{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T05:21:59Z","timestamp":1768281719737,"version":"3.49.0"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,12,6]],"date-time":"2021-12-06T00:00:00Z","timestamp":1638748800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,12,6]],"date-time":"2021-12-06T00:00:00Z","timestamp":1638748800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100003343","name":"Cambridge Commonwealth, European and International Trust","doi-asserted-by":"publisher","award":["Cambridge International Scholarship"],"award-info":[{"award-number":["Cambridge International Scholarship"]}],"id":[{"id":"10.13039\/501100003343","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000265","name":"UK Medical Research Council","doi-asserted-by":"crossref","award":["MC_UU_00002\/2"],"award-info":[{"award-number":["MC_UU_00002\/2"]}],"id":[{"id":"10.13039\/501100000265","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2022,2,15]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Bayesian modelling enables us to accommodate complex forms of data and make a comprehensive inference, but the effect of partial misspecification of the model is a concern. One approach in this setting is to modularize the model and prevent feedback from suspect modules, using a cut model. After observing data, this leads to the cut distribution which normally does not have a closed form. Previous studies have proposed algorithms to sample from this distribution, but these algorithms have unclear theoretical convergence properties. To address this, we propose a new algorithm called the stochastic approximation cut (SACut) algorithm as an alternative. The algorithm is divided into two parallel chains. The main chain targets an approximation to the cut distribution; the auxiliary chain is used to form an adaptive proposal distribution for the main chain. We prove convergence of the samples drawn by the proposed algorithm and present the exact limit. Although SACut is biased, since the main chain does not target the exact cut distribution, we prove this bias can be reduced geometrically by increasing a user-chosen tuning parameter. In addition, parallel computing can be easily adopted for SACut, which greatly reduces computation time.<\/jats:p>","DOI":"10.1007\/s11222-021-10070-2","type":"journal-article","created":{"date-parts":[[2021,12,6]],"date-time":"2021-12-06T13:02:51Z","timestamp":1638795771000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Stochastic approximation cut algorithm for inference in modularized Bayesian models"],"prefix":"10.1007","volume":"32","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7221-5877","authenticated-orcid":false,"given":"Yang","family":"Liu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert J. B.","family":"Goudie","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,12,6]]},"reference":[{"key":"10070_CR1","doi-asserted-by":"crossref","unstructured":"Atchad\u00e9, Y., Fort, G., Moulines, E., Priouret, P.: Adaptive Markov chain Monte Carlo: theory and methods. In: Barber, D., Cemgil, A.T., Chiappa, S. (Eds.) Bayesian Time Series Models, pp. 32\u201351. Cambridge University Press (2011)","DOI":"10.1017\/CBO9780511984679.003"},{"issue":"1","key":"10070_CR2","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1214\/18-AOS1712","volume":"47","author":"A Bhattacharya","year":"2019","unstructured":"Bhattacharya, A., Pati, D., Yang, Y.: Bayesian fractional posteriors. Ann. Stat. 47(1), 39\u201366 (2019)","journal-title":"Ann. Stat."},{"issue":"2","key":"10070_CR3","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1016\/j.atmosenv.2010.10.003","volume":"45","author":"M Blangiardo","year":"2011","unstructured":"Blangiardo, M., Hansell, A., Richardson, S.: A Bayesian model of time activity data to investigate health effect of air pollution in time series studies. Atmos. Environ. 