{ "id": "epkj5nkw6g", "slug": "langevin-algorithm", "title": "Langevin algorithm", "updated": "2026-09-30", "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.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" } ] }, { "id": "geo.nongeo.t2001.physical.math.statistics", "label": "Non-geographic · 2001 to 2020: Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.physical.math.statistics", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t2001", "label": "Non-geographic · 2001 to 2020", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001" }, { "id": "geo.nongeo.t2001.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.physical" }, { "id": "geo.nongeo.t2001.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.physical.math" }, { "id": "geo.nongeo.t2001.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.physical.math.statistics" } ] } ], "excerpt": "The Langevin algorithm is a Markov chain Monte Carlo method that samples a probability distribution by following the gradient of its log-density with added Gaussian noise.", "snippet": "The Langevin algorithm is a Markov chain Monte Carlo method that samples a probability distribution by following the gradient of its log-density with added Gaussian noise.", "node": "physical.math.statistics.stats_statistical_inference", "markdown": "# Langevin algorithm\n\nThe Langevin algorithm is a [Markov chain Monte Carlo](https://www.edgepedia.org/markov-chain-monte-carlo) (MCMC) method that draws samples from a probability distribution by simulating overdamped (dynamics without inertia; motion ignores momentum effects) [Langevin dynamics](https://www.edgepedia.org/langevin-dynamics): each step moves the current state along the gradient of the log-density and adds Gaussian noise of calibrated size. It produces a sequence of states whose distribution approaches the target, and it is widely used in [Bayesian statistics](https://www.edgepedia.org/bayesian-statistics), data assimilation, inverse problems, and machine learning.[1](https://www.jmlr.org/papers/volume25/23-0139/23-0139.pdf) Because each step uses gradient information, its mixing cost in dimension \\( d \\) improves on random-walk Metropolis, whose cost is \\( O(d) \\).[2](https://probability.ca/jeff/ftpdir/lang.pdf)\n\n| Key fact | Statement |\n|---|---|\n| ULA update | \\( X_{k+1} = X_k - \\gamma_{k+1} \\nabla U(X_k) + \\sqrt{2\\gamma_{k+1}}\\, Z_{k+1} \\), with \\( Z_k \\) i.i.d. \\( d \\)-dimensional standard Gaussians[3](https://alain.perso.math.cnrs.fr/data/paper/bj_2019.pdf) |\n| Target law | The driving SDE \\( dX_t = -\\nabla V(X_t)\\,dt + \\sqrt{2}\\,dB_t \\) has stationary distribution \\( \\pi \\propto \\exp(-V) \\)[4](https://bishtref.com/articles/10.1002/cpa.70032) |\n| MALA | The ULA proposal plus a Metropolis–Hastings accept–reject step, which removes the discretization bias[5](https://www.icts.res.in/sites/default/files/paap-2019-08-08-Eric%20Moulines.pdf) |\n| Optimal acceptance | In the high-dimensional scaling limit with regularity and product-target assumptions, the asymptotically optimal acceptance rate is 0.574, an optimum that need not hold for arbitrary targets[2](https://probability.ca/jeff/ftpdir/lang.pdf) |\n| Dimension scaling | MALA takes \\( O(N^{1/3}) \\) steps to explore an \\( N \\)-dimensional target versus \\( O(N) \\) for random-walk Metropolis[2](https://probability.ca/jeff/ftpdir/lang.pdf) |\n| ULA bias | The stationary distribution of ULA is \\( O(\\sqrt{d \\cdot \\gamma}) \\) from the target at step size \\( \\gamma \\)[6](https://jmlr.org/papers/volume25/23-0783/23-0783.pdf) |\n| SGLD | Replaces the full gradient with a minibatch estimate and skips the accept–reject step, enabling large-scale Bayesian learning[7](https://proceedings.neurips.cc/paper/2018/file/335cd1b90bfa4ee70b39d08a4ae0cf2d-Paper.pdf) |\n\n## How it works\n\nThe method rests on the overdamped Langevin stochastic differential equation \\( dX_t = -\\nabla V(X_t)\\,dt + \\sqrt{2}\\,dB_t \\), whose stationary distribution is \\( \\pi \\propto \\exp(-V) \\); the drift pulls samples toward high-density regions while the noise maintains the correct spread.