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These methods replace the likelihood with an unbiased particle filter estimate, similarly to particle Markov chain Monte Carlo (MCMC). As with particle MCMC, the efficiency of SMC<jats:inline-formula><jats:alternatives><jats:tex-math>$$^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow\/>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> greatly depends on the variance of the likelihood estimator, and therefore on the number of state particles used within the particle filter. We introduce novel methods to adaptively select the number of state particles within SMC<jats:inline-formula><jats:alternatives><jats:tex-math>$$^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow\/>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> using the expected squared jumping distance to trigger the adaptation, and modifying the exchange importance sampling method of Chopin et al. (J R Stat Soc: Ser B (Stat Method) 75(3):397\u2013426, 2012) to replace the current set of state particles with the new set of state particles. The resulting algorithm is fully automatic, and can significantly improve current methods. Code for our methods is available at <jats:ext-link xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" ext-link-type=\"uri\" xlink:href=\"https:\/\/github.com\/imkebotha\/adaptive-exact-approximate-smc\">https:\/\/github.com\/imkebotha\/adaptive-exact-approximate-smc<\/jats:ext-link>.<\/jats:p>","DOI":"10.1007\/s11222-023-10250-2","type":"journal-article","created":{"date-parts":[[2023,5,29]],"date-time":"2023-05-29T04:02:13Z","timestamp":1685332933000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Automatically adapting the number of state particles in SMC$$^2$$"],"prefix":"10.1007","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9222-6693","authenticated-orcid":false,"given":"Imke","family":"Botha","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Robert","family":"Kohn","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Leah","family":"South","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christopher","family":"Drovandi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,5,29]]},"reference":[{"issue":"3","key":"10250_CR1","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1111\/j.1467-9868.2009.00736.x","volume":"72","author":"C Andrieu","year":"2010","unstructured":"Andrieu, C., Doucet, A., Holenstein, R.: Particle Markov chain Monte Carlo methods. 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