45(2), 379\u2013386 (2011)","journal-title":"Atmos. Environ."},{"issue":"4","key":"10070_CR4","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1080\/10618600.1998.10474787","volume":"7","author":"SP Brooks","year":"1998","unstructured":"Brooks, S.P., Gelman, A.: General methods for monitoring convergence of iterative simulations. J. Comput. Graph. Stat. 7(4), 434\u2013455 (1998)","journal-title":"J. Comput. Graph. Stat."},{"key":"10070_CR5","unstructured":"Carmona, C.U., Nicholls, G.K.: Semi-modular inference: enhanced learning in multi-modular models by tempering the influence of components. In: Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, pp. 4226-4235. PMLR (2020)"},{"issue":"1","key":"10070_CR6","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1023\/A:1015790929604","volume":"4","author":"JC Fu","year":"2002","unstructured":"Fu, J.C., Wang, L.: A random-discretization based Monte Carlo sampling method and its applications. Methodol. Comput. Appl. Probab. 4(1), 5\u201325 (2002)","journal-title":"Methodol. Comput. Appl. Probab."},{"issue":"4","key":"10070_CR7","first-page":"457","volume":"7","author":"A Gelman","year":"1992","unstructured":"Gelman, A., Rubin, D.B.: Inference from iterative simulation using multiple sequences. Stat. Sci. 7(4), 457\u2013472 (1992)","journal-title":"Stat. Sci."},{"issue":"4","key":"10070_CR8","doi-asserted-by":"publisher","first-page":"949","DOI":"10.1198\/106186008X386102","volume":"17","author":"R Gottardo","year":"2008","unstructured":"Gottardo, R., Raftery, A.E.: Markov chain Monte Carlo with mixtures of mutually singular distributions. J. Comput. Graph. Stat. 17(4), 949\u2013975 (2008)","journal-title":"J. Comput. Graph. Stat."},{"issue":"2","key":"10070_CR9","doi-asserted-by":"publisher","first-page":"223","DOI":"10.2307\/3318737","volume":"7","author":"H Haario","year":"2001","unstructured":"Haario, H., Saksman, E., Tamminen, J.: An adaptive Metropolis algorithm. Bernoulli 7(2), 223\u2013242 (2001)","journal-title":"Bernoulli"},{"issue":"3","key":"10070_CR10","doi-asserted-by":"publisher","first-page":"383","DOI":"10.1080\/13658810802672469","volume":"24","author":"B Huang","year":"2010","unstructured":"Huang, B., Wu, B., Barry, M.: Geographically and temporally weighted regression for modeling spatio-temporal variation in house prices. Int. J. Geogr. Inf. Sci. 24(3), 383\u2013401 (2010)","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"10070_CR11","unstructured":"Jacob, P.E., Murray, L.M., Holmes, C.C., Robert, C.P.: Better together? Statistical learning in models made of modules. Preprint arXiv:1708.08719 (2017)"},{"key":"10070_CR12","doi-asserted-by":"crossref","unstructured":"Jacob, P.E., O\u2019Leary, J., Atchad\u00e9, Y.F.: Unbiased Markov chain Monte Carlo methods with couplings. J. R. Stat. Soc. B 6, 66 (2020)","DOI":"10.1111\/rssb.12336"},{"issue":"459","key":"10070_CR13","doi-asserted-by":"publisher","first-page":"807","DOI":"10.1198\/016214502388618618","volume":"97","author":"F Liang","year":"2002","unstructured":"Liang, F.: Dynamically weighted importance sampling in Monte Carlo computation. J. Am. Stat. Assoc. 97(459), 807\u2013821 (2002)","journal-title":"J. Am. Stat. Assoc."},{"issue":"9","key":"10070_CR14","doi-asserted-by":"publisher","first-page":"1007","DOI":"10.1080\/00949650902882162","volume":"80","author":"F Liang","year":"2010","unstructured":"Liang, F.: A double Metropolis\u2013Hastings sampler for spatial models with intractable normalizing constants. J. Stat. Comput. Simul. 80(9), 1007\u20131022 (2010)","journal-title":"J. Stat. Comput. Simul."},{"issue":"477","key":"10070_CR15","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1198\/016214506000001202","volume":"102","author":"F Liang","year":"2007","unstructured":"Liang, F., Liu, C., Carroll, R.J.: Stochastic approximation in Monte Carlo computation. J. Am. Stat. Assoc. 