scholarly journals Generalised free energy and active inference: can the future cause the past?

2018 ◽  
Author(s):  
Thomas Parr ◽  
Karl J Friston
Keyword(s):  
Free Energy ◽  
Energy Functional ◽  
Generative Model ◽  
Free Energy Function ◽  
Active Inference ◽  
Future Expectations ◽  
Energy Functionals ◽  
The Future ◽  
Markov Decision ◽  
Future Outcomes

AbstractWe compare two free energy functionals for active inference under Markov decision processes. One of these is a functional of beliefs about states and policies, but a function of observations, while the second is a functional of beliefs about all three. In the former (expected free energy), prior beliefs about outcomes are not part of the generative model (because they are absorbed into the prior over policies). Conversely, in the second (generalised free energy); priors over outcomes become an explicit component of the generative model. When using the free energy function, which is blind to counterfactual (i.e., future) observations, we equip the generative model with a prior over policies that ensure preferred (i.e., priors over) outcomes are realised. In other words, selected policies minimise uncertainty about future outcomes by minimising the free energy expected in the future. When using the free energy functional – that effectively treats counterfactual observations as hidden states – we show that policies are inferred or selected that realise prior preferences by minimising the free energy of future expectations. Interestingly, the form of posterior beliefs about policies (and associated belief updating) turns out to be identical under both formulations, but the quantities used to compute them are not.

scholarly journals Generalised free energy and active inference

2019 ◽  
Vol 113 (5-6) ◽  
pp. 495-513 ◽  
Author(s):  
Thomas Parr ◽  
Karl J. Friston
Keyword(s):  
Free Energy ◽  
Generative Model ◽  
Free Energy Function ◽  
Active Inference ◽  
Future Expectations ◽  
Perceptual Inference ◽  
Energy Functionals ◽  
Sensory Data ◽  
The World ◽  
The Brain

Abstract Active inference is an approach to understanding behaviour that rests upon the idea that the brain uses an internal generative model to predict incoming sensory data. The fit between this model and data may be improved in two ways. The brain could optimise probabilistic beliefs about the variables in the generative model (i.e. perceptual inference). Alternatively, by acting on the world, it could change the sensory data, such that they are more consistent with the model. This implies a common objective function (variational free energy) for action and perception that scores the fit between an internal model and the world. We compare two free energy functionals for active inference in the framework of Markov decision processes. One of these is a functional of beliefs (i.e. probability distributions) about states and policies, but a function of observations, while the second is a functional of beliefs about all three. In the former (expected free energy), prior beliefs about outcomes are not part of the generative model (because they are absorbed into the prior over policies). Conversely, in the second (generalised free energy), priors over outcomes become an explicit component of the generative model. When using the free energy function, which is blind to future observations, we equip the generative model with a prior over policies that ensure preferred (i.e. priors over) outcomes are realised. In other words, if we expect to encounter a particular kind of outcome, this lends plausibility to those policies for which this outcome is a consequence. In addition, this formulation ensures that selected policies minimise uncertainty about future outcomes by minimising the free energy expected in the future. When using the free energy functional—that effectively treats future observations as hidden states—we show that policies are inferred or selected that realise prior preferences by minimising the free energy of future expectations. Interestingly, the form of posterior beliefs about policies (and associated belief updating) turns out to be identical under both formulations, but the quantities used to compute them are not.

scholarly journals Existence of ground states for aggregation-diffusion equations

2019 ◽  
Vol 17 (03) ◽  
pp. 393-423 ◽  
Author(s):  
J. A. Carrillo ◽  
M. G. Delgadino ◽  
F. S. Patacchini
Keyword(s):  
Free Energy ◽  
Ground States ◽  
Energy Functional ◽  
Multi Agent Systems ◽  
Linear Diffusion ◽  
Sharp Condition ◽  
Macroscopic Models ◽  
Energy Functionals ◽  
Continuous Description ◽  
Degenerate Diffusions

We analyze free energy functionals for macroscopic models of multi-agent systems interacting via pairwise attractive forces and localized repulsion. The repulsion at the level of the continuous description is modeled by pressure-related terms in the functional making it energetically favorable to spread, while the attraction is modeled through nonlocal forces. We give conditions on general entropies and interaction potentials for which neither ground states nor local minimizers exist. We show that these results are sharp for homogeneous functionals with entropies leading to degenerate diffusions while they are not sharp for fast diffusions. The particular relevant case of linear diffusion is totally clarified giving a sharp condition on the interaction potential under which the corresponding free energy functional has ground states or not.