[4](https://bishtref.com/articles/10.1002/cpa.70032) Equivalently, the stated SDE has drift \\( \\nabla\\log\\pi(X_t) \\), and the density of the process is time-invariant exactly when it equals \\( \\pi \\), which is why the diffusion has \\( \\pi \\) as its invariant law; a drift of \\( (1/2)\\nabla\\log\\pi(X_t) \\) would instead pair with unit [Brownian noise](https://www.edgepedia.org/brownian-noise), \\( dX_t = (1/2)\\nabla\\log\\pi(X_t)\\,dt + dB_t \\).[8](https://ar5iv.labs.arxiv.org/html/1309.2983) Discretizing this diffusion in time with the Euler–Maruyama scheme gives a [Markov chain](https://www.edgepedia.org/markov-chain) on the state space; the gradient of the log-density supplies the drift term and an injected Gaussian term supplies the diffusion term.[9](https://wrap.warwick.ac.uk/id/eprint/52760/1/WRAP_Stuart_euclid.aoap.1353695955.pdf) With a constant step size \\( \\gamma \\), under suitable stability and regularity conditions the unadjusted chain has an invariant distribution \\( \\pi_\\gamma \\), which generally differs from \\( \\pi \\), though such an invariant law need not exist for every target and step size; when it exists, the difference is the discretization bias.[3](https://alain.perso.math.cnrs.fr/data/paper/bj_2019.pdf)\n\n## How it is done\n\nA practitioner first computes the gradient of the log-density \\( \\nabla\\log\\pi \\) (or of the negative log-posterior in Bayesian work). Each iteration then evaluates this gradient at the current state, forms the proposal \\( y = x + \\gamma \\nabla\\log\\pi(x) + \\sqrt{2\\gamma}\\,Z \\) with \\( Z \\sim N(0, I_N) \\), and, in MALA, accepts y with probability \\( \\alpha_\\gamma(x,y) = 1 \\wedge \\frac{q_\\gamma(y,x)\\pi(y)}{q_\\gamma(x,y)\\pi(x)} \\), where \\( q_\\gamma \\) is the Gaussian proposal density.[1](https://www.jmlr.org/papers/volume25/23-0139/23-0139.pdf) • [10](https://www.hairer.org/papers/mala.pdf) In the high-dimensional scaling limit the average acceptance rate should be tuned to 0.574.[2](https://probability.ca/jeff/ftpdir/lang.pdf) For stochastic-gradient versions, theory indicates a decreasing step-size sequence of the form \\( \\gamma_m \\sim m^{-1/3} \\), which yields a mean squared error decreasing at rate \\( O(m^{-1/3}) \\).[11](https://jmlr.csail.mit.edu/papers/volume17/teh16a/teh16a.pdf)\n\n## Origin\n\nMetropolized discretizations of overdamped Langevin dynamics were known as \"Smart MC\" in the chemistry literature and were later rediscovered in computational statistics as MALA.[12](https://ar5iv.labs.arxiv.org/html/1505.04905) The Metropolis-adjusted Langevin algorithm was introduced by Gareth O. Roberts and Richard L. Tweedie in 1996 in Bernoulli, in the paper \"Exponential Convergence of Langevin Distributions and Their Discrete Approximations\".[13](https://doi.org/10.2307/3318418) Stochastic gradient Langevin dynamics was introduced by [Max Welling](https://www.edgepedia.org/max-welling) and Yee Whye Teh in 2011 at ICML in the paper \"Bayesian Learning via Stochastic Gradient Langevin Dynamics\"; it adds the right amount of noise to stochastic gradient updates so the iterates converge to posterior samples as the step size is annealed.[5](https://www.icts.res.in/sites/default/files/paap-2019-08-08-Eric%20Moulines.pdf) Preconditioned stochastic gradient Langevin dynamics for deep neural networks was introduced by Chunyuan Li and colleagues in 2015 on arXiv.[14](https://doi.org/10.48550/arxiv.1512.07666) FisherMALA was introduced by Michalis K. Titsias in 2023 on arXiv.[15](https://doi.org/10.48550/arxiv.2305.14442)\n\n## Variants\n\nAdjustment and preconditioning define the main families. MALA adds the Metropolis–Hastings step to ULA; preconditioned MALA inserts a positive-definite matrix into the diffusion and suits targets with highly correlated components.[8](https://ar5iv.labs.arxiv.org/html/1309.2983) FisherMALA learns a preconditioning matrix from gradient history; at the optimum the preconditioner is proportional to the inverse [Fisher information](https://www.edgepedia.org/fisher-information) matrix \\( I^{-1} \\), with \\( I = E_{\\pi(x)} \\nabla\\log\\pi(x)\\nabla\\log\\pi(x)^\\top \\), updated at quadratic \\( O(d^2) \\) cost per iteration.