102(477), 305\u2013320 (2007)","journal-title":"J. Am. Stat. Assoc."},{"issue":"513","key":"10070_CR16","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1080\/01621459.2015.1009072","volume":"111","author":"F Liang","year":"2016","unstructured":"Liang, F., Jin, I.H., Song, Q., Liu, J.S.: An adaptive exchange algorithm for sampling from distributions with intractable normalizing constants. J. Am. Stat. Assoc. 111(513), 377\u2013393 (2016)","journal-title":"J. Am. Stat. Assoc."},{"issue":"1","key":"10070_CR17","doi-asserted-by":"publisher","first-page":"112","DOI":"10.1111\/j.2041-210X.2011.00131.x","volume":"3","author":"WA Link","year":"2012","unstructured":"Link, W.A., Eaton, M.J.: On thinning of chains in MCMC. Methods Ecol. Evol. 3(1), 112\u2013115 (2012)","journal-title":"Methods Ecol. Evol."},{"issue":"1","key":"10070_CR18","first-page":"119","volume":"4","author":"F Liu","year":"2009","unstructured":"Liu, F., Bayarri, M., Berger, J.: Modularization in Bayesian analysis, with emphasis on analysis of computer models. Bayesian Anal. 4(1), 119\u2013150 (2009)","journal-title":"Bayesian Anal."},{"issue":"1","key":"10070_CR19","doi-asserted-by":"publisher","first-page":"1417","DOI":"10.1038\/s41598-018-19772-6","volume":"8","author":"Y Liu","year":"2018","unstructured":"Liu, Y., Lam, K.-F., Wu, J.T., Lam, T.T.-Y.: Geographically weighted temporally correlated logistic regression model. Sci. Rep. 8(1), 1417 (2018)","journal-title":"Sci. Rep."},{"key":"10070_CR20","doi-asserted-by":"crossref","unstructured":"Lunn, D., Best, N., Spiegelhalter, D., Graham, G., Neuenschwander, B.: Combining MCMC with \u2018sequential\u2019 PKPD modelling. J. Pharmacokinet Phar. 36(1), 19 (2009a)","DOI":"10.1007\/s10928-008-9109-1"},{"key":"10070_CR21","doi-asserted-by":"crossref","unstructured":"Lunn, D., Spiegelhalter, D., Thomas, A., Best, N.: The BUGS project: evolution, critique and future directions. Stat. Med. 28(25), 3049\u20133067 (2009b)","DOI":"10.1002\/sim.3680"},{"key":"10070_CR22","doi-asserted-by":"crossref","unstructured":"Malefaki, S., Iliopoulos, G.: Simulation from a target distribution based on discretization and weighting. Commun. Stat. Simul. Comput. 38(4), 829\u2013845 (2009)","DOI":"10.1080\/03610910802657904"},{"issue":"3","key":"10070_CR23","doi-asserted-by":"publisher","first-page":"717","DOI":"10.1158\/1055-9965.EPI-07-2691","volume":"17","author":"D Maucort-Boulch","year":"2008","unstructured":"Maucort-Boulch, D., Franceschi, S., Plummer, M.: International correlation between human papillomavirus prevalence and cervical cancer incidence. Cancer. Epidem. Biomar. 17(3), 717\u2013720 (2008)","journal-title":"Cancer. Epidem. Biomar."},{"issue":"2","key":"10070_CR24","doi-asserted-by":"publisher","first-page":"16","DOI":"10.2202\/1557-4679.1205","volume":"6","author":"LC McCandless","year":"2010","unstructured":"McCandless, L.C., Douglas, I.J., Evans, S.J., Smeeth, L.: Cutting feedback in Bayesian regression adjustment for the propensity score. Int. J. Biostat. 6(2), 16 (2010)","journal-title":"Int. J. Biostat."},{"issue":"4","key":"10070_CR25","doi-asserted-by":"publisher","first-page":"981","DOI":"10.1214\/aoap\/1177004900","volume":"4","author":"SP Meyn","year":"1994","unstructured":"Meyn, S.P., Tweedie, R.L.: Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Probab. 4(4), 981\u20131011 (1994)","journal-title":"Ann. Appl. Probab."},{"key":"10070_CR26","doi-asserted-by":"crossref","unstructured":"Meyn, S., Tweedie, R.L., Glynn, P.W.: Markov Chains and Stochastic Stability, 2nd edn. Cambridge Mathematical Library. Cambridge University Press (2009)","DOI":"10.1017\/CBO9780511626630"},{"issue":"527","key":"10070_CR27","doi-asserted-by":"publisher","first-page":"1113","DOI":"10.1080\/01621459.2018.1469995","volume":"114","author":"JW Miller","year":"2019","unstructured":"Miller, J.W., Dunson, D.B.: Robust Bayesian inference via coarsening. J. Am. Stat. Assoc. 