# The Free Energy Principle for Perception and Action: A Deep Learning Perspective

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## Abstract

**:**

## 1. Introduction

## 2. The Free Energy Principle and Active Inference

#### 2.1. Variational Free Energy

#### 2.2. Expected Free Energy

## 3. Variational World Models

#### 3.1. Models

**Posterior Model.**The choice of the distribution is particularly important for the posterior model, which is the variational distribution. In theory, one could search for an optimal distribution of parameters for each of the environment’s transitions/observations, though that is a slow and difficult process. To speed up training, but also guaranteeing a legitimate choice of the posterior, it is possible to amortize the selection of the posterior parameters, as presented in the original VAE work [70,71]. The autoencoding amortization scheme employs the observation corresponding to a certain state to infer the parameters of the variational distribution, $q\left({s}_{t+1}\right|{s}_{t},{a}_{t},{o}_{t+1})$. This allows optimizing the parameters of the posterior to compress information optimally, as the posterior has access to the observation that the likelihood model wants to generate. In VAE terms the posterior model is typically called the “encoder”, whereas the likelihood model is dubbed the “decoder”.

**Prior Model.**The prior model can either be known or learned. To learn the prior model, one can adopt a recurrent neural network architecture, i.e., using memory cells such as long short term memories (LSTM) [99] or gated recurrent units (GRU) [100]. In other cases, the environment dynamics is known upfront, or assumptions about the prior can be made, such as assuming that the prior is a uniform probability distribution. For instance, an isotropic multivariate Gaussian $\mathcal{N}(0,I)$, with zero mean and an identity covariance matrix I, can be employed as a fixed prior, as performed in standard VAE architectures [70,71]. Alternatively, assuming the laws that govern the dynamics are known (e.g., physics laws), the environment’s physics could be exploited as a strong prior [101]. In a similar fashion, in [102], the authors used the internal state of the robot to force a known prior structure on the posterior. Finally, the prior could also be ignored/considered constant, treating the model as an entropy-regularized autoencoder [103].

#### 3.2. Uncertainty

#### 3.3. Representation

#### 3.4. Summary

## 4. Bayesian Action Selection

#### 4.1. Preferences Modeling

**Observation Preferences.**If the agent’s objective is to match a set of preferred outcomes, the preferred distribution is over the environment’s observations $p\left(o\right)$. Matching outcomes can be seen as a form of goal-directed behavior, where the agent plans its actions to achieve certain outcomes from the environment. Goal-directed behavior has been widely studied in the context of RL, both in low-dimensional [137] and visual domains [138,139]. Preferences defined in the observation space can be handy, as they just require observations from “snapshots” of the environment in the correct state. Nevertheless, artificial active inference implementations have rarely used them, as they are generally hard to match in the high-dimensional settings. Strategies that overcome such limitations [123] could be the subject of future studies.

**Internal State Preferences.**Instead of defining preferences in observation space, these could be directly instantiated in the internal state space of the agent. This form of state matching [140] assumes that the agent knows both the preferred states distribution $p\left(s\right)$ and the model in advance, or as typical in RL, that sensory states are used as internal states. Alternatively, if a set of preferred outcomes is available, preferred states can be inferred from those using an inference model $p\left(s\right|o)$. This approach has been applied in robotics simulated and realistic setups [32,68].

**Rewards as Preferences.**Another way to circumvent the problem of defining preferences is to use a reward function that represents the agent’s probability of observing the preferred outcomes. The RL problem can be cast as probabilistic inference, by introducing an optimality variable ${\mathcal{O}}_{t}$, which denotes whether the time step t is optimal [141]. The distribution over the optimality variable is defined in terms of rewards as $p({\mathcal{O}}_{t}=1|{s}_{t},{a}_{t})=exp\left(r({s}_{t},{a}_{t})\right)$. As discussed in [142], RL works alike active inference but it encodes utility value in the optimality likelihood rather than in a prior over observations. Assuming $logp\left({o}_{t}\right)=logp\left({\mathcal{O}}_{t}\right|{s}_{t},{a}_{t})$, the environment rewards can be used for active inference as well. This possibility has allowed some active inference work [33,114] to reuse reward functions from RL environments [22]. Concretely, it is possible to consider rewards as a part of the observable aspects of the environment, and define their maximum values as the preferred observations [143]. Nonetheless, defining reward functions is also problematic [144] as they are not naturally available, and this setup works well only for well-engineered environments.

**Learned Preferences.**Finally, state preferences can also be learned from previous experience using conjugate priors [91], or from expert demonstrations [68]. In a RL context, demonstrations can be used in an inverse reinforcement learning fashion [145,146], where a reward signal is inferred from correct behaviors, which is then optimized using RL techniques.

#### 4.2. Epistemics, Exploration, and Ambiguity

**Parameter-driven Exploration.**Maximizing mutual information in parameter space has been studied in RL as a way to encourage exploration, computing the information gain given by the distribution over parameters with ensembles [112,113,147] or Bayesian neural networks [53]. In particular, in [113], they use the model to both evaluate the states/actions to explore and to plan the exploratory behavior, which is close to what envisioned in active inference. Ensemble methods have also been employed in some active inference works [114,129] along with dropout [33].

**State-driven Exploration.**Maximizing mutual information between states and observations has also been studied in RL for exploration, using the Bayesian surprise signal given by the D

_{KL}divergence between the (autoencoding) posterior and the prior of the model as a reward [51]. Alternatively, the surprisal with respect to future observations has also been used in RL to generate an intrinsic motivation signal that rewards exploration [52,148,149]. In active inference, the majority of works have instead focused on using multiple samples from the likelihood model [32,33].

**Uncertainty Tradeoffs.**It is worth mentioning that, during different stages of training, uncertainty related to parameters and uncertainty related to sensory/internal states may overlap. Particularly, given that the distributions that represent the agents’ states are inferred by employing the model parameters, uncertainty in the model strongly influences uncertainty with respect to the state. This highlights the importance of considering both kinds of uncertainty, especially when the model is imperfect or its learning process is incomplete.

#### 4.3. Plans, Habits, and Search Optimization

**Plan-based policies.**Assuming a complete search over all potential sequences of actions, the plan-based method should yield the optimal policy. Unfortunately, in most domains, considering all sequences of actions is an intractable problem and more engineered random shooting methods are used to search only over the most promising sequences of actions, such as [55]. Similar methods have been employed both for RL [49] and active inference [32,68]. In particular, when the search over policies takes into account recursive beliefs about the future, this scheme is referred to as sophisticated inference [74]. Sophistication describes the degree to which an agent has beliefs about beliefs. A sophisticated agent, when evaluating a sequence of actions, instead of directly considering the sequence of outcomes, recursively evaluates outcomes in terms of the beliefs it would have when applying each action of the sequence.

