A survey of Monte Carlo methods for parameter estimation

David Luengo*, Luca Martino, Mónica Bugallo, Víctor Elvira, Simo Särkkä

*Corresponding author for this work

Research output: Contribution to journalReview Articlepeer-review

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Abstract

Statistical signal processing applications usually require the estimation of some parameters of interest given a set of observed data. These estimates are typically obtained either by solving a multi-variate optimization problem, as in the maximum likelihood (ML) or maximum a posteriori (MAP) estimators, or by performing a multi-dimensional integration, as in the minimum mean squared error (MMSE) estimators. Unfortunately, analytical expressions for these estimators cannot be found in most real-world applications, and the Monte Carlo (MC) methodology is one feasible approach. MC methods proceed by drawing random samples, either from the desired distribution or from a simpler one, and using them to compute consistent estimators. The most important families of MC algorithms are the Markov chain MC (MCMC) and importance sampling (IS). On the one hand, MCMC methods draw samples from a proposal density, building then an ergodic Markov chain whose stationary distribution is the desired distribution by accepting or rejecting those candidate samples as the new state of the chain. On the other hand, IS techniques draw samples from a simple proposal density and then assign them suitable weights that measure their quality in some appropriate way. In this paper, we perform a thorough review of MC methods for the estimation of static parameters in signal processing applications. A historical note on the development of MC schemes is also provided, followed by the basic MC method and a brief description of the rejection sampling (RS) algorithm, as well as three sections describing many of the most relevant MCMC and IS algorithms, and their combined use. Finally, five numerical examples (including the estimation of the parameters of a chaotic system, a localization problem in wireless sensor networks and a spectral analysis application) are provided in order to demonstrate the performance of the described approaches.

Original languageEnglish
Article number25
Number of pages62
JournalEurasip Journal on Advances in Signal Processing
Volume2020
Issue number1
DOIs
Publication statusPublished - 29 May 2020
MoE publication typeA2 Review article in a scientific journal

Keywords

  • Adaptive MCMC
  • Bayesian inference
  • Gibbs sampler
  • Importance sampling
  • Metropolis-Hastings algorithm
  • MH-within-Gibbs
  • Monte Carlo methods
  • Population Monte Carlo
  • Statistical signal processing

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