{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T21:40:39Z","timestamp":1790372439041,"version":"4.1.0"},"reference-count":49,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,19]],"date-time":"2021-08-19T00:00:00Z","timestamp":1629331200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100004440","name":"Wellcome","doi-asserted-by":"publisher","award":["088130\/Z\/09\/Z"],"award-info":[{"award-number":["088130\/Z\/09\/Z"]}],"id":[{"id":"10.13039\/100004440","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001866","name":"Fonds National de la Recherche Luxembourg","doi-asserted-by":"publisher","award":["13568875"],"award-info":[{"award-number":["13568875"]}],"id":[{"id":"10.13039\/501100001866","id-type":"DOI","asserted-by":"publisher"}]},{"name":"EPSRC Centre for Doctoral Training in Mathematics of Random Systems: Analysis, Modelling and Simulation","award":["EP\/S023925\/1"],"award-info":[{"award-number":["EP\/S023925\/1"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Biehl et al. (2021) present some interesting observations on an early formulation of the free energy principle. We use these observations to scaffold a discussion of the technical arguments that underwrite the free energy principle. This discussion focuses on solenoidal coupling between various (subsets of) states in sparsely coupled systems that possess a Markov blanket\u2014and the distinction between exact and approximate Bayesian inference, implied by the ensuing Bayesian mechanics.<\/jats:p>","DOI":"10.3390\/e23081076","type":"journal-article","created":{"date-parts":[[2021,8,19]],"date-time":"2021-08-19T09:58:06Z","timestamp":1629367086000},"page":"1076","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":31,"title":["Some Interesting Observations on the Free Energy Principle"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7984-8909","authenticated-orcid":false,"given":"Karl J.","family":"Friston","sequence":"first","affiliation":[{"name":"The Wellcome Centre for Human Neuroimaging, University College London, London WC1N 3AR, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0126-4588","authenticated-orcid":false,"given":"Lancelot","family":"Da Costa","sequence":"additional","affiliation":[{"name":"The Wellcome Centre for Human Neuroimaging, University College London, London WC1N 3AR, UK"},{"name":"Department of Mathematics, Imperial College London, London SW7 2AZ, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5108-5743","authenticated-orcid":false,"given":"Thomas","family":"Parr","sequence":"additional","affiliation":[{"name":"The Wellcome Centre for Human Neuroimaging, University College London, London WC1N 3AR, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Biehl, M., Pollock, F., and Kanai, R. (2021). A Technical Critique of Some Parts of the Free Energy Principle. Entropy, 23.","DOI":"10.3390\/e23030293"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"20130475","DOI":"10.1098\/rsif.2013.0475","article-title":"Life as we know it","volume":"10","author":"Friston","year":"2013","journal-title":"J. R. Soc. Interface"},{"key":"ref_3","unstructured":"Friston, K. (2019). A free energy principle for a particular physics. arXiv."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"20190159","DOI":"10.1098\/rsta.2019.0159","article-title":"Markov blankets, information geometry and stochastic thermodynamics","volume":"378","author":"Parr","year":"2019","journal-title":"Philos. Trans. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1111\/nous.12062","article-title":"The Self-Evidencing Brain","volume":"50","author":"Hohwy","year":"2014","journal-title":"No\u00fbs"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3463","DOI":"10.1007\/s11229-018-01932-w","article-title":"First principles in the life sciences: The free-energy principle, organicism, and mechanism","volume":"198","author":"Colombo","year":"2018","journal-title":"Synthese"},{"key":"ref_7","unstructured":"Metzinger, T.K., and Wiese, W. (2017). How to Knit Your Own Markov Blanket. Philosophy and Predictive Processing, MIND Group."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"20170792","DOI":"10.1098\/rsif.2017.0792","article-title":"The Markov blankets of life: Autonomy, active inference and the free energy principle","volume":"15","author":"Kirchhoff","year":"2018","journal-title":"J. R. Soc. Interface"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"021912","DOI":"10.1103\/PhysRevE.68.021912","article-title":"Fokker-Planck perspective on stochastic delay systems: Exact solutions and data analysis of biological systems","volume":"68","author":"Frank","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/j.plrev.2005.03.002","article-title":"Laws in Darwinian evolutionary theory","volume":"2","author":"Ao","year":"2005","journal-title":"Phys. Life Rev."