{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/unification-of-field-theory-and-maximum","title":"Unification of field theory and maximum entropy methods for learning probability densities","arxiv_id":"1411.5371","date":"2014-11-19","proceeding":null,"authors":["Justin B. Kinney"],"abstract":"The need to estimate smooth probability distributions (a.k.a. probability\ndensities) from finite sampled data is ubiquitous in science. Many approaches\nto this problem have been described, but none is yet regarded as providing a\ndefinitive solution. Maximum entropy estimation and Bayesian field theory are\ntwo such approaches. Both have origins in statistical physics, but the\nrelationship between them has remained unclear. Here I unify these two methods\nby showing that every maximum entropy density estimate can be recovered in the\ninfinite smoothness limit of an appropriate Bayesian field theory. I also show\nthat Bayesian field theory estimation can be performed without imposing any\nboundary conditions on candidate densities, and that the infinite smoothness\nlimit of these theories recovers the most common types of maximum entropy\nestimates. Bayesian field theory is thus seen to provide a natural test of the\nvalidity of the maximum entropy null hypothesis. Bayesian field theory also\nreturns a lower entropy density estimate when the maximum entropy hypothesis is\nfalsified. The computations necessary for this approach can be performed\nrapidly for one-dimensional data, and software for doing this is provided.\nBased on these results, I argue that Bayesian field theory is poised to provide\na definitive solution to the density estimation problem in one dimension.","url_abs":"http://arxiv.org/abs/1411.5371v5","url_pdf":"http://arxiv.org/pdf/1411.5371v5.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"unification-of-field-theory-and-maximum","repo_url":"https://github.com/jbkinney/14_maxent","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"density-estimation","task_name":"Density Estimation"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}