[16](https://papers.nips.cc/paper_files/paper/2023/file/5da6d5818a156791090c875abeca3cf8-Paper-Conference.pdf) • [15](https://doi.org/10.48550/arxiv.2305.14442) For large datasets, SGLD replaces the gradient with an unbiased minibatch estimate \\( \\nabla U_0 + (N/p)\\sum_{i \\in S}\\nabla U_i \\) and skips the accept–reject step;[7](https://proceedings.neurips.cc/paper/2018/file/335cd1b90bfa4ee70b39d08a4ae0cf2d-Paper.pdf) mSGLD removes the resulting asymptotic bias to first order in the step size,[17](https://jmlr.org/papers/volume17/15-494/15-494.pdf) and SGLDFP uses control variates referenced at the posterior mode to cut gradient variance at sublinear cost in the number of data points.[7](https://proceedings.neurips.cc/paper/2018/file/335cd1b90bfa4ee70b39d08a4ae0cf2d-Paper.pdf) Li, Chen, Carlson, and Carin combined adaptive preconditioners with SGLD for deep neural networks (pSGLD).[14](https://doi.org/10.48550/arxiv.1512.07666) Further variants include the kinetic (underdamped) algorithm with a friction parameter,[18](https://arxiv.org/pdf/2407.09301) and stochastic gradient [Hamiltonian Monte Carlo](https://www.edgepedia.org/hamiltonian-monte-carlo), which couples stochastic gradient estimates with Hamiltonian dynamics.[11](https://jmlr.csail.mit.edu/papers/volume17/teh16a/teh16a.pdf) The Metropolis-adjusted Preconditioned Langevin Algorithm (MAPLA), inspired by [natural gradient descent](https://www.edgepedia.org/natural-gradient-descent), handles constrained spaces with non-asymptotic mixing bounds,[19](https://arxiv.org/abs/2412.18701) and invariant-measure-preserving adaptive step sizes reduce steps only where numerically necessary, with a fluctuation–dissipation correction term.[20](https://arxiv.org/html/2403.11993v2)\n\n## Applications\n\nThe algorithm is described as one of the workhorses for sampling probability measures, used in Bayesian statistics, data assimilation, inverse problems, and machine learning.[1](https://www.jmlr.org/papers/volume25/23-0139/23-0139.pdf) SGLD extends it to Bayesian learning on large datasets, where one full-gradient iteration would cost \\( N \\cdot d \\) gradient evaluations; the minibatch form makes posterior sampling feasible at scale.[7](https://proceedings.neurips.cc/paper/2018/file/335cd1b90bfa4ee70b39d08a4ae0cf2d-Paper.pdf) In generative modeling, preconditioned Langevin dynamics driven by score-based generative model priors is used for infinite-dimensional linear Bayesian inverse problems, with the drift combining a learned neural-network score with the data-fit gradient.[21](https://proceedings.neurips.cc/paper_files/paper/2025/file/9aa618b7d721666ca8eadca5d161a1d4-Paper-Conference.pdf) Metropolis-adjusted diffusion models (MADM) replace the biased unadjusted Langevin corrector in diffusion-model samplers with accept–reject Langevin steps.[22](https://arxiv.org/html/2605.09654v1)\n\n## Limitations and alternatives\n\nWithout the [Metropolis](https://www.edgepedia.org/metropolis) step, discrete Langevin approximations can be transient no matter how small the step variance, and ULA may fail to converge even for light-tailed targets; MALA converges under suitable conditions, but can fail to do so geometrically fast even when the diffusion converges exponentially.[2](https://probability.ca/jeff/ftpdir/lang.pdf) • [13](https://doi.org/10.2307/3318418) • [30](https://projecteuclid.org/journals/bernoulli/volume-2/issue-4/Exponential-convergence-of-Langevin-distributions-and-their-discrete-approximations/bj/1178291835.pdf) For non-globally Lipschitz drifts, MALA may lack a spectral gap even when the SDE has one, though convergence remains exponential up to terms exponentially small in the step size.[10](https://www.hairer.org/papers/mala.pdf) For fixed precision \\( \\varepsilon \\), ULA needs \\( O(d \\cdot \\varepsilon^{-2}) \\) or \\( O(d \\cdot \\varepsilon^{-1}) \\) iterations (up to logarithmic terms) in Wasserstein or total variation distance, depending on the smoothness of U; with n iterations and step size \\( \\gamma_n \\), the Wasserstein error is \\( O(n^{-1/2}) \\) or \\( O(n^{-1}) \\).