**Habit Policies.**For habit policies, we consider a one-action version of the expected free energy $\mathcal{G}$ that can be obtained by considering one-action plans $\pi ={a}_{t}$ for all time steps:

**Hybrid Search Policies.**Finally, hybrid search schemes (c) combine the use of a learned prior with computing the expected free energy for sequences of actions. The search space is greatly limited by using the prior, which influences the choice of the nodes to select and expand. One of the most popular applications in RL of these methods is by employing variants of Monte Carlo Tree Search (MCTS) [29,153], which use both a prior over actions and estimates of the expected utility over long horizons, as in Equation (10). Similar approaches have recently been applied for active inference [33,154]. While these methods are generally applicable only for discrete action spaces, extensions of MCTS for continuous domains have been developed as well [56]. The precision parameter $\zeta $ in these methods can be used to control the influence of the prior relative to the expected free energy (computed a posteriori, with respect to a certain action/plan).

#### 4.4. Summary

## 5. Discussion and Perspectives

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Friston, K.J.; Stephan, K.E. Free-energy and the brain. Synthese
**2007**, 159, 417–458. [Google Scholar] [CrossRef] [PubMed][Green Version] - Friston, K.; FitzGerald, T.; Rigoli, F.; Schwartenbeck, P.; Doherty, J.O.; Pezzulo, G. Active inference and learning. Neurosci. Biobehav. Rev.
**2016**, 68, 862–879. [Google Scholar] [CrossRef] [PubMed][Green Version] - Parr, T.; Rees, G.; Friston, K.J. Computational Neuropsychology and Bayesian Inference. Front. Hum. Neurosci.
**2018**, 12, 61. [Google Scholar] [CrossRef][Green Version] - Demekas, D.; Parr, T.; Friston, K.J. An Investigation of the Free Energy Principle for Emotion Recognition. Front. Comput. Neurosci.
**2020**, 14, 30. [Google Scholar] [CrossRef][Green Version] - Henriksen, M. Variational Free Energy and Economics Optimizing with Biases and Bounded Rationality. Front. Psychol.
**2020**, 11, 549187. [Google Scholar] [CrossRef] - Constant, A.; Ramstead, M.J.D.; Veissière, S.P.L.; Campbell, J.O.; Friston, K.J. A variational approach to niche construction. J. R. Soc. Interface
**2018**, 15, 20170685. [Google Scholar] [CrossRef] - Bruineberg, J.; Rietveld, E.; Parr, T.; van Maanen, L.; Friston, K.J. Free-energy minimization in joint agent-environment systems: A niche construction perspective. J. Theor. Biol.
**2018**, 455, 161–178. [Google Scholar] [CrossRef] - Perrinet, L.U.; Adams, R.A.; Friston, K.J. Active inference, eye movements and oculomotor delays. Biol. Cybern.
**2014**, 108, 777–801. [Google Scholar] [CrossRef][Green Version] - Parr, T.; Friston, K.J. Active inference and the anatomy of oculomotion. Neuropsychologia
**2018**, 111, 334–343. [Google Scholar] [CrossRef] - Brown, H.; Friston, K.; Bestmann, S. Active Inference, Attention, and Motor Preparation. Front. Psychol.
**2011**, 2, 218. [Google Scholar] [CrossRef][Green Version] - Parr, T.; Friston, K.J. Working memory, attention, and salience in active inference. Sci. Rep.
**2017**, 7, 14678. [Google Scholar] [CrossRef] [PubMed] - Mirza, M.B.; Adams, R.A.; Mathys, C.D.; Friston, K.J. Scene Construction, Visual Foraging, and Active Inference. Front. Comput. Neurosci.
**2016**, 10, 56. [Google Scholar] [CrossRef] [PubMed][Green Version] - Heins, R.C.; Mirza, M.B.; Parr, T.; Friston, K.; Kagan, I.; Pooresmaeili, A. Deep Active Inference and Scene Construction. Front. Artif. Intell.
**2020**, 3, 81. [Google Scholar] [CrossRef] [PubMed] - Biehl, M.; Pollock, F.A.; Kanai, R. A Technical Critique of Some Parts of the Free Energy Principle. Entropy
**2021**, 23, 293. [Google Scholar] [CrossRef] - Friston, K.J.; Da Costa, L.; Parr, T. Some Interesting Observations on the Free Energy Principle. Entropy
**2021**, 23, 1076. [Google Scholar] [CrossRef] - Friston, K. Life as we know it. J. R. Soc. Interface
**2013**, 10, 20130475. [Google Scholar] [CrossRef][Green Version] - Kirchhoff, M.; Parr, T.; Palacios, E.; Friston, K.; Kiverstein, J. The Markov blankets of life: Autonomy, active inference and the free energy principle. J. R. Soc. Interface
**2018**, 15, 20170792. [Google Scholar] [CrossRef] - Rubin, S.; Parr, T.; Da Costa, L.; Friston, K. Future climates: Markov blankets and active inference in the biosphere. J. R. Soc. Interface
**2020**, 17, 20200503. [Google Scholar] [CrossRef] - Maturana, H.R.; Varela, F.J.; Maturana, H.R. Autopoiesis and Cognition: The Realization of the Living; D. Reidel Pub. Co.: Dordrecht, The Netherlands, 1980. [Google Scholar]
- Kirchhoff, M.D. Autopoiesis, free energy, and the life–mind continuity thesis. Synthese
**2018**, 195, 2519–2540. [Google Scholar] [CrossRef] - Blei, D.M.; Kucukelbir, A.; McAuliffe, J.D. Variational Inference: A Review for Statisticians. J. Am. Stat. Assoc.
**2017**, 112, 859–877. [Google Scholar] [CrossRef][Green Version] - Sutton, R.S.; Barto, A.G. Reinforcement Learning: An Introduction; MIT Press: Cambridge, MA, USA, 2018. [Google Scholar]
- Wise, R.A. Dopamine, learning and motivation. Nat. Rev. Neurosci.
