active_inference - Obsidian Publish

# Active Inference ## Overview Active Inference is a framework for understanding perception, learning, and decision-making based on the Free Energy Principle. It proposes that agents minimize expected free energy through both perception (inferring hidden states) and action (selecting policies that minimize expected surprise). Active inference unifies perception, learning, and decision-making under a single imperative: the minimization of variational free energy. ```mermaid graph TD subgraph "Active Inference Framework" A[Generative Model] --> B[Inference] A --> C[Learning] A --> D[Decision Making] B --> E[State Estimation] D --> F[Policy Selection] C --> G[Model Parameter Updates] E --> F F --> H[Action] H --> I[Environment] I --> J[Observations] J --> B end style A fill:#f9d,stroke:#333 style B fill:#f9d,stroke:#333 style C fill:#f9d,stroke:#333 style D fill:#f9d,stroke:#333 ``` ## Historical Development ### Origins and Evolution Active inference emerged from the Free Energy Principle proposed by Karl Friston in the early 2000s, evolving through several key stages: ```mermaid timeline title Evolution of Active Inference section Foundation 2003-2005 : Free Energy Principle : Initial formulation in neuroscience 2006-2008 : Dynamic Causal Modeling : Generative models for brain connectivity section Development 2009-2012 : Predictive Coding : Hierarchical message passing architectures 2013-2015 : First Active Inference Models : Extension to action and behavior section Maturation 2016-2018 : Expected Free Energy : Formal framework for planning and exploration 2019-2021 : Sophisticated Inference : Hierarchical planning and inference section Current Trends 2022-Present : Computational Applications : Robotics, AI systems, clinical modeling ``` The development of active inference has spanned multiple disciplines: 1. **Neuroscience Phase (2003-2010)**: Initially focused on perceptual inference in the brain 2. **Computational Phase (2010-2017)**: Development of formal mathematical frameworks 3. **Applications Phase (2017-Present)**: Extensions to robotics, psychiatry, and artificial intelligence ### Key Contributors - **Karl Friston**: Originator of the Free Energy Principle and active inference - **Christopher Buckley**: Formalized connections to control theory - **Thomas Parr**: Extended frameworks to hierarchical models - **Maxwell Ramstead**: Cultural and social applications - **Lancelot Da Costa**: Connections to stochastic optimal control ## Core Concepts ### Expected Free Energy ```math G(π) = \sum_τ G(π,τ) ``` where: - $G(π)$ is expected free energy for policy $π$ - $τ$ is future time point - $G(π,τ)$ is expected free energy at time $τ$ The expected free energy for a specific time point decomposes into: ```math G(π,τ) = \underbrace{\mathbb{E}_{q(o_τ,s_τ|π)}[\ln q(s_τ|π) - \ln p(s_τ|o_τ,π)]}_{\text{epistemic value}} + \underbrace{\mathbb{E}_{q(o_τ,s_τ|π)}[\ln q(o_τ|π) - \ln p(o_τ)]}_{\text{pragmatic value}} ``` This decomposition reveals two fundamental components: 1. **Epistemic Value**: Information gain about hidden states (exploration) 2. **Pragmatic Value**: Fulfillment of prior preferences (exploitation) ### Policy Selection ```math P(π) = σ(-γG(π)) ``` where: - $P(π)$ is policy probability - $σ$ is softmax function - $γ$ is precision parameter (inverse temperature) The parameter $γ$ controls exploration-exploitation: - Higher $γ$ → more deterministic (exploitation) - Lower $γ$ → more stochastic (exploration) ## Theoretical Foundations ### Information Geometry Perspective Active inference can be formulated in terms of information geometry, where perception and action operate on different