{ "id": "ep34gttjh1", "slug": "monte-carlo-sampling", "title": "Monte Carlo sampling", "updated": "2026-10-15", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics" }, { "id": "physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_statistical_inference" } ], "geo": [ { "id": "geo.nongeo.t1800.physical.math.statistics.stats_statistical_inference", "label": "Non-geographic · 1800 to 1945: Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics.stats_statistical_inference", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1800", "label": "Non-geographic · 1800 to 1945", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800" }, { "id": "geo.nongeo.t1800.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical" }, { "id": "geo.nongeo.t1800.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math" }, { "id": "geo.nongeo.t1800.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics" }, { "id": "geo.nongeo.t1800.physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics.stats_statistical_inference" } ] }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference", "label": "Non-geographic · 1946 to 2000: Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics" }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference" } ] } ], "excerpt": "Monte Carlo sampling is a statistical method that estimates hard-to-compute sums, integrals, and probabilities by averaging many random samples from a probability model, with error shrinking as 1/√N.", "snippet": "Monte Carlo sampling is a statistical method that estimates hard-to-compute sums, integrals, and probabilities by averaging many random samples from a probability model, with error shrinking as 1/√N.", "node": "physical.math.statistics.stats_statistical_inference", "markdown": "# Monte Carlo sampling\n\nMonte Carlo sampling is a statistical method that estimates a quantity or distribution by drawing many random samples from a probability model and averaging a function of them, yielding a point estimate accompanied by a standard error or confidence interval.[1](https://pdg.lbl.gov/2026/reviews/rpp2026-rev-monte-carlo-techniques.pdf) The method turns sums, integrals, and probabilities that are hard to compute analytically into simulation experiments whose accuracy improves predictably with sample size.[2](https://cswr.nrhstat.org/mci.html) It is used across statistics, physics, finance, computer graphics, and Bayesian computation.[2](https://cswr.nrhstat.org/mci.html) The method's founding paper described it as a statistical approach to differential and integro-differential equations in the natural sciences.[3](https://hedibert.org/wp-content/uploads/2013/12/1949MetropolisUlam.pdf)\n\n| Key fact | Value or statement | Source |\n|---|---|---|\n| Core estimator | Sample average \\( \\hat{\\mu} = (1/N) \\sum h(X_i) \\), converging to \\( E[h(X)] \\) by the law of large numbers | [2](https://cswr.nrhstat.org/mci.html) |\n| Statistical error | Proportional to \\( 1/\\sqrt{N} \\), independent of the number of dimensions | [4](https://www.nsm.buffalo.edu/~biondini/papers/handbookstatistics2015v33chapter2.pdf) |\n| 95% confidence interval | \\( \\hat{\\mu} \\pm 1.96\\, \\hat{\\sigma}/\\sqrt{N} \\) | [2](https://cswr.nrhstat.org/mci.html) |\n| Best possible rate | \\( O(n^{-1/2}) \\) cannot be improved for general square-integrable or continuous functions (Bakhvalov, 1959) | [5](https://www.math.unipd.it/~alvise/PHD_2024/LEZIONI/PHD%20course%20PDF/montecarlo.pdf) |\n| Quasi-Monte Carlo rate | Approximately \\( O((\\log N)^k N^{-1}) \\) with low-discrepancy sequences | [6](https://www.cambridge.org/core/journals/acta-numerica/article/abs/monte-carlo-and-quasimonte-carlo-methods/FE7C779B350CFEA45DB2A4CCB2DA9B5C) |\n| Rare-event cost | For event probability \\( Q \\sim 10^{-6} \\) and target coefficient of variation 0.1, about \\( N = 10^8 \\) samples are needed | [4](https://www.nsm.buffalo.edu/~biondini/papers/handbookstatistics2015v33chapter2.pdf) |\n| Common generator | The Mersenne twister (as in TRandom3) has period \\( 2^{19937} - 1 \\) | [1](https://pdg.lbl.gov/2026/reviews/rpp2026-rev-monte-carlo-techniques.pdf) |\n\n## How it works\n\nThe principle is to write the target quantity as an expectation and approximate the expectation by an empirical average. If \\( X_1, \\dots, X_N \\) are independent draws from a density \\( f \\), the estimator \\( \\hat{\\mu} = (1/N) \\sum_{i=1}^{N} h(X_i) \\) converges to \\( \\mu = E[h(X)] = \\int h(x) f(x)\\, dx \\) by the law of large numbers.