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"L25","DOI":"10.1088\/0305-4470\/37\/3\/L01","article-title":"Potential in stochastic differential equations: Novel construction","volume":"37","author":"Ao","year":"2004","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1073","DOI":"10.1088\/0253-6102\/49\/5\/01","article-title":"Emerging of Stochastic Dynamical Equalities and Steady State Thermodynamics from Darwinian Dynamics","volume":"49","author":"Ao","year":"2008","journal-title":"Commun. Theor. Phys."},{"key":"ref_13","first-page":"937860","article-title":"Free energy, value, and attractors. Comput","volume":"2012","author":"Friston","year":"2012","journal-title":"Math. Methods Med."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1779","DOI":"10.1162\/0899766041336477","article-title":"Stochastic reasoning, free energy, and information geometry","volume":"16","author":"Ikeda","year":"2004","journal-title":"Neural Comput."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Caticha, A. (2015). The Basics of Information Geometry, American Institute of Physics.","DOI":"10.1063\/1.4905960"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"100602","DOI":"10.1103\/PhysRevLett.99.100602","article-title":"Measuring Thermodynamic Length","volume":"99","author":"Crooks","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Kim, E.-J. (2018). Investigating Information Geometry in Classical and Quantum Systems through Information Length. Entropy, 20.","DOI":"10.3390\/e20080574"},{"key":"ref_18","unstructured":"Da Costa, L., Parr, T., Sengupta, B., and Friston, K. (2020). Natural selection finds natural gradient. arXiv."},{"key":"ref_19","first-page":"661","article-title":"Variational message passing","volume":"6","author":"Winn","year":"2005","journal-title":"J. Mach. Learn. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1529","DOI":"10.1080\/00018730210155133","article-title":"The Fluctuation Theorem","volume":"51","author":"Evans","year":"2002","journal-title":"Adv. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"126001","DOI":"10.1088\/0034-4885\/75\/12\/126001","article-title":"Stochastic thermodynamics, fluctuation theorems and molecular machines","volume":"75","author":"Seifert","year":"2012","journal-title":"Rep. Prog. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"263","DOI":"10.2307\/1914185","article-title":"Prospect Theory: An Analysis of Decision under Risk","volume":"47","author":"Kahneman","year":"1979","journal-title":"Econometrica"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"730","DOI":"10.1214\/aoap\/1026915623","article-title":"Risk-sensitive control and an optimal investment model II","volume":"12","author":"Fleming","year":"2002","journal-title":"Ann. Appl. Probab."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1016\/j.tics.2004.02.004","article-title":"Upper processing stages of the perception\u2013action cycle","volume":"8","author":"Fuster","year":"2004","journal-title":"Trends Cogn. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1162\/NETN_a_00018","article-title":"The graphical brain: Belief propagation and active inference","volume":"1","author":"Friston","year":"2017","journal-title":"Netw. Neurosci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"20170376","DOI":"10.1098\/rsif.2017.0376","article-title":"Uncertainty, epistemics and active inference","volume":"14","author":"Parr","year":"2017","journal-title":"J. R. Soc. Interface"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"90","DOI":"10.3389\/fncom.2018.00090","article-title":"The Anatomy of Inference: Generative Models and Brain Structure","volume":"12","author":"Parr","year":"2018","journal-title":"Front. Comput. Neurosci."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Da Costa, L., Parr, T., Sajid, N., Veselic, S., Neacsu, V., and Friston, K. (2020). Active inference on discrete state-spaces: A synthesis. arXiv.","DOI":"10.1016\/j.jmp.2020.102447"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/j.bpc.2004.12.001","article-title":"Thermodynamics of stoichiometric biochemical networks in living systems far from equilibrium","volume":"114","author":"Qian","year":"2005","journal-title":"Biophys. Chem."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"65102","DOI":"10.1063\/1.4734305","article-title":"The potential and flux landscape theory of evolution","volume":"137","author":"Zhang","year":"2012","journal-title":"J. Chem. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1386","DOI":"10.1109\/TVCG.2012.316","article-title":"The Helmholtz-Hodge Decomposition\u2014A Survey","volume":"19","author":"Bhatia","year":"2013","journal-title":"IEEE Trans. Vis. Comput. Graph."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"035201","DOI":"10.1088\/1361-6404\/ab7494","article-title":"Helmholtz\u2019s theorem for two retarded fields and its application to Maxwell\u2019s equations","volume":"41","author":"Heras","year":"2020","journal-title":"Eur. J. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Smets, P. (1998). Graphical Models for Probabilistic and Causal Reasoning. Quantified Representation of Uncertainty and Imprecision, Springer.","DOI":"10.1007\/978-94-017-1735-9"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.cortex.2015.03.025","article-title":"Active inference, communication and hermeneutics","volume":"68","author":"Friston","year":"2015","journal-title":"Cortex"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1016\/0013-4694(94)00291-R","article-title":"Electrocortical distinction of vocabulary types","volume":"94","author":"Pulvermuller","year":"1995","journal-title":"Electroencephalogr. Clin. Neurophysiol."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1109\/JPROC.2014.2306251","article-title":"Cognitive Dynamics: From Attractors to Active Inference","volume":"102","author":"Friston","year":"2014","journal-title":"Proc. IEEE"},{"key":"ref_37","unstructured":"Feynman, R.P. (1972). Statistical Mechanics, Benjamin."},{"key":"ref_38","unstructured":"Beal, M.J. (2003). Variational Algorithms for Approximate Bayesian Inference. [Ph.D. Thesis, University College London]."},{"key":"ref_39","unstructured":"Feynman, R.P. (1998). Statistical Mechanics: A Set of Lectures, Avalon Publishing."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1049\/el:19950331","article-title":"Free-energy minimisation algorithm for decoding and cryptoanalysis","volume":"31","author":"MacKay","year":"1995","journal-title":"Electron. Lett."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/j.neuroimage.2006.08.035","article-title":"Variational free energy and the Laplace approximation","volume":"34","author":"Friston","year":"2007","journal-title":"NeuroImage"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"083006","DOI":"10.1103\/PhysRevD.76.083006","article-title":"Tests of Bayesian model selection techniques for gravitational wave astronomy","volume":"76","author":"Cornish","year":"2007","journal-title":"Phys. Rev. D"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1926","DOI":"10.1126\/science.1099745","article-title":"Neuronal oscillations in cortical networks","volume":"304","author":"Buzsaki","year":"2004","journal-title":"Science"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"963","DOI":"10.1101\/SQB.1990.055.01.090","article-title":"Attentional Control of Visual Perception: Cortical and Subcortical Mechanisms","volume":"55","author":"DeSimone","year":"1990","journal-title":"Cold Spring Harb. Symp. Quant. Biol."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1146\/annurev.ne.18.030195.003011","article-title":"Visual feature integration and the temporal correlation hypothesis","volume":"18","author":"Singer","year":"1995","journal-title":"Annu. Rev. Neurosci."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Raja, V., Valluri, D., Baggs, E., Chemero, A., and Aderson, M.L. (2021, August 17). The Markov Blanket Trick: On the Scope of the Free Energy Principle and Active Inference. Available online: http:\/\/philsci-archive.pitt.edu\/18843\/.","DOI":"10.1016\/j.plrev.2021.09.001"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Bruineberg, J., Dolega, K., Dewhurst, J., and Baltieri, M. (2021, August 17). The Emperor\u2019s New Markov Blankets. Available online: http:\/\/philsci-archive.pitt.edu\/18467\/.","DOI":"10.1017\/S0140525X21002351"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Aguilera, M., Millidge, B., Tschantz, A., and Buckley, C.L. (2021). How particular is the physics of the Free Energy Principle?. arXiv.","DOI":"10.1016\/j.plrev.2021.11.001"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Da Costa, L., Friston, K., Heins, C., and Pavliotis, G.A. (2021). Bayesian Mechanics for Stationary Processes. arXiv.","DOI":"10.1098\/rspa.2021.0518"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/8\/1076\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:47:24Z","timestamp":1760165244000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/8\/1076"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,19]]},"references-count":49,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["e23081076"],"URL":"https:\/\/doi.org\/10.3390\/e23081076","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,8,19]]}}}