**2004**, 5, 483–494. [Google Scholar] [CrossRef] [PubMed] - Glimcher, P.W. Understanding dopamine and reinforcement learning: The dopamine reward prediction error hypothesis. Proc. Natl. Acad. Sci. USA
**2011**, 108, 15647–15654. [Google Scholar] [CrossRef] [PubMed][Green Version] - Silver, D.; Singh, S.; Precup, D.; Sutton, R.S. Reward is enough. Artif. Intell.
**2021**, 299, 103535. [Google Scholar] [CrossRef] - Mnih, V.; Kavukcuoglu, K.; Silver, D.; Graves, A.; Antonoglou, I.; Wierstra, D.; Riedmiller, M. Playing Atari with Deep Reinforcement Learning. arXiv
**2013**, arXiv:1312.5602. [Google Scholar] - Badia, A.P.; Piot, B.; Kapturowski, S.; Sprechmann, P.; Vitvitskyi, A.; Guo, D.; Blundell, C. Agent57: Outperforming the Atari Human Benchmark. arXiv
**2020**, arXiv:2003.13350. [Google Scholar] - Vinyals, O.; Babuschkin, I.; Czarnecki, W.M.; Mathieu, M.; Dudzik, A.; Chung, J.; Choi, D.H.; Powell, R.; Ewalds, T.; Georgiev, P.; et al. Grandmaster level in StarCraft II using multi-agent reinforcement learning. Nature
**2019**, 575, 350–354. [Google Scholar] [CrossRef] - Schrittwieser, J.; Antonoglou, I.; Hubert, T.; Simonyan, K.; Sifre, L.; Schmitt, S.; Guez, A.; Lockhart, E.; Hassabis, D.; Graepel, T.; et al. Mastering Atari, Go, chess and shogi by planning with a learned model. Nature
**2020**, 588, 604–609. [Google Scholar] [CrossRef] - Akkaya, I.; Andrychowicz, M.; Chociej, M.; Litwin, M.; McGrew, B.; Petron, A.; Paino, A.; Plappert, M.; Powell, G.; Ribas, R.; et al. Solving Rubik’s Cube with a Robot Hand. arXiv
**2019**, arXiv:1910.07113. [Google Scholar] - Ueltzhöffer, K. Deep active inference. Biol. Cybern.
**2018**, 112, 547–573. [Google Scholar] [CrossRef][Green Version] - Çatal, O.; Verbelen, T.; Nauta, J.; De Boom, C.; Dhoedt, B. Learning Perception and Planning with Deep Active Inference. In Proceedings of the ICASSP 2020—2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Barcelona, Spain, 4–8 May 2020; pp. 3952–3956. [Google Scholar] [CrossRef][Green Version]
- Fountas, Z.; Sajid, N.; Mediano, P.; Friston, K. Deep active inference agents using Monte-Carlo methods. In Advances in Neural Information Processing Systems; Larochelle, H., Ranzato, M., Hadsell, R., Balcan, M.F., Lin, H., Eds.; Curran Associates, Inc.: Red Hook, NY, USA, 2020; Volume 33, pp. 11662–11675. [Google Scholar]
- Buckley, C.L.; Kim, C.S.; McGregor, S.; Seth, A.K. The free energy principle for action and perception: A mathematical review. J. Math. Psychol.
**2017**, 81, 55–79. [Google Scholar] [CrossRef] - Da Costa, L.; Parr, T.; Sajid, N.; Veselic, S.; Neacsu, V.; Friston, K. Active inference on discrete state-spaces: A synthesis. J. Math. Psychol.
**2020**, 99, 102447. [Google Scholar] [CrossRef] [PubMed] - Lanillos, P.; Meo, C.; Pezzato, C.; Meera, A.A.; Baioumy, M.; Ohata, W.; Tschantz, A.; Millidge, B.; Wisse, M.; Buckley, C.L.; et al. Active Inference in Robotics and Artificial Agents: Survey and Challenges. arXiv
**2021**, arXiv:2112.01871. [Google Scholar] - Gershman, S.; Goodman, N. Amortized inference in probabilistic reasoning. Proc. Annu. Meet. Cogn. Sci. Soc.
**2014**, 36, 516–522. [Google Scholar] - Razavi, A.; van den Oord, A.; Vinyals, O. Generating Diverse High-Fidelity Images with VQ-VAE-2. arXiv
**2019**, arXiv:1906.00446. [Google Scholar] - Karras, T.; Aittala, M.; Laine, S.; Härkönen, E.; Hellsten, J.; Lehtinen, J.; Aila, T. Alias-Free Generative Adversarial Networks. arXiv
**2021**, arXiv:2106.12423. [Google Scholar] - Vahdat, A.; Kautz, J. NVAE: A Deep Hierarchical Variational Autoencoder. arXiv
**2021**, arXiv:2007.03898. [Google Scholar] - Zilly, J.G.; Srivastava, R.K.; Koutník, J.; Schmidhuber, J. Recurrent Highway Networks. arXiv
**2017**, arXiv:1607.03474. [Google Scholar] - Melis, G.; Kočiský, T.; Blunsom, P. Mogrifier LSTM. arXiv
**2020**, arXiv:1909.01792. [Google Scholar] - Brown, T.B.; Mann, B.; Ryder, N.; Subbiah, M.; Kaplan, J.; Dhariwal, P.; Neelakantan, A.; Shyam, P.; Sastry, G.; Askell, A.; et al. Language Models Are Few-Shot Learners. arXiv
**2020**, arXiv:2005.14165. [Google Scholar] - Xingjian, S.; Chen, Z.; Wang, H.; Yeung, D.Y.; Wong, W.K.; Woo, W.C. Convolutional LSTM network: A machine learning approach for precipitation nowcasting. Adv. Neural Inf. Process. Syst.
**2015**, 28, 802–810. [Google Scholar] - Wang, Y.; Long, M.; Wang, J.; Gao, Z.; Yu, P.S. PredRNN: Recurrent Neural Networks for Predictive Learning using Spatiotemporal LSTMs. In Advances in Neural Information Processing Systems; Guyon, I., Luxburg, U.V., Bengio, S., Wallach, H., Fergus, R., Vishwanathan, S., Garnett, R., Eds.; Curran Associates, Inc.: Red Hook, NY, USA, 2017; Volume 30. [Google Scholar]
- Denton, E.; Fergus, R. Stochastic Video Generation with a Learned Prior. arXiv
**2018**, arXiv:1802.07687. [Google Scholar] - Lotter, W.; Kreiman, G.; Cox, D. Deep Predictive Coding Networks for Video Prediction and Unsupervised Learning. arXiv
**2017**, arXiv:1605.08104. [Google Scholar] - Buesing, L.; Weber, T.; Racaniere, S.; Eslami, S.M.A.; Rezende, D.; Reichert, D.P.; Viola, F.; Besse, F.; Gregor, K.; Hassabis, D.; et al. Learning and Querying Fast Generative Models for Reinforcement Learning. arXiv
**2018**, arXiv:1802.03006. [Google Scholar] - Hafner, D.; Lillicrap, T.; Fischer, I.; Villegas, R.; Ha, D.; Lee, H.; Davidson, J. Learning Latent Dynamics for Planning from Pixels. In Proceedings of the 36th International Conference on Machine Learning, Long Beach, CA, USA, 5–9 June 2019; Chaudhuri, K., Salakhutdinov, R., Eds.; PMLR: Brookline, MA, USA, 2019; Volume 97, pp. 2555–2565. [Google Scholar]
- Ha, D.; Schmidhuber, J. Recurrent World Models Facilitate Policy Evolution. arXiv
