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Chapter 6 - Active Inference, Bayesian Optimal Design, and Expected Utility

from Part II - How Do Humans Search for Information?

Published online by Cambridge University Press:  19 May 2022

By
Noor Sajid ,
Lancelot Da Costa ,
Thomas Parr  and
Karl Friston
Edited by
Irene Cogliati Dezza ,
Eric Schulz  and
Charley M. Wu
Irene Cogliati Dezza
Affiliation:
University College London
Eric Schulz
Affiliation:
Max-Planck-Institut für biologische Kybernetik, Tübingen
Charley M. Wu
Affiliation:
Eberhard-Karls-Universität Tübingen, Germany
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Summary

Active inference, a corollary of the free energy principle, is a formal way of describing the behavior of certain kinds of random dynamical systems that have the appearance of sentience. In this chapter, we describe how active inference combines Bayesian decision theory and optimal Bayesian design principles under a single imperative to minimize expected free energy. It is this aspect of active inference that allows for the natural emergence of information-seeking behavior. When removing prior outcomes preferences from expected free energy, active inference reduces to optimal Bayesian design (i.e., information gain maximization). Conversely, active inference reduces to Bayesian decision theory in the absence of ambiguity and relative risk (i.e., expected utility maximization). Using these limiting cases, we illustrate how behaviors differ when agents select actions that optimize expected utility, expected information gain, and expected free energy. Our T-maze simulations show optimizing expected free energy produces goal-directed information-seeking behavior while optimizing expected utility induces purely exploitive behavior, and maximizing information gain engenders intrinsically motivated behavior.

Keywords

Active inferenceBayesian decision theoryoptimal Bayesian designfree energyinformation gain
Type
Chapter
Information
The Drive for Knowledge
The Science of Human Information Seeking
 , pp. 124 - 146
Publisher: Cambridge University Press
Print publication year: 2022