[2](https://cswr.nrhstat.org/mci.html) The estimator is unbiased and consistent, with root mean squared error \\( \\sigma(X)/\\sqrt{N} \\), a convergence rate of order \\( N^{-1/2} \\).[7](https://jwmi.github.io/BMS/chapter5-monte-carlo.pdf)\n\nThe central limit theorem makes the average approximately normal with variance \\( \\sigma^2/N \\), so the sample values themselves supply an error estimate: the sample variance \\( s^2 \\) gives an error of order \\( s/\\sqrt{n} \\).[8](https://artowen.su.domains/mc/Ch-intro.pdf) Bakhvalov proved in 1959 that the \\( O(n^{-1/2}) \\) rate cannot be improved for general square-integrable or continuous functions, which is the theoretical ceiling for plain [Monte Carlo](https://www.edgepedia.org/monte-carlo).[5](https://www.math.unipd.it/~alvise/PHD_2024/LEZIONI/PHD%20course%20PDF/montecarlo.pdf)\n\n## How it is done\n\nA practitioner follows four steps. First, specify the probability model: the distribution to sample from and the function \\( h \\) whose expectation answers the question. Second, generate pseudo-random numbers; von Neumann's early middle-square scheme squared an n-digit integer and extracted the middle n digits, and a modern generator should have a period around \\( N^2 \\) or \\( N^3 \\), meaning a period beyond \\( 10^{20} \\) for sample sizes near \\( 10^{10} \\).[9](https://mcnp.lanl.gov/pdf_files/Article_1987_LAS_Metropolis_125--130.pdf) • [10](https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.110/lehre/ws13/Methods_of_Monte_Carlo_Simulation/MC_all_01.pdf) Third, transform uniform draws into the target distribution; the 1949 paper describes drawing \\( x \\) uniformly and applying a precomputed function \\( y = g(x) \\) to obtain the prescribed density \\( f(x) \\).[3](https://hedibert.org/wp-content/uploads/2013/12/1949MetropolisUlam.pdf) Fourth, compute the statistic over the sample and report it with a standard error or confidence interval.[11](https://link.springer.com/chapter/10.1007/978-3-031-76074-7_15)\n\nSample size follows from the \\( 1/\\sqrt{N} \\) law. For rare-event simulation with event probability \\( Q \\ll 1 \\), the samples needed for a coefficient of variation \\( cv \\) scale as \\( N \\sim 1/(Q \\cdot cv^2) \\); estimating a probability of \\( 10^{-6} \\) with \\( cv = 0.1 \\) requires \\( 10^8 \\) samples.[4](https://www.nsm.buffalo.edu/~biondini/papers/handbookstatistics2015v33chapter2.pdf)\n\n## Origin\n\nStanislaw Ulam conceived the method at Los Alamos in 1946 while pondering solitaire combinatorics: rather than enumerate the roughly \\( 8 \\times 10^{67} \\) ways to sort a deck, he thought of laying out the game one hundred times and counting successful plays, and immediately connected the idea to neutron diffusion.[12](https://discover.lanl.gov/publications/actinide-research-quarterly/first-quarter-2023/hitting-the-jackpot-the-birth-of-the-monte-carlo-method) • [13](https://mcnp.lanl.gov/pdf_files/Article_1987_LAS_Eckhardt_131--141.pdf) He discussed the idea with [John von Neumann](https://www.edgepedia.org/john-von-neumann), whose handwritten letter to Robert Richtmyer of 11 March 1947 outlined a statistical approach to neutron diffusion in fissionable material and concluded that the statistical approach was very well suited to a digital treatment; this was the first formulation of a Monte Carlo computation for an electronic computer.[9](https://mcnp.lanl.gov/pdf_files/Article_1987_LAS_Metropolis_125--130.pdf) • [13](https://mcnp.lanl.gov/pdf_files/Article_1987_LAS_Eckhardt_131--141.pdf) [Nicholas Metropolis](https://www.edgepedia.org/nicholas-metropolis) suggested the name, a choice he linked to Ulam's uncle, who would borrow money from relatives because he \"just had to go to Monte Carlo\".