{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,24]],"date-time":"2026-08-24T23:03:56Z","timestamp":1787612636176,"version":"build-2736575974"},"reference-count":45,"publisher":"Oxford University Press (OUP)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2008,3,15]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Motivation: There often are many alternative models of a biochemical system. Distinguishing models and finding the most suitable ones is an important challenge in Systems Biology, as such model ranking, by experimental evidence, will help to judge the support of the working hypotheses forming each model.<\/jats:p><jats:p>Bayes factors are employed as a measure of evidential preference for one model over another. Marginal likelihood is a key component of Bayes factors, however computing the marginal likelihood is a difficult problem, as it involves integration of nonlinear functions in multidimensional space. There are a number of methods available to compute the marginal likelihood approximately. A detailed investigation of such methods is required to find ones that perform appropriately for biochemical modelling.<\/jats:p><jats:p>Results: We assess four methods for estimation of the marginal likelihoods required for computing Bayes factors. The Prior Arithmetic Mean estimator, the Posterior Harmonic Mean estimator, the Annealed Importance Sampling and the Annealing-Melting Integration methods are investigated and compared on a typical case study in Systems Biology. This allows us to understand the stability of the analysis results and make reliable judgements in uncertain context. We investigate the variance of Bayes factor estimates, and highlight the stability of the Annealed Importance Sampling and the Annealing-Melting Integration methods for the purposes of comparing nonlinear models.<\/jats:p><jats:p>Availability: Models used in this study are available in SBML format as the supplementary material to this article.<\/jats:p><jats:p>Contact: \u00a0vvv@dcs.gla.ac.uk<\/jats:p><jats:p>Supplementary information: Supplementary data are available at Bioinformatics online.<\/jats:p>","DOI":"10.1093\/bioinformatics\/btm607","type":"journal-article","created":{"date-parts":[[2007,12,6]],"date-time":"2007-12-06T01:14:34Z","timestamp":1196903674000},"page":"833-839","source":"Crossref","is-referenced-by-count":137,"title":["Bayesian ranking of biochemical system models"],"prefix":"10.1093","volume":"24","author":[{"given":"Vladislav","family":"Vyshemirsky","sequence":"first","affiliation":[{"name":"Department of Computing Science, University of Glasgow, Glasgow, G12 8QQ, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mark A.","family":"Girolami","sequence":"additional","affiliation":[{"name":"Department of Computing Science, University of Glasgow, Glasgow, G12 8QQ, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2007,12,5]]},"reference":[{"key":"2023020209511395400_B1","first-page":"277","article-title":"Information measures and model selection","volume":"50","author":"Akaike","year":"1983","journal-title":"Bull. Int. Stat. Inst"},{"key":"2023020209511395400_B2","doi-asserted-by":"crossref","DOI":"10.1002\/9780470316870","volume-title":"Bayesian Theory.","author":"Bernardo","year":"1994"},{"key":"2023020209511395400_B3","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1086\/508766","article-title":"Bayesian Inference from Observations of Solar-like Oscillations","volume":"654","author":"Brewer","year":"2007","journal-title":"Astrophys. J"},{"key":"2023020209511395400_B4","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1088\/1478-3967\/1\/3\/006","article-title":"The statistical mechanics of complex signaling networks: nerve growth factor signaling","volume":"1","author":"Brown","year":"2004","journal-title":"Phys. Biol"},{"key":"2023020209511395400_B5","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1147\/rd.506.0529","article-title":"An assessment of the role of computing in systems biology","volume":"90","author":"Burbeck","year":"2006","journal-title":"IBM J. RES DEV"},{"key":"2023020209511395400_B6","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1007\/3-540-36481-1_11","article-title":"Mathematical modeling of the influence of RKIP on the ERK signaling pathway","volume":"2602","author":"Cho","year":"2003","journal-title":"Lect. Notes Comput. Sci"},{"key":"2023020209511395400_B7","first-page":"543","article-title":"Sample selection in radiocarbon dating","volume":"47","author":"Christen","year":"1998","journal-title":"Appl. Stat"},{"key":"2023020209511395400_B8","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1111\/j.2517-6161.1996.tb02091.x","article-title":"Coherent analysis of forensic identification evidence","volume":"58","author":"Dawid","year":"1996","journal-title":"J. R. Stat. Soc. [Ser B]"},{"key":"2023020209511395400_B9","first-page":"149","article-title":"Modeling and simulation of genetic regulatory systems: a literature review","volume":"2602","author":"de