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References

Attias, H. (2003). Planning by Probabilistic Inference. Paper presented at the Proc. of the 9th Int. Workshop on Artificial Intelligence and Statistics. https://proceedings.mlr.press/r4/attias03a.html.Google Scholar
Auer, P. (2002). Using confidence bounds for exploitation-exploration trade-offs. Journal of Machine Learning Research, 3(Nov.), 397–422.Google Scholar
Barlow, H. (1961). Possible principles underlying the transformations of sensory messages. In Rosenblith, W. (Ed.), Sensory Communication (pp. 217–234). MIT Press.Google Scholar
Barlow, H. B. (1974). Inductive inference, coding, perception, and language. Perception, 3, 123–134.Google Scholar
Barto, A. G. (2013). Intrinsic motivation and reinforcement learning. In Baldassarre, G & Mirolli, M, Intrinsically motivated learning in natural and artificial systems (pp. 17–47). Springer.CrossRefGoogle Scholar
Barto, A., Mirolli, M., & Baldassarre, G. (2013). Novelty or Surprise? Frontiers in Psychology, 4. doi:10.3389/fpsyg.2013.00907. Retrieved from www.frontiersin.org/Journal/Abstract.aspx?s=196&name=cognitive_science&ART_DOI=10.3389/fpsyg.2013.00907.CrossRefGoogle ScholarPubMed
Beal, M. J. (2003). Variational Algorithms for Approximate Bayesian Inference. PhD. Thesis, University College London. www.proquest.com/docview/1775215626?pq-origsite=gscholar&fromopenview=true.Google Scholar
Bellemare, M. G., Srinivasan, S., Ostrovski, G., Schaul, T., Saxton, D., & Munos, R. (2016). Unifying count-based exploration and intrinsic motivation. arXiv preprint arXiv:1606.01868.Google Scholar
Berger, J. O. (2011). Statistical decision theory and Bayesian analysis. Springer.Google Scholar
Blei, D. M., Kucukelbir, A., & McAuliffe, J. D. (2017). Variational inference: A review for statisticians. Journal of the American Statistical Association, 112(518), 859–877.CrossRefGoogle Scholar
Botvinick, M., & Toussaint, M. (2012). Planning as inference. Trends in Cognitive Science., 16(10), 485–488.Google Scholar
Burda, Y., Edwards, H., Storkey, A., & Klimov, O. (2018). Exploration by random network distillation. arXiv preprint arXiv:1810.12894.Google Scholar
Çatal, O., Wauthier, S., Verbelen, T., De Boom, C., & Dhoedt, B. (2020). Deep active inference for autonomous robot navigation. arXiv preprint arXiv:2003.03220.Google Scholar
Chaloner, K., & Verdinelli, I. (1995). Bayesian experimental design: A review. Statistical Science, 273–304.Google Scholar
Cullen, M., Davey, B., Friston, K. J., & Moran, R. J. (2018). Active inference in OpenAI gym: A paradigm for computational investigations into psychiatric illness. Biological Psychiatry: Cognitive Neuroscience and Neuroimaging, 3(9), 809–818.Google Scholar
Da Costa, L., Parr, T., Sajid, N., Veselic, S., Neacsu, V., & Friston, K. (2020). Active inference on discrete state-spaces: A synthesis. Journal of Mathematical Psychology, 99, 102447. Retrieved from www.sciencedirect.com/science/article/pii/S0022249620300857.CrossRefGoogle ScholarPubMed
Da Costa, L., Sajid, N., Parr, T., Friston, K., & Smith, R. (2020). The relationship between dynamic programming and active inference: The discrete, finite-horizon case. arXiv preprint arXiv:2009.08111.Google Scholar
Fleming, W. H., & Sheu, S. J. (2002). Risk-sensitive control and an optimal investment model II. Annals of Applied Probability, 12(2), 730–767. Retrieved from https://projecteuclid.org:443/euclid.aoap/1026915623.Google Scholar
Fountas, Z., Sajid, N., Mediano, P. A., & Friston, K. (2020). Deep active inference agents using Monte-Carlo methods. arXiv preprint arXiv:2006.04176.Google Scholar
Friston, K. J. (2010). The free-energy principle: A unified brain theory? Nature Reviews Neuroscience, 11(2), 127–138. http://dx.doi.org/10.1038/nrn2787.Google Scholar
Friston, K. (2019). A free energy principle for a particular physics. arXiv preprint arXiv:1906.10184.Google Scholar
Friston, K., Da Costa, L., Hafner, D., Hesp, C., & Parr, T. (2020). Sophisticated inference. arXiv preprint arXiv:2006.04120.Google Scholar
Friston, K. J., Daunizeau, J., Kilner, J., & Kiebel, S. J. (2010). Action and behavior: A free-energy formulation. Biological Cybernetics, 102(3), 227–260.CrossRefGoogle ScholarPubMed
Friston, K., FitzGerald, T., Rigoli, F., Schwartenbeck, P., O’Doherty, J., & Pezzulo, G. (2016). Active inference and learning. Neuroscience and Biobehavioral Reviews, 68, 862–879. Retrieved from http://www.ncbi.nlm.nih.gov/pubmed/27375276.Google Scholar
Friston, K., FitzGerald, T., Rigoli, F., Schwartenbeck, P., & Pezzulo, G. (2017). Active inference: A process theory. Neural Computation, 29(1), 1–49. Retrieved from https://www.ncbi.nlm.nih.gov/pubmed/27870614.Google Scholar