**2018**, arXiv:1809.01999. [Google Scholar] - Mazzaglia, P.; Catal, O.; Verbelen, T.; Dhoedt, B. Self-Supervised Exploration via Latent Bayesian Surprise. arXiv
**2021**, arXiv:2104.07495. [Google Scholar] - Pathak, D.; Agrawal, P.; Efros, A.A.; Darrell, T. Curiosity-driven Exploration by Self-supervised Prediction. arXiv
**2017**, arXiv:1705.05363. [Google Scholar] - Houthooft, R.; Chen, X.; Duan, Y.; Schulman, J.; De Turck, F.; Abbeel, P. VIME: Variational Information Maximizing Exploration. In Proceedings of the 30th International Conference on Neural Information Processing Systems, NIPS’16, Barcelona, Spain, 5–10 December 2016; pp. 1117–1125. [Google Scholar]
- Çatal, O.; Leroux, S.; De Boom, C.; Verbelen, T.; Dhoedt, B. Anomaly Detection for Autonomous Guided Vehicles using Bayesian Surprise. In Proceedings of the 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Las Vegas, NV, USA, 24 October–24 January 2020; pp. 8148–8153. [Google Scholar] [CrossRef]
- Hansen, N. The CMA Evolution Strategy: A Tutorial. arXiv
**2016**, arXiv:1604.00772. [Google Scholar] - Hubert, T.; Schrittwieser, J.; Antonoglou, I.; Barekatain, M.; Schmitt, S.; Silver, D. Learning and Planning in Complex Action Spaces. arXiv
**2021**, arXiv:2104.06303. [Google Scholar] - Von Helmholtz, H. Handbuch der Physiologischen Optik: Mit 213 in den Text Eingedruckten Holzschnitten und 11 Tafeln; Wentworth Press: Sydney, Australia, 1867; Volume 9. [Google Scholar]
- Friston, K. The free-energy principle: A rough guide to the brain? Trends Cogn. Sci.
**2009**, 13, 293–301. [Google Scholar] [CrossRef] - Ramstead, M.J.; Kirchhoff, M.D.; Friston, K.J. A tale of two densities: Active inference is enactive inference. Adapt. Behav.
**2020**, 28, 225–239. [Google Scholar] [CrossRef][Green Version] - Friston, K.J.; Parr, T.; de Vries, B. The graphical brain: Belief propagation and active inference. Netw. Neurosci.
**2017**, 1, 381–414. [Google Scholar] [CrossRef] [PubMed] - Friston, K.J.; Daunizeau, J.; Kiebel, S.J. Reinforcement Learning or Active Inference? PLoS ONE
**2009**, 4, e6421. [Google Scholar] [CrossRef] [PubMed][Green Version] - Karl, F. A Free Energy Principle for Biological Systems. Entropy
**2012**, 14, 2100–2121. [Google Scholar] [CrossRef] - Schwartenbeck, P.; Passecker, J.; Hauser, T.U.; FitzGerald, T.H.; Kronbichler, M.; Friston, K.J. Computational mechanisms of curiosity and goal-directed exploration. eLife
**2019**, 8, e41703. [Google Scholar] [CrossRef] - Friston, K.J.; Lin, M.; Frith, C.D.; Pezzulo, G.; Hobson, J.A.; Ondobaka, S. Active Inference, Curiosity and Insight. Neural Comput.
**2017**, 29, 2633–2683. [Google Scholar] [CrossRef] - Friston, K.; Rigoli, F.; Ognibene, D.; Mathys, C.; Fitzgerald, T.; Pezzulo, G. Active inference and epistemic value. Cogn. Neurosci.
**2015**, 6, 187–214. [Google Scholar] [CrossRef] - Hafner, D.; Lillicrap, T.; Norouzi, M.; Ba, J. Mastering Atari with Discrete World Models. arXiv
**2021**, arXiv:2010.02193. [Google Scholar] - Hafner, D.; Lillicrap, T.P.; Ba, J.; Norouzi, M. Dream to Control: Learning Behaviors by Latent Imagination. In Proceedings of the ICLR Conference, Addis Abeba, Ethiopia, 26 April–1 May 2020. [Google Scholar]
- Çatal, O.; Nauta, J.; Verbelen, T.; Simoens, P.; Dhoedt, B. Bayesian policy selection using active inference. arXiv
**2019**, arXiv:1904.08149. [Google Scholar] - Çatal, O.; Verbelen, T.; Van de Maele, T.; Dhoedt, B.; Safron, A. Robot navigation as hierarchical active inference. Neural Netw.
**2021**, 142, 192–204. [Google Scholar] [CrossRef] - Kingma, D.P.; Welling, M. Auto-Encoding Variational Bayes. arXiv
**2014**, arXiv:1312.6114. [Google Scholar] - Rezende, D.J.; Mohamed, S.; Wierstra, D. Stochastic Backpropagation and Approximate Inference in Deep Generative Models. In Proceedings of the 31st International Conference on Machine Learning (ICML), Beijing, China, 21–26 June 2014; Volume 32, pp. 1278–1286. [Google Scholar]
- Tishby, N.; Pereira, F.C.; Bialek, W. The information bottleneck method. arXiv
**2000**, arXiv:physics/0004057. [Google Scholar] - Alemi, A.A.; Fischer, I.; Dillon, J.V.; Murphy, K. Deep Variational Information Bottleneck. arXiv
**2019**, arXiv:1612.00410. [Google Scholar] - Friston, K.; Da Costa, L.; Hafner, D.; Hesp, C.; Parr, T. Sophisticated Inference. Neural Comput.
**2021**, 33, 713–763. [Google Scholar] [CrossRef] - Hornik, K. Approximation capabilities of multilayer feedforward networks. Neural Netw.
**1991**, 4, 251–257. [Google Scholar] [CrossRef] - Heiden, E.; Millard, D.; Coumans, E.; Sheng, Y.; Sukhatme, G.S. NeuralSim: Augmenting Differentiable Simulators with Neural Networks. arXiv
**2021**, arXiv:2011.04217. [Google Scholar] - Freeman, C.D.; Frey, E.; Raichuk, A.; Girgin, S.; Mordatch, I.; Bachem, O. Brax—A Differentiable Physics Engine for Large Scale Rigid Body Simulation. arXiv
**2021**, arXiv:2106.13281. [Google Scholar] - Lovejoy, W.S. A survey of algorithmic methods for partially observed Markov decision processes. Ann. Oper. Res.
**1991**, 28, 47–65. [Google Scholar] [CrossRef] - Roy, N.; Gordon, G.; Thrun, S. Finding Approximate POMDP solutions Through Belief Compression. J. Artif. Intell. Res.
**2005**, 23, 1–40. [Google Scholar] [CrossRef] - Kurniawati, H.; Hsu, D.; Lee, W.S. Sarsop: Efficient point-based pomdp planning by approximating optimally reachable belief spaces. In Robotics: Science and Systems; Citeseer: Pennsylvania, PA, USA, 2008; Volume 2008. [Google Scholar]
- Heess, N.; Hunt, J.J.; Lillicrap, T.P.; Silver, D. Memory-based control with recurrent neural networks. arXiv
