{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T21:36:21Z","timestamp":1773524181634,"version":"3.50.1"},"reference-count":45,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,4,15]],"date-time":"2021-04-15T00:00:00Z","timestamp":1618444800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We conduct a case study in which we empirically illustrate the performance of different classes of Bayesian inference methods to estimate stochastic volatility models. In particular, we consider how different particle filtering methods affect the variance of the estimated likelihood. We review and compare particle Markov Chain Monte Carlo (MCMC), RMHMC, fixed-form variational Bayes, and integrated nested Laplace approximation to estimate the posterior distribution of the parameters. Additionally, we conduct the review from the point of view of whether these methods are (1) easily adaptable to different model specifications; (2) adaptable to higher dimensions of the model in a straightforward way; (3) feasible in the multivariate case. We show that when using the stochastic volatility model for methods comparison, various data-generating processes have to be considered to make a fair assessment of the methods. Finally, we present a challenging specification of the multivariate stochastic volatility model, which is rarely used to illustrate the methods but constitutes an important practical application.<\/jats:p>","DOI":"10.3390\/e23040466","type":"journal-article","created":{"date-parts":[[2021,4,15]],"date-time":"2021-04-15T12:11:00Z","timestamp":1618488660000},"page":"466","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["\u201cExact\u201d and Approximate Methods for Bayesian Inference: Stochastic Volatility Case Study"],"prefix":"10.3390","volume":"23","author":[{"given":"Yuliya","family":"Shapovalova","sequence":"first","affiliation":[{"name":"Institute for Computing and Information Sciences, Radboud University Nijmegen, Toernooiveld 212, 6525 EC Nijmegen, The Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Shephard, N., and Torben, G. (2009). Stochastic volatility: Origins and overview. Handbook of Financial Time Series, Springer.","DOI":"10.1007\/978-3-540-71297-8_10"},{"key":"ref_2","unstructured":"Platanioti, K., McCoy, E., and Stephens, D. (2005). A Review of Stochastic Volatility: Univariate and Multivariate Models, Imperial College London. Technical Report, Working Paper."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1080\/07474930600713564","article-title":"Multivariate stochastic volatility: A review","volume":"25","author":"Asai","year":"2006","journal-title":"Econom. Rev."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1111\/j.1467-9868.2009.00736.x","article-title":"Particle Markov chain Monte Carlo methods","volume":"72","author":"Andrieu","year":"2010","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1111\/j.1467-9868.2010.00765.x","article-title":"Riemann Manifold Langevin and Hamiltonian Monte Carlo methods","volume":"73","author":"Girolami","year":"2011","journal-title":"J. R. Stat. Soc. Ser. B (Stat. Methodol.)"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"837","DOI":"10.1214\/13-BA858","article-title":"Fixed-form variational posterior approximation through stochastic linear regression","volume":"8","author":"Salimans","year":"2013","journal-title":"Bayesian Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1111\/j.1467-9868.2008.00700.x","article-title":"Approximate Bayesian inference for latent Gaussian models by using Integrated Nested Laplace Approximations","volume":"71","author":"Rue","year":"2009","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Mandelbrot, B.B. (1997). The variation of certain speculative prices. Fractals and Scaling in Finance, Springer.","DOI":"10.1007\/978-1-4757-2763-0"},{"key":"ref_9","unstructured":"Black, F. Studies of stock price volatility changes. Proceedings of the 1976 Meeting of the Business and Economic Statistics Section."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1086\/260062","article-title":"The pricing of options and corporate liabilities","volume":"81","author":"Black","year":"1973","journal-title":"J. Political Econ."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1111\/j.1540-6261.1987.tb02568.x","article-title":"The pricing of options on assets with stochastic volatilities","volume":"42","author":"Hull","year":"1987","journal-title":"J. Financ."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"143","DOI":"10.2307\/2330709","article-title":"Option pricing when the variance is changing","volume":"22","author":"Johnson","year":"1987","journal-title":"J. Financ. Quant. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1016\/0304-405X(87)90009-2","article-title":"Option values under stochastic volatility: Theory and empirical estimates","volume":"19","author":"Wiggins","year":"1987","journal-title":"J. Financ. Econ."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1093\/rfs\/6.2.327","article-title":"A closed-form solution for options with stochastic volatility with applications to bond and currency options","volume":"6","author":"Heston","year":"1993","journal-title":"Rev. Financ. Stud."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Campbell, J.Y., Champbell, J.J., Campbell, J.W., Lo, A.W., Lo, A.W., and MacKinlay, A.C. (1997). The Econometrics of Financial Markets, Princeton University Press.","DOI":"10.1515\/9781400830213"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1080\/07350015.1996.10524672","article-title":"Estimation of an asymmetric stochastic volatility model for asset returns","volume":"14","author":"Harvey","year":"1996","journal-title":"J. Bus. Econ. Stat."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/j.jeconom.2003.09.001","article-title":"Bayesian analysis of stochastic volatility models with fat-tails and correlated errors","volume":"122","author":"Jacquier","year":"2004","journal-title":"J. Econom."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/j.jeconom.2004.08.002","article-title":"On leverage in a stochastic volatility model","volume":"127","author":"Yu","year":"2005","journal-title":"J. Econom."