[9](https://mcnp.lanl.gov/pdf_files/Article_1987_LAS_Metropolis_125--130.pdf) The first calculations ran on the ENIAC in April and May 1948, after its upgrade to a stored-program machine.[12](https://discover.lanl.gov/publications/actinide-research-quarterly/first-quarter-2023/hitting-the-jackpot-the-birth-of-the-monte-carlo-method) • [14](https://eniacinaction.com/wp-content/uploads/2014/02/LosAlamosBetsOnENIAC.pdf) The method's public debut came with the paper \"The Monte Carlo method\" in the [Journal of the American Statistical Association](https://www.edgepedia.org/journal-of-the-american-statistical-association).[3](https://hedibert.org/wp-content/uploads/2013/12/1949MetropolisUlam.pdf) • [5](https://www.math.unipd.it/~alvise/PHD_2024/LEZIONI/PHD%20course%20PDF/montecarlo.pdf) Enrico Fermi had independently invented the fundamentals of random sampling in the 1930s while studying neutron moderation in Italy, keeping the work unpublished.[12](https://discover.lanl.gov/publications/actinide-research-quarterly/first-quarter-2023/hitting-the-jackpot-the-birth-of-the-monte-carlo-method)\n\n## Variants\n\nPlain Monte Carlo draws independent samples. When direct sampling is impossible, several named variants apply. [Importance sampling](https://www.edgepedia.org/importance-sampling) samples from a proposal density \\( g \\) and corrects with weights \\( w(X_i) = f(X_i)/g(X_i) \\); the variance is minimized when \\( g(x) \\propto |h(x)| f(x) \\), and in one worked example importance sampling needed about 11 times fewer observations than plain Monte Carlo for the same precision, though in another example its variance was about 50% larger, so a poor proposal can lose precision.[2](https://cswr.nrhstat.org/mci.html) • [1](https://pdg.lbl.gov/2026/reviews/rpp2026-rev-monte-carlo-techniques.pdf) When densities are known only up to proportionality constants, self-normalized importance sampling uses weights normalized to sum to one.[7](https://jwmi.github.io/BMS/chapter5-monte-carlo.pdf)\n\n[Rejection sampling](https://www.edgepedia.org/rejection-sampling) chooses a proposal \\( q \\) and constant \\( c \\) with \\( c q(x) \\geq \\tilde{p}(x) \\), samples a candidate and a uniform threshold, and rejects when the threshold exceeds the target density; its efficiency equals \\( 1/C \\), so the envelope constant should stay close to 1.[7](https://jwmi.github.io/BMS/chapter5-monte-carlo.pdf) • [1](https://pdg.lbl.gov/2026/reviews/rpp2026-rev-monte-carlo-techniques.pdf) [Markov chain Monte Carlo](https://www.edgepedia.org/markov-chain-monte-carlo) samples from a chain whose limiting distribution is the target. The [Metropolis](https://www.edgepedia.org/metropolis) algorithm, introduced in the 1953 paper \"Equation of State Calculations by Fast Computing Machines\" by Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller, samples configurations with probability \\( \\exp(-E/(k \\cdot T)) \\) and weights them evenly, accepting downhill moves always and uphill moves with probability \\( \\exp(-\\Delta E/(k \\cdot T)) \\).[15](https://doi.org/10.1063/1.1699114) • [16](https://materias.df.uba.ar/compua2017c1/files/2012/07/Metropolis.pdf) Hastings generalized the method in 1970 to arbitrary proposals, with the acceptance ratio \\( \\alpha = \\min[1,\\, p(\\theta) q(\\theta_0;\\theta) / (p(\\theta_0) q(\\theta;\\theta_0))] \\), reducing to \\( \\min[1,\\, p(\\theta)/p(\\theta_0)] \\) for symmetric proposals; because the chain depends on \\( p \\) only through ratios, the normalizing constant need not be known.[17](https://doi.org/10.1093/biomet/57.1.97) • [18](https://probability.ca/hastings/hastings.pdf) • [1](https://pdg.lbl.gov/2026/reviews/rpp2026-rev-monte-carlo-techniques.pdf) Gibbs sampling, also known as the heat bath method, samples from the conditional distributions of the joint and can be viewed as a Metropolis method with those conditionals as proposals.