**2015**, arXiv:1512.04455. [Google Scholar] - Rumelhart, D.E.; Hinton, G.E.; Williams, R.J. Learning representations by back-propagating errors. Nature
**1986**, 323, 533–536. [Google Scholar] [CrossRef] - Bengio, Y.; Léonard, N.; Courville, A. Estimating or Propagating Gradients through Stochastic Neurons for Conditional Computation. arXiv
**2013**, arXiv:1308.3432. [Google Scholar] - Glynn, P.W. Likelilood ratio gradient estimation: An overview. In Proceedings of the 19th Conference on Winter Simulation, Atlanta, GA, USA, 14–16 December 1987; pp. 366–375. [Google Scholar]
- Williams, R.J. Simple Statistical Gradient-Following Algorithms for Connectionist Reinforcement Learning. Mach. Learn.
**1992**, 8, 229–256. [Google Scholar] [CrossRef][Green Version] - Van de Maele, T.; Verbelen, T.; Çatal, O.; De Boom, C.; Dhoedt, B. Active Vision for Robot Manipulators Using the Free Energy Principle. Front. Neurorobot.
**2021**, 15, 14. [Google Scholar] [CrossRef] [PubMed] - Lee, A.X.; Nagabandi, A.; Abbeel, P.; Levine, S. Stochastic Latent Actor-Critic: Deep Reinforcement Learning with a Latent Variable Model. arXiv
**2020**, arXiv:1907.00953. [Google Scholar] - Igl, M.; Zintgraf, L.; Le, T.A.; Wood, F.; Whiteson, S. Deep Variational Reinforcement Learning for POMDPs. arXiv
**2018**, arXiv:1806.02426. [Google Scholar] - Rolfe, J.T. Discrete variational autoencoders. arXiv
**2016**, arXiv:1609.02200. [Google Scholar] - Ozair, S.; Li, Y.; Razavi, A.; Antonoglou, I.; van den Oord, A.; Vinyals, O. Vector Quantized Models for Planning. arXiv
**2021**, arXiv:2106.04615. [Google Scholar] - Sajid, N.; Tigas, P.; Zakharov, A.; Fountas, Z.; Friston, K. Exploration and preference satisfaction trade-off in reward-free learning. arXiv
**2021**, arXiv:2106.04316. [Google Scholar] - Serban, I.V.; Ororbia, A.G.; Pineau, J.; Courville, A. Piecewise Latent Variables for Neural Variational Text Processing. In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, Copenhagen, Denmark, 7–11 September 2017; Association for Computational Linguistics: Copenhagen, Denmark, 2017; pp. 422–432. [Google Scholar] [CrossRef]
- Rezende, D.J.; Mohamed, S. Variational Inference with Normalizing Flows. arXiv
**2016**, arXiv:1505.05770. [Google Scholar] - Salimans, T.; Kingma, D.P.; Welling, M. Markov Chain Monte Carlo and Variational Inference: Bridging the Gap. arXiv
**2015**, arXiv:1410.6460. [Google Scholar] - LeCun, Y.; Bottou, L.; Bengio, Y.; Haffner, P. Gradient-based learning applied to document recognition. Proc. IEEE
**1998**, 86, 2278–2324. [Google Scholar] [CrossRef][Green Version] - Dosovitskiy, A.; Beyer, L.; Kolesnikov, A.; Weissenborn, D.; Zhai, X.; Unterthiner, T.; Dehghani, M.; Minderer, M.; Heigold, G.; Gelly, S.; et al. An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale. arXiv
**2021**, arXiv:2010.11929. [Google Scholar] - Rosenblatt, F. The perceptron: A probabilistic model for information storage and organization in the brain. Psychol. Rev.
**1958**, 65, 386. [Google Scholar] [CrossRef] [PubMed][Green Version] - Scarselli, F.; Gori, M.; Tsoi, A.C.; Hagenbuchner, M.; Monfardini, G. The Graph Neural Network Model. IEEE Trans. Neural Netw.
**2009**, 20, 61–80. [Google Scholar] [CrossRef][Green Version] - Hochreiter, S.; Schmidhuber, J. Long Short-Term Memory. Neural Comput.
**1997**, 9, 1735–1780. [Google Scholar] [CrossRef] - Chung, J.; Gulcehre, C.; Cho, K.; Bengio, Y. Empirical evaluation of gated recurrent neural networks on sequence modeling. In Proceedings of the NIPS 2014 Workshop on Deep Learning, Montreal, QC, Canada, 12–13 December 2014. [Google Scholar]
- Toth, P.; Rezende, D.J.; Jaegle, A.; Racanière, S.; Botev, A.; Higgins, I. Hamiltonian Generative Networks. arXiv
**2020**, arXiv:1909.13789. [Google Scholar] - Sancaktar, C.; van Gerven, M.A.J.; Lanillos, P. End-to-End Pixel-Based Deep Active Inference for Body Perception and Action. In Proceedings of the 2020 Joint IEEE 10th International Conference on Development and Learning and Epigenetic Robotics (ICDL-EpiRob), Valparaiso, Chile, 26–30 October 2020. [Google Scholar] [CrossRef]
- Ghosh, P.; Sajjadi, M.S.M.; Vergari, A.; Black, M.; Schölkopf, B. From Variational to Deterministic Autoencoders. arXiv
**2020**, arXiv:1903.12436. [Google Scholar] - Friston, K.; Schwartenbeck, P.; Fitzgerald, T.; Moutoussis, M.; Behrens, T.; Dolan, R. The anatomy of choice: Active inference and agency. Front. Hum. Neurosci.
**2013**, 7, 598. [Google Scholar] [CrossRef][Green Version] - Parr, T.; Benrimoh, D.A.; Vincent, P.; Friston, K.J. Precision and False Perceptual Inference. Front. Integr. Neurosci.
**2018**, 12, 39. [Google Scholar] [CrossRef] - Parr, T.; Friston, K.J. Uncertainty, epistemics and active inference. J. R. Soc. Interface
**2017**, 14, 20170376. [Google Scholar] [CrossRef] - Higgins, I.; Matthey, L.; Pal, A.; Burgess, C.P.; Glorot, X.; Botvinick, M.M.; Mohamed, S.; Lerchner, A. Beta-VAE: Learning Basic Visual Concepts with a Constrained Variational Framework. In Proceedings of the ICLR Conference, Toulon, France, 24–26 April 2017. [Google Scholar]
- Razavi, A.; van den Oord, A.; Poole, B.; Vinyals, O. Preventing Posterior Collapse with delta-VAEs. arXiv