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1111\/1467-937X.00050","article-title":"Stochastic volatility: Likelihood inference and comparison with ARCH models","volume":"65","author":"Kim","year":"1998","journal-title":"Rev. Econ. Stud."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1049\/ip-f-2.1993.0015","article-title":"Novel approach to nonlinear\/non-Gaussian Bayesian state estimation","volume":"140","author":"Gordon","year":"1993","journal-title":"IEE Proc. F (Radar Signal Process.)"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Doucet, A., de Freitas, N., and Gordon, N. (2001). Sequential Monte Carlo Methods in Practice, Springer.","DOI":"10.1007\/978-1-4757-3437-9"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2385","DOI":"10.1214\/009053604000000698","article-title":"Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference","volume":"32","author":"Chopin","year":"2004","journal-title":"Ann. Stat."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1080\/01621459.1999.10474153","article-title":"Filtering via simulation: Auxiliary particle filters","volume":"94","author":"Pitt","year":"1999","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1016\/j.jeconom.2015.03.047","article-title":"Particle efficient importance sampling","volume":"190","author":"Scharth","year":"2016","journal-title":"J. Econom."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1385","DOI":"10.1016\/j.jeconom.2007.02.007","article-title":"Efficient high-dimensional importance sampling","volume":"141","author":"Richard","year":"2007","journal-title":"J. Econom."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1636","DOI":"10.1080\/01621459.2016.1222291","article-title":"The iterated auxiliary particle filter","volume":"112","author":"Guarniero","year":"2017","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1498","DOI":"10.1016\/j.spl.2008.01.032","article-title":"A note on auxiliary particle filters","volume":"78","author":"Johansen","year":"2008","journal-title":"Stat. Probab. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"S\u00e4rkk\u00e4, S. (2013). Bayesian Filtering and Smoothing, Cambridge University Press.","DOI":"10.1017\/CBO9781139344203"},{"key":"ref_29","first-page":"110","article-title":"Weak convergence and optimal scaling of random walk Metropolis algorithms","volume":"7","author":"Roberts","year":"1997","journal-title":"Ann. Appl. Probab."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Robert, C.P., and Casella, G. (2005). Monte Carlo Statistical Methods, Springer. Springer Texts in Statistics.","DOI":"10.1007\/978-1-4757-4145-2"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1016\/0370-2693(87)91197-X","article-title":"Hybrid Monte Carlo","volume":"195","author":"Duane","year":"1987","journal-title":"Phys. Lett. B"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Neal, R.M. (2011). MCMC Using Hamiltonian Dynamics. Handbook of Markov Chain Monte Carlo, CRC Press.","DOI":"10.1201\/b10905-6"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Betancourt, M. (2017). A Conceptual Introduction to Hamiltonian Monte Carlo. arXiv.","DOI":"10.3150\/16-BEJ810"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1111\/1467-9868.00123","article-title":"Optimal scaling of discrete approximations to Langevin diffusions","volume":"60","author":"Roberts","year":"1998","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"530","DOI":"10.3102\/1076998615606113","article-title":"Stan: A probabilistic programming language for Bayesian inference and optimization","volume":"40","author":"Gelman","year":"2015","journal-title":"J. Educ. Behav. Stat."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"e55","DOI":"10.7717\/peerj-cs.55","article-title":"Probabilistic programming in Python using PyMC3","volume":"2","author":"Salvatier","year":"2016","journal-title":"PeerJ Comput. Sci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1080\/1351847X.2010.495475","article-title":"Estimating stochastic volatility models using integrated nested Laplace approximations","volume":"17","author":"Martino","year":"2011","journal-title":"Eur. J. Financ."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1080\/03610918.2013.790444","article-title":"Bayesian Estimation and Prediction of Stochastic Volatility Models via INLA","volume":"44","author":"Ehlers","year":"2015","journal-title":"Commun. Stat.-Simul. Comput."},{"key":"ref_39","unstructured":"Martino, S. (2007). Approximate Bayesian Inference for Multivariate Stochastic Volatility Models, Department of Mathematical Sciences, Norwegian University of Science and Technology. Technical Report."},{"key":"ref_40","unstructured":"\u0160m\u00eddl, V., and Quinn, A. (2006). The Variational Bayes Method in Signal Processing, Springer Science & Business Media."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/0304-4076(93)01569-8","article-title":"Quasi-maximum likelihood estimation of stochastic volatility models","volume":"63","author":"Ruiz","year":"1994","journal-title":"J. Econom."},{"key":"ref_42","unstructured":"Naesseth, C., Linderman, S., Ranganath, R., and Blei, D. (2018, January 9\u201311). Variational Sequential Monte Carlo. Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics, Playa Blanca, Lanzarote, Canary Islands, Spain."},{"key":"ref_43","first-page":"417","article-title":"Approximate marginals in latent Gaussian models","volume":"12","author":"Cseke","year":"2011","journal-title":"J. Mach. Learn. Res."},{"key":"ref_44","first-page":"1593","article-title":"The No-U-Turn sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo","volume":"15","author":"Hoffman","year":"2014","journal-title":"J. Mach. Learn. Res."},{"key":"ref_45","unstructured":"Stan Development Team (2021, April 06). RStan: The R Interface to Stan. R Package Version 2.21.2. Available online: http:\/\/mc-stan.org\/."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/4\/466\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:48:29Z","timestamp":1760161709000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/4\/466"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,15]]},"references-count":45,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,4]]}},"alternative-id":["e23040466"],"URL":"https:\/\/doi.org\/10.3390\/e23040466","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,15]]}}}