114(527), 1113\u20131125 (2019)","journal-title":"J. Am. Stat. Assoc."},{"issue":"2","key":"10070_CR28","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1093\/biomet\/93.2.451","volume":"93","author":"J M\u00f8ller","year":"2006","unstructured":"M\u00f8ller, J., Pettitt, A.N., Reeves, R., Berthelsen, K.K.: An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants. Biometrika 93(2), 451\u2013458 (2006)","journal-title":"Biometrika"},{"key":"10070_CR29","unstructured":"Murray, I., Ghahramani, Z., MacKay, D.J.C.: MCMC for doubly-intractable distributions. In: Proceedings of the Twenty-Second Conference on Uncertainty in Artificial Intelligence, UAI\u201906, pp. 359\u2013366. AUAI Press, Arlington, VA, USA (2006)"},{"issue":"17","key":"10070_CR30","doi-asserted-by":"publisher","first-page":"2695","DOI":"10.1002\/sim.2129","volume":"24","author":"T Nakaya","year":"2005","unstructured":"Nakaya, T., Fotheringham, A.S., Brunsdon, C., Charlton, M.: Geographically weighted Poisson regression for disease association mapping. Stat. Med. 24(17), 2695\u20132717 (2005)","journal-title":"Stat. Med."},{"issue":"523","key":"10070_CR31","doi-asserted-by":"publisher","first-page":"1372","DOI":"10.1080\/01621459.2018.1448824","volume":"113","author":"J Park","year":"2018","unstructured":"Park, J., Haran, M.: Bayesian inference in the presence of intractable normalizing functions. J. Am. Stat. Assoc. 113(523), 1372\u20131390 (2018)","journal-title":"J. Am. Stat. Assoc."},{"issue":"1","key":"10070_CR32","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1007\/s11222-014-9503-z","volume":"25","author":"M Plummer","year":"2015","unstructured":"Plummer, M.: Cuts in Bayesian graphical models. Stat. Comput. 25(1), 37\u201343 (2015)","journal-title":"Stat. Comput."},{"issue":"1","key":"10070_CR33","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1093\/biomet\/83.1.95","volume":"83","author":"GO Roberts","year":"1996","unstructured":"Roberts, G.O., Tweedie, R.L.: Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83(1), 95\u2013110 (1996)","journal-title":"Biometrika"},{"issue":"10","key":"10070_CR34","doi-asserted-by":"publisher","first-page":"1621","DOI":"10.1016\/j.jspi.2013.05.013","volume":"143","author":"SG Walker","year":"2013","unstructured":"Walker, S.G.: Bayesian inference with misspecified models. J. Stat. Plan. Inference 143(10), 1621\u20131633 (2013)","journal-title":"J. Stat. Plan. Inference"},{"key":"10070_CR35","doi-asserted-by":"crossref","unstructured":"Zigler, C.M.: The central role of Bayes\u2019 theorem for joint estimation of causal effects and propensity scores. Am. Stat. 70(1), 47\u201354 (2016)","DOI":"10.1080\/00031305.2015.1111260"}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-021-10070-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11222-021-10070-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-021-10070-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,13]],"date-time":"2024-09-13T21:24:38Z","timestamp":1726262678000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11222-021-10070-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,6]]},"references-count":35,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,2,15]]}},"alternative-id":["10070"],"URL":"https:\/\/doi.org\/10.1007\/s11222-021-10070-2","relation":{},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"value":"0960-3174","type":"print"},{"value":"1573-1375","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,12,6]]},"assertion":[{"value":"22 April 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 November 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 December 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"7"}}