# Parr & Friston (2019): Generalised Free Energy and Active Inference ## Citation Parr, T., & Friston, K. J. (2019). Generalised free energy and active inference. *Biological Cybernetics*, 113(5-6), 495-513. ## Motivation The original Active Inference framework used two distinct objectives—variational free energy ($F$) for perception and expected free energy ($G$) for planning—creating a theoretical asymmetry. This paper resolves that by introducing generalised free energy as a single unifying objective. ## Key Contributions 1. **Generalised free energy** $\mathcal{F}$ that unifies variational and expected free energy 2. **Formal proof** that $G$ arises naturally from $\mathcal{F}$ when marginalizing over future observations 3. **Temporal depth**: Multi-step policy evaluation as iterated free energy minimization 4. **Risk-sensitive planning**: Natural incorporation of risk aversion through precision over policies 5. **Connection to information geometry**: Generalised free energy as a natural gradient flow ## Core Equations ### Generalised Free Energy ```math \mathcal{F}(\pi) = \underbrace{F(q(s_\tau|\pi), o_{\leq t})}_{\text{past: variational FE}} + \underbrace{\mathbb{E}_{q(o_{>t}|\pi)}[F(q(s_\tau|\pi), o_{>t})]}_{\text{future: expected FE}} ``` ### Epistemic-Pragmatic Decomposition ```math \begin{aligned} G(\pi, \tau) &= \underbrace{-\mathbb{E}_{q(o_\tau|\pi)}[D_{KL}[q(s_\tau|o_\tau, \pi)||q(s_\tau|\pi)]]}_{\text{negative epistemic value (info gain)}} \\ &+ \underbrace{\mathbb{E}_{q(s_\tau|\pi)}[H[p(o_\tau|s_\tau)]]}_{\text{ambiguity}} + \underbrace{D_{KL}[q(o_\tau|\pi)||p(o_\tau)]}_{\text{risk (pragmatic)}} \end{aligned} ``` ### Unified Objective for Perception and Planning ```mermaid graph TD subgraph "Generalised Free Energy" GF[Generalised F] --> VF[Variational F: Perception] GF --> EF[Expected F: Planning] VF --> BU[Belief Updating] EF --> PS[Policy Selection] end style GF fill:#bbf,stroke:#333 ``` ## Theoretical Impact This work resolves a key theoretical gap: prior formulations required separate justifications for perceptual inference (minimize $F$) and action selection (minimize $G$). The generalised formulation shows both emerge from a single variational principle applied at different temporal scales. ### Implications for Implementation | Aspect | Before (2017) | After (2019) | | --- | --- | --- | | Perception | Minimize $F$ | Minimize $\mathcal{F}$ (present) | | Planning | Minimize $G$ | Minimize $\mathcal{F}$ (future) | | Theoretical basis | Two objectives | One unified objective | | Risk sensitivity | Ad hoc | Naturally emerges | | Temporal depth | Fixed | Flexible | ## Frequently Used in Conjunction With - [[friston_2017]] — Original process theory that this paper extends - [[knowledge_base/cognitive/active_inference]] — Core Active Inference framework - [[knowledge_base/mathematics/expected_free_energy]] — Expected free energy derivation - [[knowledge_base/mathematics/variational_free_energy]] — Variational free energy - [[knowledge_base/mathematics/free_energy_principle]] — Free energy principle ## References - Parr, T., & Friston, K. J. (2019). Generalised free energy and active inference. *Biological Cybernetics*, 113(5-6), 495-513. - Friston, K. J., et al. (2017). Active inference: A process theory. *Neural Computation*, 29(1), 1-49. - Da Costa, L., et al. (2020). Active inference on discrete state-spaces.