scholarly journals Canonical neural networks perform active inference

2022 ◽  
Vol 5 (1) ◽  
Author(s):  
Takuya Isomura ◽  
Hideaki Shimazaki ◽  
Karl J. Friston
Keyword(s):  
Neural Networks ◽  
Process Model ◽  
Active Inference ◽  
Hebbian Plasticity ◽  
Neuronal Mechanisms ◽  
Markov Decision ◽  
Rate Coding ◽  
Mathematical Analyses ◽  
Future Outcomes ◽  
And Control

AbstractThis work considers a class of canonical neural networks comprising rate coding models, wherein neural activity and plasticity minimise a common cost function—and plasticity is modulated with a certain delay. We show that such neural networks implicitly perform active inference and learning to minimise the risk associated with future outcomes. Mathematical analyses demonstrate that this biological optimisation can be cast as maximisation of model evidence, or equivalently minimisation of variational free energy, under the well-known form of a partially observed Markov decision process model. This equivalence indicates that the delayed modulation of Hebbian plasticity—accompanied with adaptation of firing thresholds—is a sufficient neuronal substrate to attain Bayes optimal inference and control. We corroborated this proposition using numerical analyses of maze tasks. This theory offers a universal characterisation of canonical neural networks in terms of Bayesian belief updating and provides insight into the neuronal mechanisms underlying planning and adaptive behavioural control.

scholarly journals Incompressible limit for a magnetostrictive energy functional

2013 ◽  
Vol 61 (4) ◽  
pp. 1025-1030
Author(s):  
B. Gambin ◽  
W. Bielski
Keyword(s):  
Free Energy ◽  
Energy Functional ◽  
Elastic Behaviour ◽  
Incompressible Limit ◽  
Elastic Deformations ◽  
Energy Functionals ◽  
Second Gradient ◽  
Non Local ◽  
Γ Convergence ◽  
Non Locality

Abstract The modern materials undergoing large elastic deformations and exhibiting strong magnetostrictive effect are modelled here by free energy functionals for nonlinear and non-local magnetoelastic behaviour. The aim of this work is to prove a new theorem which claims that a sequence of free energy functionals of slightly compressible magnetostrictive materials with a non-local elastic behaviour, converges to an energy functional of a nearly incompressible magnetostrictive material. This convergence is referred to as a Γ -convergence. The non-locality is limited to non-local elastic behaviour which is modelled by a term containing the second gradient of deformation in the energy functional.

scholarly journals Sophisticated Inference

2021 ◽  
Vol 33 (3) ◽  
pp. 713-763
Author(s):  
Karl Friston ◽  
Lancelot Da Costa ◽  
Danijar Hafner ◽  
Casper Hesp ◽  
Thomas Parr
Keyword(s):  
Free Energy ◽  
Information Gain ◽  
Energy Functional ◽  
Decision Problems ◽  
Tree Search ◽  
Value Functions ◽  
Active Inference ◽  
States Of Affairs ◽  
Variational Free Energy ◽  
Belief States

Active inference offers a first principle account of sentient behavior, from which special and important cases—for example, reinforcement learning, active learning, Bayes optimal inference, Bayes optimal design—can be derived. Active inference finesses the exploitation-exploration dilemma in relation to prior preferences by placing information gain on the same footing as reward or value. In brief, active inference replaces value functions with functionals of (Bayesian) beliefs, in the form of an expected (variational) free energy. In this letter, we consider a sophisticated kind of active inference using a recursive form of expected free energy. Sophistication describes the degree to which an agent has beliefs about beliefs. We consider agents with beliefs about the counterfactual consequences of action for states of affairs and beliefs about those latent states. In other words, we move from simply considering beliefs about “what would happen if I did that” to “what I would believe about what would happen if I did that.” The recursive form of the free energy functional effectively implements a deep tree search over actions and outcomes in the future. Crucially, this search is over sequences of belief states as opposed to states per se. We illustrate the competence of this scheme using numerical simulations of deep decision problems.