Jong","year":"2003","journal-title":"Lect. Notes Comput. Sci"},{"key":"2023020209511395400_B10","volume-title":"Bayesian Methods for Nonlinear Classification and Regression.","author":"Denison","year":"2002"},{"key":"2023020209511395400_B11","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1038\/35002125","article-title":"A synthetic oscillatory network of transcriptional regulators","volume":"403","author":"Elowitz","year":"2000","journal-title":"Nature"},{"key":"2023020209511395400_B12","article-title":"Marginal likelihood estimation via power posteriors","author":"Friel","year":"2006","journal-title":"Technical report"},{"key":"2023020209511395400_B13","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1214\/ss\/1028905934","article-title":"Simulating normalising constants: from importance sampling to bridge sampling to path sampling","volume":"13","author":"Gelman","year":"1998","journal-title":"Stat. Sci"},{"key":"2023020209511395400_B14","doi-asserted-by":"crossref","DOI":"10.1201\/9780429258411","volume-title":"Bayesian Data Analysis.","author":"Gelman","year":"1995"},{"key":"2023020209511395400_B15","doi-asserted-by":"crossref","first-page":"995","DOI":"10.7326\/0003-4819-130-12-199906150-00008","article-title":"Toward evidence-based medical statistics. 1 : the P value fallacy","volume":"130","author":"Goodman","year":"1999","journal-title":"Ann. Intern. Med"},{"key":"2023020209511395400_B16","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1016\/j.jtbi.2007.01.022","article-title":"Signal transduction network motifs and biological memory","volume":"246","author":"Han","year":"2007","journal-title":"J. Theor. Biol"},{"key":"2023020209511395400_B17","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1093\/biomet\/57.1.97","article-title":"Monte Carlo sampling methods using Markov chains and thier applications","volume":"57","author":"Hastings","year":"1970","journal-title":"Biometrika"},{"key":"2023020209511395400_B18","doi-asserted-by":"crossref","first-page":"2596","DOI":"10.1093\/bioinformatics\/btm367","article-title":"Bayesian inference for dynamic transcriptional regulation; the hes1 system as a case study","volume":"23","author":"Heron","year":"2007","journal-title":"Bioinformatics"},{"key":"2023020209511395400_B19","doi-asserted-by":"crossref","first-page":"3067","DOI":"10.1093\/bioinformatics\/btl485","article-title":"COPASI\u2014a COmplex PAthway SImulator","volume":"22","author":"Hoops","year":"2006","journal-title":"Bioinformatics"},{"key":"2023020209511395400_B20","doi-asserted-by":"crossref","first-page":"2271","DOI":"10.1093\/bioinformatics\/btg313","article-title":"Sensitivity and specificity of inferring genetic regulatory interactions from microarray experiments with dynamic Bayesian networks","volume":"19","author":"Husmeier","year":"2003","journal-title":"Bioinformatics"},{"key":"2023020209511395400_B21","doi-asserted-by":"crossref","first-page":"1986","DOI":"10.1093\/bioinformatics\/btm283","article-title":"Non-parametric quantification of protein lysate arrays","volume":"23","author":"Hu","year":"2007","journal-title":"Bioinformatics"},{"key":"2023020209511395400_B22","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511790423","volume-title":"Probability Theory: The Logic Of Science.","author":"Jaynes","year":"2003"},{"key":"2023020209511395400_B23","volume-title":"Theory of Probability.","author":"Jeffreys","year":"1961","edition":"3rd"},{"key":"2023020209511395400_B24","doi-asserted-by":"crossref","first-page":"18169","DOI":"10.1074\/jbc.M008870200","article-title":"Identification of the mechanisms regulating the differential activation of the MAPK cascade by epidermal growth factor and nerve growth factor in PC12 cells","volume":"276","author":"Kao","year":"2001","journal-title":"J. Biol. Chem"},{"key":"2023020209511395400_B25","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1126\/science.220.4598.671","article-title":"Optimization by simulated annealing","volume":"220","author":"Kirkpatrick","year":"1983","journal-title":"Science"},{"key":"2023020209511395400_B26","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1080\/10635150500433722","article-title":"Computing Bayes factors using thermodynamic integration","volume":"55","author":"Lartillot","year":"2006","journal-title":"Syst. Biol"},{"key":"2023020209511395400_B27","first-page":"648","article-title":"Estimating Bayes factors via posterior simulation with the Laplace-Metropolis estimator","volume":"92","author":"Lewis","year":"1997","journal-title":"J. Am. Stat. Assoc"},{"key":"2023020209511395400_B28","volume-title":"Information Theory, Inference, and Learning Algorithms.","author":"MacKay","year":"2003"},{"key":"2023020209511395400_B29","article-title":"Bayes factors for nonlinear hypotheses and likelihood distributions","author":"McCulloch","year":"1991","journal-title":"Technical Report 101"},{"issue":"Suppl. 