[3](https://alain.perso.math.cnrs.fr/data/paper/bj_2019.pdf) Its stationary bias is \\( O(\\sqrt{d \\cdot \\gamma}) \\) at step size \\( \\gamma \\).[6](https://jmlr.org/papers/volume25/23-0783/23-0783.pdf) MALA, by contrast, needs at most \\( \\text{poly-log}(1/\\varepsilon) \\) iterations with a constant step size in the strongly log-concave setting.[6](https://jmlr.org/papers/volume25/23-0783/23-0783.pdf) For log-concave densities with condition number \\( \\kappa \\), MALA requires \\( O(\\kappa \\cdot d \\log(1/\\delta)) \\) steps for total variation error \\( \\delta \\), against \\( O(\\kappa^{2} \\cdot d/\\delta^{2}) \\) for ULA.[23](https://proceedings.mlr.press/v75/dwivedi18a/dwivedi18a.pdf) Although ULA's error for the full variable vector scales poorly with dimension, iterations proportional to a lower power of d often suffice for all k-marginals, an effect called delocalization of bias.[4](https://bishtref.com/articles/10.1002/cpa.70032) SGLD inherits the subsampling problem: with constant step size scaled as \\( 1/N \\), its invariant measure departs significantly from the posterior and behaves like SGD because of the high variance of stochastic gradients;[7](https://proceedings.neurips.cc/paper/2018/file/335cd1b90bfa4ee70b39d08a4ae0cf2d-Paper.pdf) controlling this bias forces step sizes so small that the cost of reaching a target accuracy is roughly the same for all batch sizes.[24](https://ar5iv.labs.arxiv.org/html/1706.02692) Lower bounds show there are posteriors for which any SGLD algorithm returns extremely poor samples until it has seen each data point at least once on average.[25](https://ar5iv.labs.arxiv.org/html/2412.01952) Langevin dynamics is also not robust to L^2 (more generally L^p) score estimation error in high dimensions, remaining far from the target on sub-exponential timescales even with exponentially small score error; earlier work such as Das et al. 2023 and Huang et al. 2024 studied [L-infinity](https://www.edgepedia.org/l-infinity) bounds on the score estimate, which do not arise naturally with score matching.[26](https://arxiv.org/pdf/2603.11319) Against random-walk Metropolis, gradient information improves the dimension dependence of mixing from \\( O(d) \\) to \\( O(d^{1/3}) \\).[6](https://jmlr.org/papers/volume25/23-0783/23-0783.pdf) Against Hamiltonian Monte Carlo, published comparisons are indirect: the unbiased kinetic estimator UBUBU can be 2–3 times more efficient than randomized Hamiltonian Monte Carlo in reported experiments.[27](https://arxiv.org/abs/2311.05025) Second-order variants that use Hessian information improve on first-order Langevin Monte Carlo in ill-conditioned settings,[28](https://arxiv.org/html/1710.00095v4) and error bounds extend to targets that are smooth and log-concave but not strongly log-concave via kinetic variants KLMC and KLMC2.[29](https://arxiv.org/html/1906.08530v3)\n\n## References\n\n1. [Optimal Scaling for the Proximal Langevin Algorithm in High Dimensions](https://www.jmlr.org/papers/volume25/23-0139/23-0139.pdf)\n2. [Optimal Scaling for Various Metropolis-Hastings Algorithms (Roberts & Rosenthal)](https://probability.ca/jeff/ftpdir/lang.pdf)\n3. [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://alain.perso.math.cnrs.fr/data/paper/bj_2019.pdf)\n4. [Convergence of Unadjusted Langevin in High Dimensions: Delocalization of Bias](https://bishtref.com/articles/10.1002/cpa.70032)\n5. [The Langevin MCMC: Theory and Methods (Moulines lecture notes)](https://www.icts.res.in/sites/default/files/paap-2019-08-08-Eric%20Moulines.pdf)\n6. [On the Computational Complexity of Metropolis-Adjusted Langevin Algorithms for Bayesian Posterior Sampling](https://jmlr.org/papers/volume25/23-0783/23-0783.pdf)\n7. [The promises and pitfalls of Stochastic Gradient Langevin Dynamics (NeurIPS 2018)](https://proceedings.neurips.cc/paper/2018/file/335cd1b90bfa4ee70b39d08a4ae0cf2d-Paper.pdf)\n8. [Langevin diffusions and the Metropolis-adjusted Langevin algorithm](https://ar5iv.labs.arxiv.org/html/1309.2983)\n9. [Optimal scaling and diffusion