Time-Independent Plasticity Related to Critical Point of Free Energy Function and Functional

2014 ◽  
Vol 136 (2) ◽  
Author(s):  
Q. Yang ◽  
Y. R. Liu ◽  
X. Q. Feng ◽  
S. W. Yu
Keyword(s):  
Free Energy ◽  
Critical Point ◽  
Thermodynamic Equilibrium ◽  
Energy Function ◽  
Hessian Matrix ◽  
Energy Functional ◽  
Internal Variables ◽  
Free Energy Function ◽  
Free Energy Functional ◽  
Appl Math

In this paper, time-independent plasticity is addressed within the thermodynamic framework with internal variables by Rice (1971, “Inelastic Constitutive Relations for Solids: An Internal Variable Theory and Its Application to Metal Plasticity,” J. Mech. Phys. Solids, 19, pp. 433–455). It is shown in this paper that the existence of a free energy function along with thermodynamic equilibrium conditions directly leads to associated flow rules. The time-independent inelastic behaviors can be fully determined by the Hessian matrix at the nondegenerate critical point of the free energy function. The normality rule of Hill and Rice (1973, “Elastic Potentials and the Structure of Inelastic Constitutive Laws,” SIAM J. Appl. Math., 25, pp. 448–461) or the Il'yushin (1961, “On a Postulate of Plasticity,” J. Appl. Math. Mech. 25, pp. 746–750) postulate is just a stability requirement of the thermodynamic equilibrium. The existence of a free energy functional which is not a direct function of the internal variables, along with thermodynamic equilibrium conditions also leads to associated flow rules. The time-independent inelastic behaviors with the free energy functional can be fully determined by the quasi Hessian matrix at the quasi critical point of the free energy functional. With the free energy functional, the thermodynamic forces conjugate to the internal variables are nonconservative and are constructed based on Darboux theorem. Based on the constructed nonconservative forces, it is shown that there may exist several possible thermodynamic equilibrium mechanisms for the thermodynamic system of the material sample. Therefore, the associated flow rules based on free energy functionals may degenerate into nonassociated flow rules. The symmetry of the conjugate forces plays a central role for the characteristics of time-independent plasticity.

scholarly journals Bayesian theories of consciousness: a review in search for a minimal unifying model

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Wiktor Rorot
Keyword(s):  
Free Energy ◽  
Conscious Experience ◽  
Generative Model ◽  
Predictive Processing ◽  
Active Inference ◽  
Energy Principle ◽  
Free Energy Principle

Abstract The goal of the paper is to review existing work on consciousness within the frameworks of Predictive Processing, Active Inference, and Free Energy Principle. The emphasis is put on the role played by the precision and complexity of the internal generative model. In the light of those proposals, these two properties appear to be the minimal necessary components for the emergence of conscious experience—a Minimal Unifying Model of consciousness.

scholarly journals Whence the Expected Free Energy?

2021 ◽  
pp. 1-36
Author(s):  
Beren Millidge ◽  
Alexander Tschantz ◽  
Christopher L. Buckley
Keyword(s):  
Free Energy ◽  
Exploratory Behavior ◽  
Energy Minimization ◽  
Intrinsic Value ◽  
Natural Extension ◽  
Active Inference ◽  
Exploration And Exploitation ◽  
Free Energy Minimization ◽  
Variational Free Energy ◽  
The Future

The expected free energy (EFE) is a central quantity in the theory of active inference. It is the quantity that all active inference agents are mandated to minimize through action, and its decomposition into extrinsic and intrinsic value terms is key to the balance of exploration and exploitation that active inference agents evince. Despite its importance, the mathematical origins of this quantity and its relation to the variational free energy (VFE) remain unclear. In this letter, we investigate the origins of the EFE in detail and show that it is not simply ”the free energy in the future.” We present a functional that we argue is the natural extension of the VFE but actively discourages exploratory behavior, thus demonstrating that exploration does not directly follow from free energy minimization into the future. We then develop a novel objective, the free energy of the expected future (FEEF), which possesses both the epistemic component of the EFE and an intuitive mathematical grounding as the divergence between predicted and desired futures.

Coarse Grained Free Energy Functional for Lennard-Jones Systems

1999 ◽  
Vol 580 ◽  
Author(s):  
M. E. Gracheva ◽  
J. M. Rickman ◽  
J. D. Gunton ◽  
D. C. Coffey
Keyword(s):  
Free Energy ◽  
Chemical Potential ◽  
Canonical Ensemble ◽  
Particle Density ◽  
Energy Functional ◽  
Distribution Functions ◽  
Coarse Grained ◽  
Free Energy Function ◽  
Gas Phases ◽  
Free Energy Functional

AbstractResults are presented for the coarse grained distribution function and Ginzburg-Landau free energy function for coexistence of liquid and gas phases. These distribution functions were obtained by two different methods: 1) the compilation of particle density information from different coarse-grained cells using the canonical ensemble, and 2) the compilation of energy and density information from a single simulation cell by tuning the chemical potential using the grand-canonical ensemble. Both methods permit the calculation of a coarse-grained free energy functional which links the atomic and mesoscopic length scales.

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