[19](https://www.cs.cmu.edu/~motionplanning/papers/sbp_papers/kalman/intro_montecarlo.pdf)\n\nSequential Monte Carlo (particle filtering) performs approximate [Bayesian inference](https://www.edgepedia.org/bayesian-inference) on a hidden state evolving over time, using sequentially adaptive proposals to combat weight degeneracy; its foundations lie in sequential importance sampling and in sampling/importance resampling, introduced by Rubin in 1987.[20](https://doi.org/10.2307/2289460) • [21](https://arxiv.org/pdf/1903.04797v3.pdf) • [22](https://ar5iv.labs.arxiv.org/html/2102.05407) Particle Markov chain Monte Carlo, introduced by Andrieu, Doucet, and Holenstein in 2010, uses SMC algorithms to design efficient high-dimensional proposals for MCMC, with the particle independent Metropolis-Hastings, particle marginal Metropolis-Hastings, and particle Gibbs samplers.[23](https://doi.org/10.1111/j.1467-9868.2009.00736.x)\n\nSimpler variance-reduction tools include antithetic variates, control variates, and stratified sampling; stratification can never increase the variance and improves the convergence rate.[24](https://people.smp.uq.edu.au/DirkKroese/ps/montecarlo.pdf) • [25](https://graphics.cs.kuleuven.be/publications/phdniels/files/split/chapter3.pdf)\n\n## Applications\n\nIn von Neumann's 1947 plan for the ENIAC, each punched card represented one neutron at one moment, with random numbers deciding the distance traveled before a collision and the collision type (absorption, scattering, or fission producing up to four daughter neutrons).[14](https://eniacinaction.com/wp-content/uploads/2014/02/LosAlamosBetsOnENIAC.pdf) In finance, the most cited demonstration of quasi-Monte Carlo is Paskov and Traub's 1995 use of low-discrepancy QMC in 360 dimensions to value parcels of mortgage-backed obligations, successful to a degree that caused universal surprise.[5](https://www.math.unipd.it/~alvise/PHD_2024/LEZIONI/PHD%20course%20PDF/montecarlo.pdf) Recent work combines Monte Carlo with generative models: normalizing flows and diffusion models are increasingly used as flexible proposal distributions for sampling targets known only up to a normalization constant,[26](https://arxiv.org/pdf/2608.07648) and a 2025 Royal Society survey shows that methods using pre-trained diffusion models as priors for Bayesian inverse problems primarily employ a twisting mechanism for intermediate distributions, with Monte Carlo then sampling from the twisted distributions without additional training.[27](https://royalsocietypublishing.org/rsta/article/383/2299/20240331/234786/Bridging-diffusion-posterior-sampling-and-Monte)\n\n## Limitations and alternatives\n\nPlain Monte Carlo converges as \\( O(N^{-1/2}) \\) independent of dimension, which makes it robust but slow.[6](https://www.cambridge.org/core/journals/acta-numerica/article/abs/monte-carlo-and-quasimonte-carlo-methods/FE7C779B350CFEA45DB2A4CCB2DA9B5C) Deterministic quadrature is far faster in low dimensions: [Simpson's rule](https://www.edgepedia.org/simpsons-rule) converges as \\( 1/n^4 \\) for a smooth one-dimensional integrand, but a ten-dimensional integral with 20 points per coordinate needs \\( 20^{10} \\approx 10^{13} \\) points, where Monte Carlo might reach similar accuracy with about \\( 10^6 \\).[28](https://archive.math.arizona.edu/tgk/mc/book_chap1.pdf) • [29](https://math.nyu.edu/~goodman/teaching/SciComp2003/Book/monteCarlo.pdf) Quasi-Monte Carlo replaces pseudo-random points with deterministic low-discrepancy sequences such as the Halton sequence, attaining error decay close to \\( n^{-1} \\), or approximately \\( O((\\log N)^k N^{-1}) \\); it performs best in roughly 5 to 50 dimensions and requires a fixed dimension.[6](https://www.cambridge.org/core/journals/acta-numerica/article/abs/monte-carlo-and-quasimonte-carlo-methods/FE7C779B350CFEA45DB2A4CCB2DA9B5C) • [10](https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.110/lehre/ws13/Methods_of_Monte_Carlo_Simulation/MC_all_01.pdf) Randomized QMC retains independent replication for error estimation and can be even more accurate than QMC itself.[8](https://artowen.su.domains/mc/Ch-intro.pdf) QMC's advantage disappears for integrands with discontinuities, where total variation is infinite.