Friston, K. J., Lin, M., Frith, C. D., Pezzulo, G., Hobson, J. A., & Ondobaka, S. (2017). Active inference, curiosity and insight. Neural Computation, 29(10), 2633–2683. Friston, K. J., Parr, T., & de Vries, B. (2017). The graphical brain: Belief propagation and active inference. Network Neuroscience, 1(4), 381–414. Retrieved from http://www.ncbi.nlm.nih.gov/pubmed/29417960.CrossRefGoogle ScholarPubMed
Friston, K. J., Parr, T., Yufik, Y., Sajid, N., Price, C. J., & Holmes, E. (2020). Generative models, linguistic communication and active inference. Neuroscience & Biobehavioral Reviews, 118, 42–64. https://doi.org/10.1016/j.neubiorev.2020.07.005.Google Scholar
Friston, K. J., Rigoli, F., Ognibene, D., Mathys, C., Fitzgerald, T., & Pezzulo, G. (2015). Active inference and epistemic value. Cognitive Neuroscience, 6(4), 187–224. Retrieved from http://dx.doi.org/10.1080/17588928.2015.1020053.Google Scholar
Friston, K. J., Rosch, R., Parr, T., Price, C., & Bowman, H. (2018). Deep temporal models and active inference. Neuroscience and Biobehavioral Reviews, 90, 486–501.Google Scholar
Friston, K., Schwartenbeck, P., FitzGerald, T., Moutoussis, M., Behrens, T., & Dolan, R. J. (2014). The anatomy of choice: dopamine and decision-making. Philosophical Transactions of the Royal Society B: Biological Sciences, 369 (1655). Retrieved from http://www.ncbi.nlm.nih.gov/pubmed/25267823.Google Scholar
Gottlieb, J., Oudeyer, P.-Y., Lopes, M., & Baranes, A. (2013). Information-seeking, curiosity, and attention: computational and neural mechanisms. Trends in Cognitive Science, 17(11), 585–593. Retrieved from https://www.sciencedirect.com/science/article/pii/S1364661313002052.CrossRefGoogle ScholarPubMed
Harsanyi, J. C. (1978). Bayesian decision theory and utilitarian ethics. The American Economic Review, 68(2), 223–228. Retrieved from www.jstor.org/stable/1816692.Google Scholar
Houthooft, R., Chen, X., Duan, Y., Schulman, J., De Turck, F., & Abbeel, P. (2016). Vime: Variational information maximizing exploration. Advances in Neural Information Processing Systems, 29, 1109–1117.Google Scholar
Itti, L., & Baldi, P. (2009). Bayesian surprise attracts human attention. Vision Research, 49(10), 1295–1306.Google Scholar
Jaynes, E. T. (1957). Information theory and statistical mechanics. Physical Review, 106(4), 620.Google Scholar
Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263–291.Google Scholar
Kaplan, R., & Friston, K. J. (2018). Planning and navigation as active inference. Biological Cybernetics, 112(4), 323–343.CrossRefGoogle ScholarPubMed
Laureiro-Martínez, D., Brusoni, S., & Zollo, M. (2010). The neuroscientific foundations of the exploration−exploitation dilemma. Journal of Neuroscience, Psychology, and Economics, 3(2), 95.Google Scholar
Lindley, D. V. (1956). On a measure of the information provided by an experiment. The Annals of Mathematical Statistics, 986–1005.Google Scholar
Linsker, R. (1990). Perceptual neural organization: some approaches based on network models and information theory. Annual Review of Neuroscience, 13, 257–281.Google Scholar
Millidge, B., Tschantz, A., & Buckley, C. L. (2020). Whence the expected free energy? arXiv preprint arXiv:2004.08128.Google Scholar
Mirza, M. B., Adams, R. A., Mathys, C. D., & Friston, K. J. (2016). Scene construction, visual foraging, and active inference. Frontiers in Computational Neuroscience, 10 (56). Retrieved from http://journal.frontiersin.org/Article/10.3389/fncom.2016.00056/abstract. Mitchell, T., Sacks, J., & Ylvisaker, D. (1994). Asymptotic Bayes criteria for nonparametric response surface design. The Annals of Statistics, 22(2), 634–651.CrossRefGoogle ScholarPubMed
Optican, L., & Richmond, B. J. (1987). Temporal encoding of two-dimensional patterns by single units in primate inferior cortex. II Information theoretic analysis. Journal of Neurophysiology, 57, 132–146.Google Scholar
Parr, T. (2019). The computational neurology of active vision. UCL (Unpublished doctoral thesis, University College London). https://discovery.ucl.ac.uk/id/eprint/10084391/Google Scholar
Parr, T., Da Costa, L., & Friston, K. (2020). Markov blankets, information geometry and stochastic thermodynamics. Philosophical Transactions of the Royal Society A, 378(2164), 20190159.Google Scholar
Parr, T., & Friston, K. J. (2019a). Attention or salience? Current Opinion in Psychology, 29, 1–5.Google Scholar
Parr, T., & Friston, K. J. (2019b). Generalised free energy and active inference. Biological Cybernetics, 113(5–6), 495–513.Google Scholar