limits for the Langevin algorithm in high dimensions](https://wrap.warwick.ac.uk/id/eprint/52760/1/WRAP_Stuart_euclid.aoap.1353695955.pdf)\n10. [Non-asymptotic mixing of the MALA algorithm (Hairer, Stuart, Vollmer)](https://www.hairer.org/papers/mala.pdf)\n11. [Consistency and Fluctuations For Stochastic Gradient Langevin Dynamics](https://jmlr.csail.mit.edu/papers/volume17/teh16a/teh16a.pdf)\n12. [Improving dynamical properties of metropolized discretizations of overdamped Langevin dynamics](https://ar5iv.labs.arxiv.org/html/1505.04905)\n13. [Gareth O. Roberts, Richard L. Tweedie (1996). Exponential Convergence of Langevin Distributions and Their Discrete Approximations. Bernoulli.](https://doi.org/10.2307/3318418)\n14. [Li, Chunyuan and colleagues (2015). Preconditioned Stochastic Gradient Langevin Dynamics for Deep Neural Networks. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.1512.07666)\n15. [Titsias, Michalis K. (2023). Optimal Preconditioning and Fisher Adaptive Langevin Sampling. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.2305.14442)\n16. [Optimal Preconditioning and Fisher Adaptive Langevin Sampling (FisherMALA)](https://papers.nips.cc/paper_files/paper/2023/file/5da6d5818a156791090c875abeca3cf8-Paper-Conference.pdf)\n17. [Exploration of the (Non-)Asymptotic Bias and Variance of Stochastic Gradient Langevin Dynamics](https://jmlr.org/papers/volume17/15-494/15-494.pdf)\n18. [Convergence in total variation for the kinetic Langevin algorithm](https://arxiv.org/pdf/2407.09301)\n19. [High-accuracy sampling from constrained spaces with the Metropolis-adjusted Preconditioned Langevin Algorithm](https://arxiv.org/abs/2412.18701)\n20. [Adaptive stepsize algorithms for Langevin dynamics](https://arxiv.org/html/2403.11993v2)\n21. [Preconditioned Langevin Dynamics with Score-Based Generative Models for Infinite-Dimensional Linear Bayesian Inverse Problems](https://proceedings.neurips.cc/paper_files/paper/2025/file/9aa618b7d721666ca8eadca5d161a1d4-Paper-Conference.pdf)\n22. [Metropolis-Adjusted Diffusion Models](https://arxiv.org/html/2605.09654v1)\n23. [Log-concave sampling: Metropolis-Hastings algorithms are fast! (Dwivedi et al.)](https://proceedings.mlr.press/v75/dwivedi18a/dwivedi18a.pdf)\n24. [The True Cost of SGLD](https://ar5iv.labs.arxiv.org/html/1706.02692)\n25. [No Free Lunch for Stochastic Gradient Langevin Dynamics (arXiv 2412.01952)](https://ar5iv.labs.arxiv.org/html/2412.01952)\n26. [Robustness of Langevin dynamics to score estimation error (lower bounds)](https://arxiv.org/pdf/2603.11319)\n27. [Unbiased Kinetic Langevin Monte Carlo with Inexact gradients](https://arxiv.org/abs/2311.05025)\n28. [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://arxiv.org/html/1710.00095v4)\n29. [Bounding the error of discretized Langevin algorithms for non-strongly log-concave targets](https://arxiv.org/html/1906.08530v3)\n30. [projecteuclid.org](https://projecteuclid.org/journals/bernoulli/volume-2/issue-4/Exponential-convergence-of-Langevin-distributions-and-their-discrete-approximations/bj/1178291835.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 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026 · Last review: Sep 30, 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": [], "url": "https://www.edgepedia.org/langevin-algorithm", "markdown_url": "https://www.edgepedia.org/langevin-algorithm.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": "\"Langevin algorithm\", Edgepedia (EdgeChat), https://www.edgepedia.org/langevin-algorithm. Edgepedia Community License 1.0.", "credit_md": "\"[Langevin algorithm](https://www.edgepedia.org/langevin-algorithm)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/langevin-algorithm](https://www.edgepedia.org/langevin-algorithm). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Langevin algorithm\", Edgepedia (EdgeChat), https://www.edgepedia.org/langevin-algorithm. Edgepedia Community License 1.0.", "speakable": "The Langevin algorithm is a Markov chain Monte Carlo method that samples a probability distribution by following the gradient of its log-density with added Gaussian noise." }