[25](https://graphics.cs.kuleuven.be/publications/phdniels/files/split/chapter3.pdf)\n\nHigh-variance integrands and rare events inflate the constant in the \\( 1/\\sqrt{N} \\) rate, which is why rare-event probabilities demand enormous sample sizes.[4](https://www.nsm.buffalo.edu/~biondini/papers/handbookstatistics2015v33chapter2.pdf) Importance sampling fails through weight degeneracy when the proposal deviates from the target; the theoretically optimal proposal yields zero variance but depends on the unknown target quantity and is usually not computable.[2](https://cswr.nrhstat.org/mci.html) • [24](https://people.smp.uq.edu.au/DirkKroese/ps/montecarlo.pdf) • [7](https://jwmi.github.io/BMS/chapter5-monte-carlo.pdf) MCMC produces correlated samples, making it hard to assess convergence or how long to wait for effectively independent draws; a random-walk Metropolis method with step size \\( \\ell \\) needs about \\( (L/\\ell)^2 \\) steps to cover a distance \\( L \\).[19](https://www.cs.cmu.edu/~motionplanning/papers/sbp_papers/kalman/intro_montecarlo.pdf) • [1](https://pdg.lbl.gov/2026/reviews/rpp2026-rev-monte-carlo-techniques.pdf) [Metastability](https://www.edgepedia.org/metastability) in multimodal targets is a central difficulty; protein folding is a case where local MCMC is essentially impractical.[26](https://arxiv.org/pdf/2608.07648) Rejection sampling degrades exponentially with dimension: sampling a uniform point in the unit ball takes about 400 expected trials at dimension 10 and about \\( 3 \\times 10^{20} \\) at dimension 40.[29](https://math.nyu.edu/~goodman/teaching/SciComp2003/Book/monteCarlo.pdf) • [10](https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.110/lehre/ws13/Methods_of_Monte_Carlo_Simulation/MC_all_01.pdf) Generator quality matters: the widely used Mersenne twister, despite its period of \\( 2^{19937} - 1 \\), fails some stringent TestU01 tests.[1](https://pdg.lbl.gov/2026/reviews/rpp2026-rev-monte-carlo-techniques.pdf)\n\n## References\n\n1. [Monte Carlo Techniques (Particle Data Group review, revised September 2025)](https://pdg.lbl.gov/2026/reviews/rpp2026-rev-monte-carlo-techniques.pdf)\n2. [Chapter 7 Monte Carlo integration, Computational Statistics with R](https://cswr.nrhstat.org/mci.html)\n3. [The Monte Carlo Method (N. Metropolis and S. Ulam, JASA 44, Sep. 1949, pp. 335-341)](https://hedibert.org/wp-content/uploads/2013/12/1949MetropolisUlam.pdf)\n4. [Simulation and Importance Sampling (Biondini, Handbook of Statistics 2015)](https://www.nsm.buffalo.edu/~biondini/papers/handbookstatistics2015v33chapter2.pdf)\n5. [Introduction to Monte Carlo and Quasi-Monte Carlo methods (Padova PhD course notes)](https://www.math.unipd.it/~alvise/PHD_2024/LEZIONI/PHD%20course%20PDF/montecarlo.pdf)\n6. [Monte Carlo and quasi-Monte Carlo methods (Caflisch, Acta Numerica 1998)](https://www.cambridge.org/core/journals/acta-numerica/article/abs/monte-carlo-and-quasimonte-carlo-methods/FE7C779B350CFEA45DB2A4CCB2DA9B5C)\n7. [Chapter 5: Monte Carlo Approximation (J. W. Miller)](https://jwmi.github.io/BMS/chapter5-monte-carlo.pdf)\n8. [Monte Carlo Theory, Methods and Practice, Introduction and Chapter 2 (Art Owen)](https://artowen.su.domains/mc/Ch-intro.pdf)\n9. [The Beginning of the Monte Carlo Method (N. Metropolis, Los Alamos Science, 1987)](https://mcnp.lanl.gov/pdf_files/Article_1987_LAS_Metropolis_125--130.pdf)\n10. [Methods of Monte Carlo Simulation (Ulm University course notes)](https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.110/lehre/ws13/Methods_of_Monte_Carlo_Simulation/MC_all_01.pdf)\n11. [Monte Carlo Methods (Albert & Rizzo, R by Example, Springer, 2024)](https://link.springer.com/chapter/10.1007/978-3-031-76074-7_15)\n12. [Hitting the Jackpot: The Birth of the Monte Carlo Method (Los Alamos Actinide Research Quarterly, First Quarter 2023)](https://discover.lanl.gov/publications/actinide-research-quarterly/first-quarter-2023/hitting-the-jackpot-the-birth-of-the-monte-carlo-method)\n13. [Stan Ulam, John von Neumann, and the Monte Carlo Method (R. Eckhardt, Los Alamos Science, 