Parr, T., Markovic, D., Kiebel, S. J., & Friston, K. J. (2019). Neuronal message passing using Mean-field, Bethe, and Marginal approximations. Scientific Reports, 9(1), 1889. Retrieved from https://doi.org/10.1038/s41598-018-38246-3.Google Scholar
Pathak, D., Agrawal, P., Efros, A. A., & Darrell, T. (2017). Curiosity-driven exploration by self-supervised prediction. Paper presented at the International Conference on Machine Learning.CrossRefGoogle Scholar
Pukelsheim, F. (2006). Optimal design of experiments: SIAM.Google Scholar
Russo, D., Van Roy, B., Kazerouni, A., Osband, I., & Wen, Z. (2017). A tutorial on Thompson sampling. arXiv preprint arXiv:1707.02038.Google Scholar
Sacks, J., Welch, W. J., Mitchell, T. J., & Wynn, H. P. (1989). Design and analysis of computer experiments. Statistical Science, 4(4), 409–423.Google Scholar
Sajid, N., Ball, P. J., Parr, T., & Friston, K. J. (2021). Active inference: Demystified and compared. Neural Computation, 33(3), 674–712.Google Scholar
Savage, L. J. (1972). The foundations of statistics: Courier Corporation.Google Scholar
Schmidhuber, J. (1991a). Curious model-building control systems. In Proc. International Joint Conference on Neural Networks, Singapore. IEEE, 2, 1458–1463. https://mediatum.ub.tum.de/doc/814953/file.pdf.Google Scholar
Schmidhuber, J. (1991b). A possibility for implementing curiosity and boredom in model-building neural controllers. Paper presented at the Proc. of the international conference on simulation of adaptive behavior: From animals to animats. https://mediatum.ub.tum.de/doc/814958/file.pdfCrossRefGoogle Scholar
Schmidhuber, J. (2006). Developmental robotics, optimal artificial curiosity, creativity, music, and the fine arts. Connection Science, 18(2), 173–187. https://doi.org/10.1080/09540090600768658.Google Scholar
Schulz, E., & Gershman, S. J. (2019). The algorithmic architecture of exploration in the human brain. Current Opinion in Neurobiology, 55, 7–14.Google Scholar
Schwartenbeck, P., Passecker, J., Hauser, T. U., FitzGerald, T. H., Kronbichler, M., & Friston, K. J. (2019). Computational mechanisms of curiosity and goal-directed exploration. eLife, 8, e.41707. https://doi.org/10.7554/eLife.41703.Google Scholar
Shewry, M. C., & Wynn, H. P. (1987). Maximum entropy sampling. Journal of Applied Statistics, 14(2), 165–170.CrossRefGoogle Scholar
Stone, M. (1959). Application of a measure of information to the design and comparison of regression experiments. The Annals of Mathematical Statistics, 30(1), 55–70.CrossRefGoogle Scholar
Sun, Y., Gomez, F., & Schmidhuber, J. (2011). Planning to be surprised: Optimal Bayesian exploration in dynamic environments. In Schmidhuber, J., Thórisson, K. R., & Looks, M. (Eds.), Artificial General Intelligence: 4th International Conference, AGI 2011, Mountain View, CA, USA, August 3–6,2011. Proceedings (pp. 41–51). Springer.Google Scholar
Sutton, R. S., & Barto, A. G. (1998). Introduction to Reinforcement Learning: MIT Press.CrossRefGoogle Scholar
Todorov, E. (2008). General duality between optimal control and estimation. In 2008 47th IEEE Conference on Decision and Control (pp. 4286–4292). IEEE.Google Scholar
Tschantz, A., Seth, A. K., & Buckley, C. L. (2020). Learning action-oriented models through active inference. PLoS Computational Biology, 16(4), e1007805. Retrieved from https://doi.org/10.1371/journal.pcbi.1007805.CrossRefGoogle ScholarPubMed
van den Broek, J. L., Wiegerinck, W. A. J. J., & Kappen, H. J. (2010). Risk-sensitive path integral control. UAI, 6, 1–8.Google Scholar
van der Himst, O., & Lanillos, P. (2020). Deep Active Inference for Partially Observable MDPs. In International Workshop on Active Inference (pp. 61–71). Springer.Google Scholar
Vasconcelos, M., Monteiro, T., & Kacelnik, A. (2015). Irrational choice and the value of information. Scientific Reports, 5(1), 13874. Retrieved from https://doi.org/10.1038/srep13874.CrossRefGoogle ScholarPubMed
Vértes, E., & Sahani, M. (2018). Flexible and accurate inference and learning for deep generative models. arXiv preprint arXiv:1805.11051.Google Scholar
Von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior. Princeton University Press.Google Scholar
Wilson, R. C., Geana, A., White, J. M., Ludvig, E. A., & Cohen, J. D. (2014). Humans use directed and random exploration to solve the explore–exploit dilemma. Journal of Experimental Psychology: General, 143(6), 2074.Google Scholar
Zintgraf, L., Shiarlis, K., Igl, M., Schulze, S., Gal, Y., Hofmann, K., & Whiteson, S. (2019). VariBAD: A very good method for Bayes-adaptive deep RL via meta-learning. arXiv preprint arXiv:1910.08348.Google Scholar

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