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(2021) proposed a Gibbs sampler for the M4PLM, but it is time\u2010consuming. In this paper, a mixture\u2010modelling\u2010based Bayesian MH\u2010RM (MM\u2010MH\u2010RM) algorithm is proposed for the M4PLM to obtain the maximum a posteriori (MAP) estimates. In a comparison of the MM\u2010MH\u2010RM algorithm to the original MH\u2010RM algorithm, two simulation studies and an empirical example demonstrated that the MM\u2010MH\u2010RM algorithm possessed the benefits of the mixture\u2010modelling approach and could produce more robust estimates with guaranteed convergence rates and fast computation. The MATLAB codes for the MM\u2010MH\u2010RM algorithm are available in the online appendix.<\/jats:p>","DOI":"10.1111\/bmsp.12300","type":"journal-article","created":{"date-parts":[[2023,2,3]],"date-time":"2023-02-03T01:06:26Z","timestamp":1675386386000},"page":"585-604","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["<scp>Mixture\u2010modelling\u2010based<\/scp>\n                    Bayesian\n                    <scp>MH\u2010RM<\/scp>\n                    algorithm for the multidimensional\n                    <scp>4PLM<\/scp>"],"prefix":"10.1111","volume":"76","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2480-8800","authenticated-orcid":false,"given":"Shaoyang","family":"Guo","sequence":"first","affiliation":[{"name":"Shanghai Institute of AI for Education and Department of Educational Psychology of Faculty of Education East China Normal University  Shanghai China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yanlei","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Educational Science Liaocheng University  Liaocheng China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1232-0020","authenticated-orcid":false,"given":"Chanjin","family":"Zheng","sequence":"additional","affiliation":[{"name":"Shanghai Institute of AI for Education and Department of Educational Psychology of Faculty of Education East China Normal University  Shanghai China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guiyu","family":"Li","sequence":"additional","affiliation":[{"name":"Faculty of Education, Institute of Curriculum &amp; Instruction East China Normal University  Shanghai China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2023,2,2]]},"reference":[{"key":"e_1_2_11_2_1","doi-asserted-by":"publisher","DOI":"10.1002\/j.2333\u20108504.1981.tb01255.x"},{"key":"e_1_2_11_3_1","doi-asserted-by":"publisher","DOI":"10.1007\/bf02296195"},{"key":"e_1_2_11_4_1","unstructured":"Cai L.(2008).A Metropolis\u2010Hastings Robbins\u2010Monro algorithm for maximum likelihood nonlinear latent structure analysis with a comprehensive measurement model. 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