{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:30:34Z","timestamp":1760243434780,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2013,4,8]],"date-time":"2013-04-08T00:00:00Z","timestamp":1365379200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Studies of learning algorithms typically concentrate on situations where potentially ever growing training sample is available. Yet, there can be situations (e.g., detection of differentially expressed genes on unreplicated data or estimation of time delay in non-stationary gravitationally lensed photon streams) where only extremely small samples can be used in order to perform an inference. On unreplicated data, the inference has to be performed on the smallest sample possible\u2014sample of size 1. We study whether anything useful can be learnt in such extreme situations by concentrating on a Bayesian approach that can account for possible prior information on expected counts. We perform a detailed information theoretic study of such Bayesian estimation and quantify the effect of Bayesian averaging on its first two moments. Finally, to analyze potential benefits of the Bayesian approach, we also consider Maximum Likelihood (ML) estimation as a baseline approach. We show both theoretically and empirically that the Bayesian model averaging can be potentially beneficial.<\/jats:p>","DOI":"10.3390\/e15041202","type":"journal-article","created":{"date-parts":[[2013,4,8]],"date-time":"2013-04-08T12:01:31Z","timestamp":1365422491000},"page":"1202-1220","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Pushing for the Extreme: Estimation of Poisson Distribution from Low Count Unreplicated Data\u2014How Close Can We Get?"],"prefix":"10.3390","volume":"15","author":[{"given":"Peter","family":"Ti\u0148o","sequence":"first","affiliation":[{"name":"School of Computer Science, The University of Birmingham, Birmingham, B15 2TT, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2013,4,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Varuzza, L., Gruber, A., and de B. Pereira, C. (2008). Significance tests for comparing digital gene expression profiles. Nat. 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Methodology and analysis of optical photometric Data","volume":"385","author":"Press","year":"1992","journal-title":"Astrophys. J."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"e310","DOI":"10.1186\/1471-2105-10-310","article-title":"Basic properties and information theory of audic-claverie statistic for analyzing cDNA arrays","volume":"10","year":"2009","journal-title":"BMC Bioinform."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Liu, D., Zhang, H., Polycarpou, M., Alippi, C., and He, H. (\u2013, January 29). One-shot Learning of Poisson Distributions in cDNA Array Analysis. Advances in Neural Networks, Proceedings of the 8th International Symposium on Neural Networks (ISNN 2011), Guilin, China. Lecture Notes in Computer Science (LNCS 6676).","DOI":"10.1007\/978-3-642-21111-9"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1534\/genetics.110.114983","article-title":"Statistical design and analysis of RNA sequencing data","volume":"185","author":"Auer","year":"2010","journal-title":"Genetics"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/15\/4\/1202\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:45:58Z","timestamp":1760219158000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/15\/4\/1202"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,4,8]]},"references-count":13,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2013,4]]}},"alternative-id":["e15041202"],"URL":"https:\/\/doi.org\/10.3390\/e15041202","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2013,4,8]]}}}