manifolds: ```math \begin{aligned} \text{Perceptual Inference}: \theta^{perc}_{t+1} &= \theta^{perc}_t - \kappa \mathcal{G}^{-1}\nabla_{\theta^{perc}}F \\ \text{Active Inference}: \theta^{act}_{t+1} &= \theta^{act}_t - \kappa \mathcal{G}^{-1}\nabla_{\theta^{act}}G \end{aligned} ``` where: - $\mathcal{G}$ is the Fisher information matrix - $\kappa$ is a learning rate - $\nabla_{\theta^{perc}}F$ is the gradient of variational free energy - $\nabla_{\theta^{act}}G$ is the gradient of expected free energy ### Derivation of Expected Free Energy Starting from the variational free energy: ```math \begin{aligned} F &= \mathbb{E}_{q(s)}[\ln q(s) - \ln p(o,s)] \\ &= \mathbb{E}_{q(s)}[\ln q(s) - \ln p(s|o) - \ln p(o)] \\ &= D_{KL}[q(s)||p(s|o)] - \ln p(o) \end{aligned} ``` The expected free energy extends this to future time points: ```math \begin{aligned} G(π,τ) &= \mathbb{E}_{q(o_τ,s_τ|π)}[\ln q(s_τ|π) - \ln p(o_τ,s_τ|π)] \\ &= \mathbb{E}_{q(o_τ,s_τ|π)}[\ln q(s_τ|π) - \ln p(s_τ|o_τ,π) - \ln p(o_τ)] \\ &= \mathbb{E}_{q(o_τ|π)}[D_{KL}[q(s_τ|π)||p(s_τ|o_τ,π)]] - \mathbb{E}_{q(o_τ|π)}[\ln p(o_τ)] \end{aligned} ``` With further algebraic manipulation and assumptions about conditional independence: ```math \begin{aligned} G(π,τ) &= \underbrace{\mathbb{E}_{q(o_τ,s_τ|π)}[\ln q(s_τ|π) - \ln q(s_τ|o_τ,π)]}_{\text{Expected information gain (epistemic value)}} + \underbrace{\mathbb{E}_{q(o_τ|π)}[-\ln p(o_τ)]}_{\text{Expected surprise (pragmatic value)}} \end{aligned} ``` ### Connection to Variational Inference The relationship between perception (variational inference) and action (active inference) can be formalized as: ```math \begin{aligned} \text{Perception}: q^*(s) &= \argmin_q F[q,o] \\ \text{Action}: a^* &= \argmin_a \mathbb{E}_{q(s_t)}[F[q,a]] \end{aligned} ``` This mathematical formulation shows that both perception and action serve the same objective: minimizing free energy. ## Mathematical Framework ### Discrete Time Active Inference The discrete time formulation uses VFE for both perception and action: ```math \begin{aligned} & \text{Perception (State Inference):} \\ & F_t = \text{KL}[q(s_t)||p(s_t|o_{1:t})] - \ln p(o_t|o_{1:t-1}) \\ & \text{Action Selection:} \\ & a_t^* = \argmin_a \mathbb{E}_{q(s_t)}[F_{t+1}(a)] \\ & \text{Policy Selection:} \\ & P(\pi) = \sigma(-\gamma G(\pi)) \text{ where } G(\pi) = \sum_\tau G(\pi,\tau) \end{aligned} ``` Where the expected free energy for each policy and time point is: ```math \begin{aligned} G(\pi,\tau) &= \mathbb{E}_{q(o_\tau,s_\tau|\pi)}[\ln q(s_\tau|\pi) - \ln p(s_\tau|o_\tau)] \\ &+ \mathbb{E}_{q(o_\tau,s_\tau|\pi)}[\ln q(o_\tau|\pi) - \ln p(o_\tau)] \end{aligned} ``` ### Continuous Time Active Inference The continuous time extension incorporates path integral formulation: ```math \begin{aligned} & \text{State Dynamics:} \\ & dF = \frac{\partial F}{\partial s}ds + \frac{1}{2}\text{tr}\left(\frac{\partial^2 F}{\partial s^2}D\right)dt \\ & \text{Action Selection:} \\ & a^* = \argmin_a \int_t^{t+dt} \mathcal{L}(s(\tau), \dot{s}(\tau), a) d\tau \\ & \text{Policy Selection:} \\ & P(\pi) = \sigma(-\gamma \int_t^{t+T} \mathcal{L}_\pi d\tau) \end{aligned} ``` Where the Lagrangian $\mathcal{L}$ contains both prediction error and information-theoretic terms. ### Unified Framework The relationship between discrete and continuous time is bridged by: ```math \begin{aligned} & F_{\text{discrete}} = -\ln p(o_t|s_t) - \ln p(s_t|s_{t-1}, a_{t-1}) + \ln q(s_t) \\ & F_{\text{continuous}} = \int_t^{t+dt} [\mathcal{L}(s,\dot{s},a) + \text{KL}[q(s(\tau))||p(s(\tau)|o(\tau))]] d\tau \end{aligned} ``` ## Neural Implementation ### Predictive Coding Architecture ```mermaid graph TD subgraph "Hierarchical Neural