**2019**, arXiv:1901.03416. [Google Scholar] - Blundell, C.; Cornebise, J.; Kavukcuoglu, K.; Wierstra, D. Weight Uncertainty in Neural Networks. arXiv
**2015**, arXiv:1505.05424. [Google Scholar] - Gal, Y.; Ghahramani, Z. Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning. arXiv
**2016**, arXiv:1506.02142. [Google Scholar] - Lakshminarayanan, B.; Pritzel, A.; Blundell, C. Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles. arXiv
**2017**, arXiv:1612.01474. [Google Scholar] - Pathak, D.; Gandhi, D.; Gupta, A. Self-Supervised Exploration via Disagreement. arXiv
**2019**, arXiv:1906.04161. [Google Scholar] - Sekar, R.; Rybkin, O.; Daniilidis, K.; Abbeel, P.; Hafner, D.; Pathak, D. Planning to Explore via Self-Supervised World Models. In Proceedings of the ICML Conference, Virtual Conference, 12–18 July 2020. [Google Scholar]
- Tschantz, A.; Millidge, B.; Seth, A.K.; Buckley, C.L. Reinforcement Learning through Active Inference. arXiv
**2020**, arXiv:2002.12636. [Google Scholar] - Van den Oord, A.; Li, Y.; Vinyals, O. Representation Learning with Contrastive Predictive Coding. arXiv
**2019**, arXiv:1807.03748. [Google Scholar] - Caron, M.; Misra, I.; Mairal, J.; Goyal, P.; Bojanowski, P.; Joulin, A. Unsupervised Learning of Visual Features by Contrasting Cluster Assignments. arXiv
**2021**, arXiv:2006.09882. [Google Scholar] - Grill, J.B.; Strub, F.; Altché, F.; Tallec, C.; Richemond, P.H.; Buchatskaya, E.; Doersch, C.; Pires, B.A.; Guo, Z.D.; Azar, M.G.; et al. Bootstrap your own latent: A new approach to self-supervised Learning. arXiv
**2020**, arXiv:2006.07733. [Google Scholar] - Chen, X.; He, K. Exploring Simple Siamese Representation Learning. arXiv
**2020**, arXiv:2011.10566. [Google Scholar] - Zbontar, J.; Jing, L.; Misra, I.; LeCun, Y.; Deny, S. Barlow Twins: Self-Supervised Learning via Redundancy Reduction. arXiv
**2021**, arXiv:2103.03230. [Google Scholar] - Chen, Z.; Bei, Y.; Rudin, C. Concept whitening for interpretable image recognition. Nat. Mach. Intell.
**2020**, 2, 772–782. [Google Scholar] [CrossRef] - Schwarzer, M.; Anand, A.; Goel, R.; Hjelm, R.D.; Courville, A.; Bachman, P. Data-Efficient Reinforcement Learning with Self-Predictive Representations. arXiv
**2021**, arXiv:2007.05929. [Google Scholar] - Ma, X.; Chen, S.; Hsu, D.; Lee, W.S. Contrastive Variational Model-Based Reinforcement Learning for Complex Observations. In Proceedings of the 4th Conference on Robot Learning, Virtual Conference, 16–18 November 2020. [Google Scholar]
- Mazzaglia, P.; Verbelen, T.; Dhoedt, B. Contrastive Active Inference. In Proceedings of the Advances in Neural Information Processing Systems, Virtual Conference, 6–14 December 2021. [Google Scholar]
- Çatal, O.; Wauthier, S.; De Boom, C.; Verbelen, T.; Dhoedt, B. Learning Generative State Space Models for Active Inference. Front. Comput. Neurosci.
**2020**, 14, 103. [Google Scholar] [CrossRef] - Friston, K.J.; Rosch, R.; Parr, T.; Price, C.; Bowman, H. Deep temporal models and active inference. Neurosci. Biobehav. Rev.
**2017**, 77, 388–402. [Google Scholar] [CrossRef] - Millidge, B. Deep Active Inference as Variational Policy Gradients. arXiv
**2019**, arXiv:1907.03876. [Google Scholar] [CrossRef][Green Version] - Saxena, V.; Ba, J.; Hafner, D. Clockwork Variational Autoencoders. arXiv
**2021**, arXiv:2102.09532. [Google Scholar] - Wu, B.; Nair, S.; Martin-Martin, R.; Fei-Fei, L.; Finn, C. Greedy Hierarchical Variational Autoencoders for Large-Scale Video Prediction. arXiv
**2021**, arXiv:2103.04174. [Google Scholar] - Tschantz, A.; Baltieri, M.; Seth, A.K.; Buckley, C.L. Scaling Active Inference. In Proceedings of the 2020 International Joint Conference on Neural Networks (IJCNN), Glasgow, UK, 19–24 July 2020; pp. 1–8. [Google Scholar] [CrossRef]
- Kaiser, L.; Babaeizadeh, M.; Milos, P.; Osinski, B.; Campbell, R.H.; Czechowski, K.; Erhan, D.; Finn, C.; Kozakowski, P.; Levine, S.; et al. Model-Based Reinforcement Learning for Atari. arXiv
**2020**, arXiv:1903.00374. [Google Scholar] - Srinivas, A.; Laskin, M.; Abbeel, P. CURL: Contrastive Unsupervised Representations for Reinforcement Learning. arXiv
**2020**, arXiv:2004.04136. [Google Scholar] - Pezzulo, G.; Rigoli, F.; Friston, K.J. Hierarchical Active Inference: A Theory of Motivated Control. Trends Cogn. Sci.
**2018**, 22, 294–306. [Google Scholar] [CrossRef] [PubMed][Green Version] - Zakharov, A.; Guo, Q.; Fountas, Z. Variational Predictive Routing with Nested Subjective Timescales. arXiv
**2021**, arXiv:2110.11236. [Google Scholar] - Wauthier, S.T.; Çatal, O.; De Boom, C.; Verbelen, T.; Dhoedt, B. Sleep: Model Reduction in Deep Active Inference. In Active Inference; Verbelen, T., Lanillos, P., Buckley, C.L., De Boom, C., Eds.; Springer International Publishing: Cham, Switzerland, 2020; pp. 72–83. [Google Scholar]
- Pezzulo, G.; Rigoli, F.; Friston, K. Active Inference, homeostatic regulation and adaptive behavioural control. Prog. Neurobiol.
