Entropy 2025, 27, 62 30 of 33
generative models, or even (deep learning-based) amortised inference models. These vari-
ous extensions could provide valuable tools for using AIF models in both theoretical and
applied research.
Author Contributions: Conceptualisation, S.W.N., J.E.L. and P .T.W.; methodology, S.W.N., J.E.L. and
P .T.W.; software, S.W.N., J.E.L. and P .T.W.; formal analysis, S.W.N. and J.E.L.; writing—original draft
preparation, S.W.N. and J.E.L.; writing—review and editing, C.H., K.F., C.M. and P .T.W.; visualisation,
S.W.N. and J.E.L.; supervision, C.M. and P .T.W.; project administration, P .T.W. All authors read and
agreed to the published version of this manuscript.
Funding: C.M. acknowledges funding from Aarhus Universitets Forskningsfonds (grant no. AUFF-
E-2019-7-10) and from the Carlsberg Foundation (grant no. CF21-0439).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: The original data presented in this study are openly available in
ActiveInferenceJuliaPaper at URL: https://osf.io/j3k5q/.
Conflicts of Interest: The authors declare no conflicts of interest. The funders had no role in the design
of this study; in the collection, analyses or interpretation of data; in the writing of this manuscript; or
in the decision to publish the results.
Abbreviations
The following abbreviations are used in this manuscript:
AIF Active inference
FEP Free energy principle
VFE Variational free energy
EFE Expected free energy
MCMC Markov Chain Monte Carlo
POMDP Partially Observed Markov Decision Process
References
1. Parr, T.; Pezzulo, G.; Friston, K.J. Active Inference: The Free Energy Principle in Mind, Brain, and Behavior; The MIT Press: Cambridge,
MA, USA, 2022. [CrossRef]
2. Friston, K.; FitzGerald, T.; Rigoli, F.; Schwartenbeck, P .; O’Doherty, J.; Pezzulo, G. Active inference and learning. Neurosci.
Biobehav. Rev.2016, 68, 862–879. [CrossRef]
3. Friston, K.; FitzGerald, T.; Rigoli, F.; Schwartenbeck, P .; Pezzulo, G. Active inference: A process theory. Neural Comput.2017,
29, 1–49. [CrossRef]
4. Friston, K.J.; Stephan, K.E. Free-energy and the brain. Synthese 2007, 159, 417–458. [CrossRef] [PubMed]
5. Friston, K. The free-energy principle: A unified brain theory? Nat. Rev. Neurosci.2010, 11, 127–138. [CrossRef] [PubMed]
6. Friston, K. The free-energy principle: A rough guide to the brain? Trends Cogn. Sci.2009, 13, 293–301. [CrossRef] [PubMed]
7. Friston, K. A free energy principle for a particular physics. arXiv 2019, arXiv:1906.10184. [CrossRef]
8. Friston, K.; Da Costa, L.; Sajid, N.; Heins, C.; Ueltzhöffer, K.; Pavliotis, G.A.; Parr, T. The free energy principle made simpler but
not too simple. Phys. Rep.2023, 1024, 1–29. [CrossRef]
9. Friston, K.; Kiebel, S. Predictive coding under the free-energy principle. Philos. Trans. R. Soc. B Biol. Sci.2009, 364, 1211–1221.
[CrossRef] [PubMed]
10. Karl, F. A Free Energy Principle for Biological Systems. Entropy 2012, 14, 2100–2121. [CrossRef]
11. Corcoran, A.W.; Pezzulo, G.; Hohwy, J. From allostatic agents to counterfactual cognisers: Active inference, biological regulation,
and the origins of cognition. Biol. Philos.2020, 35, 32. [CrossRef]
12. Heins, C.; Millidge, B.; Da Costa, L.; Mann, R.P .; Friston, K.J.; Couzin, I.D. Collective behavior from surprise minimization. Proc.
Natl. Acad. Sci. USA2024, 121, e2320239121. [CrossRef] [PubMed]
13. Patzelt, E.H.; Hartley, C.A.; Gershman, S.J. Computational Phenotyping: Using Models to Understand Individual Differences in
Personality, Development, and Mental Illness. Personal. Neurosci.2018, 1, e18. [CrossRef] [PubMed]