Network" A[Sensory Input] --> B[Level 1 Error Units] B --> C[Level 1 State Units] C --> D[Level 2 Error Units] D --> E[Level 2 State Units] E --> F[Level 3 Error Units] F --> G[Level 3 State Units] C --> B E --> D G --> F end style A fill:#f9f,stroke:#333 style B fill:#bbf,stroke:#333 style C fill:#bfb,stroke:#333 ``` The neural implementation follows a hierarchical predictive coding architecture: ```math \begin{aligned} \text{Error Units: } & \varepsilon_l = \mu_l - g(\mu_{l+1}) \\ \text{State Updates: } & \dot{\mu}_l = -\frac{\partial F}{\partial \mu_l} = \kappa_l(\varepsilon_{l-1} - \varepsilon_l) \\ \text{Precision Updates: } & \dot{\pi}_l = -\frac{\partial F}{\partial \pi_l} = \frac{1}{2}(\varepsilon_l^2 - \pi_l^{-1}) \end{aligned} ``` ### Message Passing Implementation Neural populations implement message passing through: 1. **Forward Messages (Bottom-up)** ```math m_{l \rightarrow l+1} = \Pi_l \varepsilon_l ``` 2. **Backward Messages (Top-down)** ```math m_{l+1 \rightarrow l} = \frac{\partial g}{\partial \mu_{l+1}} \Pi_{l+1} \varepsilon_{l+1} ``` 3. **Lateral Interactions** ```math m_{l \leftrightarrow l} = \Pi_l \frac{\partial^2 F}{\partial \mu_l^2} ``` ## Markov Blanket Formulation ### Active Inference and Markov Blankets ```mermaid graph TD subgraph "Active Inference through Markov Blankets" E[External States η] --> S[Sensory States s] S --> I[Internal States μ] I --> A[Active States a] A --> E S --> A I --> P[Predictions] P --> S A --> O[Outcomes] O --> S end style E fill:#f9f,stroke:#333 style S fill:#bbf,stroke:#333 style I fill:#bfb,stroke:#333 style A fill:#fbb,stroke:#333 ``` ### Dynamical Formulation ```math \begin{aligned} & \text{State Dynamics:} \\ & \begin{bmatrix} \dot{η} \\ \dot{s} \\ \dot{a} \\ \dot{μ} \end{bmatrix} = \begin{bmatrix} f_η(η,s,a) \\ f_s(η,s) \\ f_a(s,a,μ) \\ f_μ(s,a,μ) \end{bmatrix} + ω \\ & \text{where } ω \text{ is random fluctuation} \end{aligned} ``` ### Conditional Dependencies ```math \begin{aligned} & \text{Sensory States:} \quad p(s|η,μ) = p(s|η) \\ & \text{Active States:} \quad p(a|s,η,μ) = p(a|s,μ) \\ & \text{Flow:} \quad \dot{x} = (Q - \Gamma)\nabla_x F \end{aligned} ``` ## Technical Foundations ### Information Geometry of Active Inference ```mermaid graph TD subgraph "Geometric Structure" M[Statistical Manifold] --> B[Belief Space] B --> P[Policy Space] P --> A[Action Space] A --> O[Outcome Space] O --> M end style M fill:#f9f,stroke:#333 style B fill:#bbf,stroke:#333 style P fill:#bfb,stroke:#333 ``` ### Variational Derivation ```math \begin{aligned} & \text{Expected Free Energy:} \\ & G(π) = \mathbb{E}_{q(o,s|π)}[\ln q(s|π) - \ln p(o,s|π)] \\ & \text{Policy Selection:} \\ & q(π) = σ(-γG(π)) \\ & \text{State Estimation:} \\ & \dot{μ} = -\kappa \frac{\partial F}{\partial μ} \end{aligned} ``` ### Path Integral Policy ```math \begin{aligned} & \text{Optimal Policy:} \\ & π^* = \argmin_π \int_t^{t+T} \mathcal{L}(s,a,π)dt \\ & \text{where:} \\ & \mathcal{L}(s,a,π) = G(π) + \frac{1}{2}||a||^2_R \end{aligned} ``` ## Mathematical Connections ### Differential Geometry Framework ```math \begin{aligned} & \text{Metric Tensor:} \\ & g_{ij} = \mathbb{E}_q\left[\frac{\partial \ln q}{\partial θ^i}\frac{\partial \ln q}{\partial θ^j}\right] \\ & \text{Natural Gradient Flow:} \\ & \dot{θ} = -g^{ij}\frac{\partial G}{\partial θ^j} \\ & \text{Christoffel Symbols:} \\ & \Gamma^k_{ij} = \frac{1}{2}g^{kl}(\partial_ig_{jl} + \partial_jg_{il} - \partial_lg_{ij}) \end{aligned} ``` ### Stochastic Optimal Control ```mermaid graph LR subgraph "Control Theory Connection" V[Value Function] --> Q[Q-Function] Q --> A[Action Selection] A --> S[State Transition] S --> R[Reward] R --> V end style V fill:#f9f,stroke:#333 