**2015**, 134, 17–35. [Google Scholar] [CrossRef] [PubMed][Green Version] - Millidge, B.; Tschantz, A.; Buckley, C.L. Whence the Expected Free Energy? arXiv
**2020**, arXiv:2004.08128. [Google Scholar] [CrossRef] - Andrychowicz, M.; Wolski, F.; Ray, A.; Schneider, J.; Fong, R.; Welinder, P.; McGrew, B.; Tobin, J.; Pieter Abbeel, O.; Zaremba, W. Hindsight Experience Replay. In Advances in Neural Information Processing Systems; Guyon, I., Luxburg, U.V., Bengio, S., Wallach, H., Fergus, R., Vishwanathan, S., Garnett, R., Eds.; Curran Associates, Inc.: Red Hook, NY, USA, 2017; Volume 30. [Google Scholar]
- Warde-Farley, D.; de Wiele, T.V.; Kulkarni, T.D.; Ionescu, C.; Hansen, S.; Mnih, V. Unsupervised Control through Non-Parametric Discriminative Rewards. In Proceedings of the 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, 6–9 May 2019. [Google Scholar]
- Mendonca, R.; Rybkin, O.; Daniilidis, K.; Hafner, D.; Pathak, D. Discovering and Achieving Goals via World Models. arXiv
**2021**, arXiv:2110.09514. [Google Scholar] - Lee, L.; Eysenbach, B.; Parisotto, E.; Xing, E.; Levine, S.; Salakhutdinov, R. Efficient Exploration via State Marginal Matching. arXiv
**2020**, arXiv:1906.05274. [Google Scholar] - Levine, S. Reinforcement Learning and Control as Probabilistic Inference: Tutorial and Review. arXiv
**2018**, arXiv:1805.00909. [Google Scholar] - Millidge, B.; Tschantz, A.; Seth, A.K.; Buckley, C.L. On the Relationship between Active Inference and Control as Inference. arXiv
**2020**, arXiv:2006.12964. [Google Scholar] - Sajid, N.; Ball, P.J.; Parr, T.; Friston, K.J. Active Inference: Demystified and Compared. Neural Comput.
**2021**, 33, 674–712. [Google Scholar] [CrossRef] - Clark, J.; Amodei, D. Faulty Reward Functions in the Wild; OpenAI: San Francisco, CA, USA, 2016. [Google Scholar]
- Ziebart, B.D.; Maas, A.L.; Bagnell, J.A.; Dey, A.K. Maximum entropy inverse reinforcement learning. In Proceedings of the Twenty-Third AAAI Conference on Artificial Intelligence, Chicago, IL, USA, 13–17 July 2008; Volume 8, pp. 1433–1438. [Google Scholar]
- Abbeel, P.; Ng, A.Y. Apprenticeship Learning via Inverse Reinforcement Learning. In Proceedings of the Twenty-First International Conference on Machine Learning, ICML’04, Banff, AB, Canada, 4–8 July 2004; Association for Computing Machinery: New York, NY, USA, 2004; p. 1. [Google Scholar] [CrossRef]
- Shyam, P.; Jaśkowski, W.; Gomez, F. Model-Based Active Exploration. arXiv
**2019**, arXiv:1810.12162. [Google Scholar] - Achiam, J.; Sastry, S. Surprise-Based Intrinsic Motivation for Deep Reinforcement Learning. arXiv
**2017**, arXiv:1703.01732. [Google Scholar] - Burda, Y.; Edwards, H.; Pathak, D.; Storkey, A.J.; Darrell, T.; Efros, A.A. Large-Scale Study of Curiosity-Driven Learning. In Proceedings of the 7th International Conference on Learning Representations, ICLR, New Orleans, LA, USA, 6–9 May 2019. [Google Scholar]
- Schulman, J.; Wolski, F.; Dhariwal, P.; Radford, A.; Klimov, O. Proximal Policy Optimization Algorithms. arXiv
**2017**, arXiv:1707.06347. [Google Scholar] - Haarnoja, T.; Zhou, A.; Abbeel, P.; Levine, S. Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor. In Proceedings of the 35th International Conference on Machine Learning, Stockholm, Sweden, 10–15 July 2018; Dy, J., Krause, A., Eds.; PMLR: Brookline, MA, USA, 2018; Volume 80, pp. 1861–1870. [Google Scholar]
- Eysenbach, B.; Levine, S. Maximum Entropy RL (Provably) Solves Some Robust RL Problems. arXiv
**2021**, arXiv:2103.06257. [Google Scholar] - Silver, D.; Hubert, T.; Schrittwieser, J.; Antonoglou, I.; Lai, M.; Guez, A.; Lanctot, M.; Sifre, L.; Kumaran, D.; Graepel, T.; et al. Mastering Chess and Shogi by Self-Play with a General Reinforcement Learning Algorithm. arXiv
**2017**, arXiv:1712.01815. [Google Scholar] - Maisto, D.; Gregoretti, F.; Friston, K.; Pezzulo, G. Active Tree Search in Large POMDPs. arXiv
**2021**, arXiv:2103.13860. [Google Scholar] - Clavera, I.; Fu, V.; Abbeel, P. Model-Augmented Actor-Critic: Backpropagating through Paths. arXiv
**2020**, arXiv:2005.08068. [Google Scholar] - Pardo, F.; Tavakoli, A.; Levdik, V.; Kormushev, P. Time Limits in Reinforcement Learning. In Proceedings of the 35th International Conference on Machine Learning, Stockholm, Sweden, 10–15 July 2018; Dy, J., Krause, A., Eds.; PMLR: Brookline, MA, USA, 2018; Volume 80, pp. 4045–4054. [Google Scholar]
- Mhaskar, H.; Liao, Q.; Poggio, T. When and Why Are Deep Networks Better than Shallow Ones? In Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, AAAI’17, San Francisco, CA, USA, 4–9 February 2017; AAAI Press: Palo Alto, CA, USA, 2017; pp. 2343–2349. [Google Scholar]
- Novak, R.; Bahri, Y.; Abolafia, D.A.; Pennington, J.; Sohl-Dickstein, J. Sensitivity and Generalization in Neural Networks: An Empirical Study. arXiv
**2018**, arXiv:1802.08760. [Google Scholar] - Colbrook, M.J.; Antun, V.; Hansen, A.C. Can stable and accurate neural networks be computed?—On the barriers of deep learning and Smale’s 18th problem. arXiv
**2021**, arXiv:2101.08286. [Google Scholar] - Ben-David, S.; Hrubeš, P.; Moran, S.; Shpilka, A.; Yehudayoff, A. Learnability can be undecidable. Nat. Mach. Intell.
**2019**, 1, 44–48. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**The free energy functional minimized by active inference takes two forms: variational free energy, with respect to past experience, and expected free energy, for selecting future behaviors. For each of the two, an (amortized) Bayesian optimization scheme is followed that needs to consider several aspects, as summarized in the diagram. The numbering indicates the section of the paper discussing each aspect.