2","key":"2023020209511395400_B30","doi-asserted-by":"crossref","first-page":"ii122","DOI":"10.1093\/bioinformatics\/btg1069","article-title":"Artificial gene networks for objective comparison of analysis algorithms","volume":"19","author":"Mendes","year":"2003","journal-title":"Bioinformatics"},{"key":"2023020209511395400_B31","article-title":"Probabilistic inference using Markov Chain Monte Carlo methods","author":"Neal","year":"1993","journal-title":"Technical Report CRG-TR-93-1"},{"key":"2023020209511395400_B32","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1023\/A:1008923215028","article-title":"Annealed importance sampling","volume":"11","author":"Neal","year":"2001","journal-title":"Statistics and Computing"},{"key":"2023020209511395400_B33","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1111\/j.2517-6161.1994.tb01956.x","article-title":"Approximate Bayesian inference by the weighted likelihood bootstrap","volume":"3","author":"Newton","year":"1994","journal-title":"JRSS Ser. B"},{"key":"2023020209511395400_B34","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/BF01406511","article-title":"A Monte Carlo method for high dimensional integration","volume":"55","author":"Ogata","year":"1989","journal-title":"Num. Math"},{"key":"2023020209511395400_B35","doi-asserted-by":"crossref","first-page":"145","DOI":"10.2307\/2095483","article-title":"Choosing models for cross-classifications","volume":"51","author":"Raftery","year":"1986","journal-title":"Am. Sociol. Rev"},{"key":"2023020209511395400_B36","first-page":"1","article-title":"Estimating the integrated likelihood via posterior simulation using the harmonic mean identity","volume":"8","author":"Raftery","year":"2007","journal-title":"Bayesian Stat"},{"key":"2023020209511395400_B37","volume-title":"Gaussian Processes for Machine Learning.","author":"Rasmussen","year":"2006"},{"issue":"Suppl. 2","key":"2023020209511395400_B38","doi-asserted-by":"crossref","first-page":"S2","DOI":"10.1186\/1471-2105-8-S2-S2","article-title":"Bayesian model-based inference of transcription factor activity","volume":"8","author":"Rogers","year":"2007","journal-title":"BMC Bioinformatics"},{"key":"2023020209511395400_B39","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1128\/MMBR.68.2.320-344.2004","article-title":"ERK and p38 MAPK-activated protein kinases: a family of protein kinases with diverse biological functions","volume":"68","author":"Roux","year":"2004","journal-title":"Microbiol. Mol. Biol. Rev"},{"key":"2023020209511395400_B40","doi-asserted-by":"crossref","first-page":"370","DOI":"10.1038\/nbt0402-370","article-title":"Computational modelling of the dynamics of the MAP kinase cascade activated by surface and internalised EGF receptors","volume":"20","author":"Schoeberl","year":"2002","journal-title":"Nat. Biotechnol"},{"key":"2023020209511395400_B41","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1111\/1467-9868.00353","article-title":"Bayesian measures of model complexity and fit (with discussion)","volume":"64","author":"Spiegelhalter","year":"2002","journal-title":"JRSS Ser. B"},{"key":"2023020209511395400_B42","volume-title":"Computational Analysis of Biochemical Systems.","author":"Voit","year":"2000"},{"key":"2023020209511395400_B43","doi-asserted-by":"crossref","DOI":"10.1186\/1742-4682-4-30","article-title":"A continuum mathematical model of endothelial layer maintenance and senescence","volume":"4","author":"Wang","year":"2007","journal-title":"Theor. Biol. Med. Model"},{"key":"2023020209511395400_B44","doi-asserted-by":"crossref","first-page":"2523","DOI":"10.1093\/bioinformatics\/btl391","article-title":"Comparative evaluation of reverse engineering gene regulatory networks with relevance networks, graphical Gaussian models and Bayesian networks","volume":"22","author":"Werhli","year":"2006","journal-title":"Bioinformatics"},{"key":"2023020209511395400_B45","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1093\/bib\/bbm007","article-title":"Bayesian methods in bioinformatics and computational systems biology","volume":"8","author":"Wilkinson","year":"2007","journal-title":"Briefings in bioinformatics"}],"container-title":["Bioinformatics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/bioinformatics\/article-pdf\/24\/6\/833\/49045615\/bioinformatics_24_6_833.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/bioinformatics\/article-pdf\/24\/6\/833\/49045615\/bioinformatics_24_6_833.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,1,23]],"date-time":"2025-01-23T07:50:11Z","timestamp":1737618611000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/bioinformatics\/article\/24\/6\/833\/192524"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,12,5]]},"references-count":45,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2008,3,15]]}},"URL":"https:\/\/doi.org\/10.1093\/bioinformatics\/btm607","relation":{},"ISSN":["1367-4811","1367-4803"],"issn-type":[{"value":"1367-4811","type":"electronic"},{"value":"1367-4803","type":"print"}],"subject":[],"published-other":{"date-parts":[[2008,3,15]]},"published":{"date-parts":[[2007,12,5]]}}}