likelihoods at fixed parameter values, there is still a question of how to find the maximum likelihood estimate. In this article we propose an efficient Gaussian-process-based method for exploring the parameter space using estimates of the likelihood from a sequential Monte Carlo sampler. Our method accounts for the inherent Monte Carlo variability of the estimated likelihood, and does not require knowledge of gradients. The procedure adds potential parameter values by maximizing the so-called expected improvement, leveraging the fact that the likelihood function is assumed to be smooth. Our simulations demonstrate that our method has significant computational and efficiency gains over existing grid- and gradient-based techniques. Our method is applied to modeling the closing stock price of three technology firms.

Motivation & Objective

• To address the challenge of maximizing intractable likelihoods in SDEs due to the absence of closed-form transition densities.
• To reduce computational burden in MLE by replacing grid-based or gradient-based optimization with a sequential, adaptive sampling strategy.
• To account for Monte Carlo variability in SMC-estimated likelihoods while avoiding reliance on gradient information.
• To provide a robust, uncertainty-quantified method for MLE in complex SDE models using Bayesian optimization principles.

Proposed method

• Models the log-likelihood function as a Gaussian process (GP) to capture its smoothness and uncertainty over the parameter space.
• Uses sequential kriging to iteratively select new parameter values by maximizing the expected improvement (EI) criterion, balancing exploration and exploitation.
• Employs sequential Monte Carlo (SMC) to estimate the likelihood at each selected parameter value, accounting for Monte Carlo variability.
• Updates the GP posterior after each new likelihood estimate, incorporating uncertainty in the likelihood approximation.
• Applies a Bayesian framework to learn GP hyperparameters, allowing for prior knowledge integration and natural uncertainty quantification.
• Leverages parallel computing via multi-core R routines to enhance computational efficiency, particularly for Monte Carlo and importance sampling steps.

Experimental results

Research questions

• RQ1 Can a sequential, gradient-free optimization method efficiently locate the MLE in SDEs with intractable likelihoods?
• RQ2 How can Monte Carlo variability in SMC-estimated likelihoods be effectively modeled and accounted for in the optimization process?
• RQ3 Does using expected improvement in a kriging framework outperform grid-based or gradient-based methods in terms of computational efficiency and convergence speed?
• RQ4 To what extent does the proposed SKBO method provide reliable uncertainty quantification for the MLE in SDE models?

Key findings

• The SKBO method achieves significant computational and efficiency gains over grid-based and gradient-based optimization techniques in MLE for SDEs.
• The method successfully locates the MLE by adaptively sampling the parameter space, avoiding local maxima through global search via expected improvement.
• Even with a naive GP parameterization, the method delivers strong performance, indicating robustness to model misspecification in initial assumptions.
• The approach naturally incorporates uncertainty in likelihood estimates and provides a coherent framework for quantifying uncertainty in the MLE.
• The method is highly parallelizable, suggesting potential for further speedups using multi-core or GPU computing architectures.