**Figure 2.**The external environment states $\eta $ are the hidden causes of sensorial states o (observations). The environment attempts to represents such hidden causes through its internal model states s. Crucially, internal states may or may not correspond to external states, which means that hidden causes in the brain do not need to be represented in the same way as in the environment. Active states a (actions), which are developed according to internal states, allow the agent to condition the environment states.

**Figure 3.**The diagram illustrates the interplay between the different factors that compose the graphical model. (1) Policy precision; (2) beliefs about policies; (3) transition probabilities, also known as dynamics; (4) parameters of the likelihood mapping; (5) likelihood model.

**Figure 4.**Representation learning approaches for world models with a latent dynamics. On the left, the base approach with the likelihood-model that reconstructs sensory information. On the right: (

**a**) Task-oriented representation; (

**b**) State-consistent representation; (

**c**) Memory-equipped model (memory cell indicated with $\mathcal{M}$); (

**d**) Hierarchical states structure.

**Figure 5.**Different approaches for selecting actions. Blue circles represent the path selected by the agent. (

**a**) Deep search via action plans: the path selected has the lowest free energy. (

**b**) Habit learning via state-action policies: the agent always samples from the same conditional distribution. (

**c**) Tree search guided by value and policy: the agent selects the actions according to the prior and the expected free energy.

**Table 1.**Implementation and design choices for learning the variational world model of the agent. The table displays one or two examples for each aspect–modality pair, both in the active inference and in the more general active inference literature, when applicable.

Modality | Active Inference | Deep Learning | |
---|---|---|---|

States distribution | Gaussian | [32,33,114,123,129] | [49,87] |

Categorical | [91] | [66,90] | |

Others | - | [92,93,94] | |

Prior model | No prior | - | [103] |

Known prior | [86,102,114] | [70,71,101] | |

Learned prior | [31,33,68,91,123] | [48,49,66] | |

Uncertainty | Precision | [33] | [107,108] |

Ensemble | [91,114] | [112,113] | |

Dropout | [33] | [130] | |

Representation | Task-oriented | - | [29] |

State consistency | [123] | [116,119,121,122,131] | |

Memory-equipped | [32,124] | [47,48,49,50,66] | |

Hierarchical | [132] | [127,128] |

**Table 2.**Implementation and design choices for the action selection process, minimizing the expected free energy. The table displays one or two examples for each aspect–modality pair, both in the active inference and in the deep learning (mainly, reinforcement learning) literature, when applicable. * All active inference methods generally consider hidden states exploration.

Modality | Active Inference | Deep Learning | |
---|---|---|---|

Preferences | Observations | [123] | [137,138,139] |

States | [68] | [140] | |

Rewards | [33,114,126,129] | [66,67,87,155] | |

Learned | [91] | [145,146] | |

Exploration | Hidden states | * | [51,52,148,149] |

Likelihood parameters | [33,91,114] | [112,113] | |

Action selection | Action plans | [32,68] | [55] |

State-action policy | [123,126] | [150,151] | |

Amortized search | [33,154] | [29] |

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Mazzaglia, P.; Verbelen, T.; Çatal, O.; Dhoedt, B.
The Free Energy Principle for Perception and Action: A Deep Learning Perspective. *Entropy* **2022**, *24*, 301.
https://doi.org/10.3390/e24020301

**AMA Style**

Mazzaglia P, Verbelen T, Çatal O, Dhoedt B.
The Free Energy Principle for Perception and Action: A Deep Learning Perspective. *Entropy*. 2022; 24(2):301.
https://doi.org/10.3390/e24020301

**Chicago/Turabian Style**

Mazzaglia, Pietro, Tim Verbelen, Ozan Çatal, and Bart Dhoedt.
2022. "The Free Energy Principle for Perception and Action: A Deep Learning Perspective" *Entropy* 24, no. 2: 301.
https://doi.org/10.3390/e24020301