1987)](https://mcnp.lanl.gov/pdf_files/Article_1987_LAS_Eckhardt_131--141.pdf)\n14. [Los Alamos Bets on ENIAC: Nuclear Monte Carlo Origins (Haigh, Priestley & Rope, IEEE Annals of the History of Computing, 2014)](https://eniacinaction.com/wp-content/uploads/2014/02/LosAlamosBetsOnENIAC.pdf)\n15. [Nicholas Metropolis and colleagues (1953). Equation of State Calculations by Fast Computing Machines. The Journal of Chemical Physics.](https://doi.org/10.1063/1.1699114)\n16. [Equation of State Calculations by Fast Computing Machines (Metropolis, Rosenbluth, Rosenbluth, Teller, Teller, 1953)](https://materias.df.uba.ar/compua2017c1/files/2012/07/Metropolis.pdf)\n17. [W. K. Hastings (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika.](https://doi.org/10.1093/biomet/57.1.97)\n18. [Monte Carlo sampling methods using Markov chains and their applications (W. K. Hastings, Biometrika 1970)](https://probability.ca/hastings/hastings.pdf)\n19. [Introduction to Monte Carlo Methods (chapter, D. J. C. MacKay)](https://www.cs.cmu.edu/~motionplanning/papers/sbp_papers/kalman/intro_montecarlo.pdf)\n20. [Donald B. Rubin (1987). The Calculation of Posterior Distributions by Data Augmentation: Comment: A Noniterative Sampling/Importance Resampling Alternative to the Data Augmentation Algorithm for Creating a Few Imputations When Fractions of Missing Information Are Modest: The SIR Algorithm. Journal of the American Statistical Association.](https://doi.org/10.2307/2289460)\n21. [Elements of Sequential Monte Carlo (Naesseth, Lindsten, Schön)](https://arxiv.org/pdf/1903.04797v3.pdf)\n22. [Advances in Importance Sampling (review)](https://ar5iv.labs.arxiv.org/html/2102.05407)\n23. [Christophe Andrieu, Arnaud Doucet, Roman Holenstein (2010). Particle Markov Chain Monte Carlo Methods. Journal of the Royal Statistical Society Series B (Statistical Methodology).](https://doi.org/10.1111/j.1467-9868.2009.00736.x)\n24. [Monte Carlo Methods (Kroese handbook chapter)](https://people.smp.uq.edu.au/DirkKroese/ps/montecarlo.pdf)\n25. [Monte Carlo Integration (dissertation chapter, KU Leuven computer graphics)](https://graphics.cs.kuleuven.be/publications/phdniels/files/split/chapter3.pdf)\n26. [Leveraging generative models to assist Monte Carlo sampling (arXiv review)](https://arxiv.org/pdf/2608.07648)\n27. [Bridging diffusion posterior sampling and Monte Carlo methods: a survey (Phil. Trans. R. Soc. A, 2025)](https://royalsocietypublishing.org/rsta/article/383/2299/20240331/234786/Bridging-diffusion-posterior-sampling-and-Monte)\n28. [Monte Carlo: Methods, Chapter 1 Introduction (T. G. Kurtz, U. Arizona)](https://archive.math.arizona.edu/tgk/mc/book_chap1.pdf)\n29. [Principles of Scientific Computing, Monte Carlo (NYU, Goodman)](https://math.nyu.edu/~goodman/teaching/SciComp2003/Book/monteCarlo.pdf)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Statistical inference, estimation, sampling, and testing*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://www.nsm.buffalo.edu/~biondini/papers/handbookstatistics2015v33chapter2.pdf", "https://www.cs.cmu.edu/~motionplanning/papers/sbp_papers/kalman/intro_montecarlo.pdf", "https://math.nyu.edu/~goodman/teaching/SciComp2003/Book/monteCarlo.pdf" ], "url": "https://www.edgepedia.org/monte-carlo-sampling", "markdown_url": "https://www.edgepedia.org/monte-carlo-sampling.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Monte Carlo sampling\", Edgepedia (EdgeChat), https://www.edgepedia.org/monte-carlo-sampling. Edgepedia Community License 1.0.", "credit_md": "\"[Monte Carlo sampling](https://www.edgepedia.org/monte-carlo-sampling)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/monte-carlo-sampling](https://www.edgepedia.org/monte-carlo-sampling). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Monte Carlo sampling\", Edgepedia (EdgeChat), https://www.edgepedia.org/monte-carlo-sampling. Edgepedia Community License 1.0.", "speakable": "Monte Carlo sampling is a statistical method that estimates hard-to-compute sums, integrals, and probabilities by averaging many random samples from a probability model, with error shrinking as 1/√N." }