# A Survey of Monte Carlo Methods for Parameter Estimation

arXiv: https://arxiv.org/abs/2107.11820 · HTML: https://mlsanity.com/paper/2107.11820/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 362 references that are in the map.
- First posted on arXiv 25 Jul 2021.

## Builds on (most cited first, 30 of 35)

- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [A Kernel Test of Goodness of Fit](https://mlsanity.com/paper/1602.02964/) (2016)
- [Measuring Sample Quality with Kernels](https://mlsanity.com/paper/1703.01717/) (2017)
- [Measuring Sample Quality with Stein's Method](https://mlsanity.com/paper/1506.03039/) (2015)
- [Control functionals for Monte Carlo integration](https://mlsanity.com/paper/1410.2392/) (2014)
- [On Markov chain Monte Carlo methods for tall data](https://mlsanity.com/paper/1505.02827/) (2015)
- [Asymptotically Exact, Embarrassingly Parallel MCMC](https://mlsanity.com/paper/1311.4780/) (2013)
- [Measuring Sample Quality with Diffusions](https://mlsanity.com/paper/1611.06972/) (2016)
- [Firefly Monte Carlo: Exact MCMC with Subsets of Data](https://mlsanity.com/paper/1403.5693/) (2014)
- [Particle Gibbs with Ancestor Sampling](https://mlsanity.com/paper/1401.0604/) (2014)
- [The Geometric Foundations of Hamiltonian Monte Carlo](https://mlsanity.com/paper/1410.5110/) (2014)
- [Nested Sequential Monte Carlo Methods](https://mlsanity.com/paper/1502.02536/) (2015)
- [Layered Adaptive Importance Sampling](https://mlsanity.com/paper/1505.04732/) (2015)
- [Parallelizing MCMC via Weierstrass Sampler](https://mlsanity.com/paper/1312.4605/) (2013)
- [Noisy Monte Carlo: Convergence of Markov chains with approximate transition kernels](https://mlsanity.com/paper/1403.5496/) (2014)
- [Parallelizing MCMC with Random Partition Trees](https://mlsanity.com/paper/1506.03164/) (2015)
- [Adaptive, delayed-acceptance MCMC for targets with expensive likelihoods](https://mlsanity.com/paper/1509.00172/) (2015)
- [Optimizing The Integrator Step Size for Hamiltonian Monte Carlo](https://mlsanity.com/paper/1411.6669/) (2014)
- [Statistically efficient thinning of a Markov chain sampler](https://mlsanity.com/paper/1510.07727/) (2015)
- [High-dimensional Filtering using Nested Sequential Monte Carlo](https://mlsanity.com/paper/1612.09162/) (2016)
- [Gradient Importance Sampling](https://mlsanity.com/paper/1507.05781/) (2015)
- [The Recycling Gibbs Sampler for Efficient Learning](https://mlsanity.com/paper/1611.07056/) (2016)
- [Group Importance Sampling for Particle Filtering and MCMC](https://mlsanity.com/paper/1704.02771/) (2017)
- [Metropolis Sampling](https://mlsanity.com/paper/1704.04629/) (2017)
- [A Review of Multiple Try MCMC algorithms for Signal Processing](https://mlsanity.com/paper/1801.09065/) (2018)
- [Air Markov Chain Monte Carlo](https://mlsanity.com/paper/1801.09309/) (2018)
- [Adaptive Independent Sticky MCMC algorithms](https://mlsanity.com/paper/1308.3779/) (2013)
- [Big Learning with Bayesian Methods](https://mlsanity.com/paper/1411.6370/) (2014)
- [GPU-accelerated Gibbs sampling: a case study of the Horseshoe Probit model](https://mlsanity.com/paper/1608.04329/) (2016)

## Built on by (most cited first, 23 of 23)

- [A survey of Monte Carlo methods for noisy and costly densities with application to reinforcement learning and ABC](https://mlsanity.com/paper/2108.00490/) (2021)
- [Cooperation in the Latent Space: The Benefits of Adding Mixture Components in Variational Autoencoders](https://mlsanity.com/paper/2209.15514/) (2022)
- [Bayesian Safety Validation for Failure Probability Estimation of Black-Box Systems](https://mlsanity.com/paper/2305.02449/) (2023)
- [Unrolled Variational Bayesian Algorithm for Image Blind Deconvolution](https://mlsanity.com/paper/2110.07202/) (2021)
- [ISFL: Federated Learning for Non-i.i.d. Data with Local Importance Sampling](https://mlsanity.com/paper/2210.02119/) (2022)
- [PINNverse: Accurate parameter estimation in differential equations from noisy data with constrained physics-informed neural networks](https://mlsanity.com/paper/2504.05248/) (2025)
- [Optimality in Noisy Importance Sampling](https://mlsanity.com/paper/2201.02432/) (2022)
- [Data-efficient and Interpretable Inverse Materials Design using a Disentangled Variational Autoencoder](https://mlsanity.com/paper/2409.06740/) (2024)
- [Covid19 Reproduction Number: Credibility Intervals by Blockwise Proximal Monte Carlo Samplers](https://mlsanity.com/paper/2203.09142/) (2022)
- [Ai-Sampler: Adversarial Learning of Markov kernels with involutive maps](https://mlsanity.com/paper/2406.02490/) (2024)
- [Adaptive posterior distributions for uncertainty analysis of covariance matrices in Bayesian inversion problems for multioutput signals](https://mlsanity.com/paper/2501.01148/) (2025)
- [CoCoAFusE: Beyond Mixtures of Experts via Model Fusion](https://mlsanity.com/paper/2505.01105/) (2025)
- [Radon--Wasserstein Gradient Flows for Interacting-Particle Sampling in High Dimensions](https://mlsanity.com/paper/2602.05227/) (2026)
- [Toward Unlimited Self-Learning MCMC with Parallel Adaptive Annealing](https://mlsanity.com/paper/2211.14024/) (2022)
- [Ensemble Differential Evolution with Simulation-Based Hybridization and Self-Adaptation for Inventory Management Under Uncertainty](https://mlsanity.com/paper/2309.12852/) (2023)
- [Pathspace Kalman Filters with Dynamic Process Uncertainty for Analyzing Time-course Data](https://mlsanity.com/paper/2402.04498/) (2024)
- [Accelerating Convergence of Stein Variational Gradient Descent via Deep Unfolding](https://mlsanity.com/paper/2402.15125/) (2024)
- [Estimating Multi-chirp Parameters using Curvature-guided Langevin Monte Carlo](https://mlsanity.com/paper/2501.18178/) (2025)
- [Accurate and Reliable Uncertainty Estimates for Deterministic Predictions Extensions to Under and Overpredictions](https://mlsanity.com/paper/2604.08755/) (2026)
- [Truncated Neural Likelihood Estimation for Simulation-Based Inference in State-Space Models](https://mlsanity.com/paper/2605.21805/) (2026)
- [From Graphs to Gradients: Physics-Inspired Structural Attribution for Cyber-Physical IoT Systems and Beyond](https://mlsanity.com/paper/2607.05563/) (2026)
- [Gaussian Volumetric Representation for Efficient Shear-Warp Visualization](https://mlsanity.com/paper/2607.25377/) (2026)
- [Repulsive normalizing flow mixtures for adaptive importance sampling: reliability analysis of complex systems](https://mlsanity.com/paper/2609.21160/) (2026)

## Abstract (arXiv)

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 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.