style Q fill:#bbf,stroke:#333 style A fill:#bfb,stroke:#333 ``` ### Hamiltonian Mechanics ```math \begin{aligned} & \text{Hamiltonian:} \\ & H(s,p) = p^T f(s) + \frac{1}{2}p^T \Sigma p + V(s) \\ & \text{Hamilton's Equations:} \\ & \dot{s} = \frac{\partial H}{\partial p}, \quad \dot{p} = -\frac{\partial H}{\partial s} \\ & \text{Action:} \\ & S = \int_0^T (p^T \dot{s} - H(s,p))dt \end{aligned} ``` ## Computational Frameworks ### Deep Active Inference ```python class DeepActiveInference: def __init__(self, state_dim: int, obs_dim: int, action_dim: int): """Initialize deep active inference model.""" self.encoder = ProbabilisticEncoder(obs_dim, state_dim) self.decoder = ProbabilisticDecoder(state_dim, obs_dim) self.transition = StateTransitionModel(state_dim, action_dim) self.policy = PolicyNetwork(state_dim, action_dim) def infer_state(self, obs: torch.Tensor) -> Distribution: """Infer latent state from observation.""" return self.encoder(obs) def predict_next_state(self, state: torch.Tensor, action: torch.Tensor) -> Distribution: """Predict next state given current state and action.""" return self.transition(state, action) def compute_free_energy(self, obs: torch.Tensor, state: Distribution) -> torch.Tensor: """Compute variational free energy.""" # Reconstruction term expected_llh = self.decoder.expected_log_likelihood(obs, state) # KL divergence term kl_div = kl_divergence(state, self.prior) return -expected_llh + kl_div ``` ### Hierarchical Implementation ```python class HierarchicalActiveInference: def __init__(self, level_dims: List[int], temporal_scales: List[float]): """Initialize hierarchical active inference model.""" self.levels = nn.ModuleList([ ActiveInferenceLevel(dim_in, dim_out, scale) for dim_in, dim_out, scale in zip(level_dims[:-1], level_dims[1:], temporal_scales) ]) def update_beliefs(self, obs: torch.Tensor) -> List[Distribution]: """Update beliefs across hierarchy.""" # Bottom-up pass prediction_errors = [] current_input = obs for level in self.levels: state_dist = level.infer_state(current_input) pred_error = level.compute_prediction_error(current_input) prediction_errors.append(pred_error) current_input = state_dist.mean # Top-down pass for level_idx in reversed(range(len(self.levels))): self.levels[level_idx].update_state(prediction_errors[level_idx]) return [level.get_state_distribution() for level in self.levels] ``` ## Advanced Applications ### 1. Computational Psychiatry ```mermaid graph TD subgraph "Psychiatric Conditions" A[Normal Function] --> B[Precision Disruption] B --> C[Schizophrenia] B --> D[Autism] A --> E[Prior Disruption] E --> F[Depression] E --> G[Anxiety] end style A fill:#f9f,stroke:#333 style B fill:#bbf,stroke:#333 style E fill:#bfb,stroke:#333 ``` ### 2. Robotics and Control ```mermaid graph LR subgraph "Robot Control Architecture" A[Sensory Input] --> B[State Estimation] B --> C[Policy Selection] C --> D[Motor Commands] D --> E[Environment] E --> A end style A fill:#f9f,stroke:#333 style C fill:#bbf,stroke:#333 style D fill:#bfb,stroke:#333 ``` ### 3. Social Cognition ```mermaid graph TD subgraph "Multi-Agent Active Inference" A[Agent 1 Model] --> B[Joint Action Space] C[Agent 2 Model] --> B B --> D[Collective Behavior] D --> E[Environmental Change] E --> A E --> C end style A fill:#f9f,stroke:#333 style B fill:#bbf,stroke:#333 style D fill:#bfb,stroke:#333 ``` ## Connection to Other Frameworks ### Reinforcement Learning Active inference offers an alternative to reinforcement learning with several key differences: ```mermaid graph TD subgraph "Comparison with Reinforcement Learning" A[Objective Function] --> B[RL: Expected Reward] A --> C[AI: Expected Free Energy] D[State Representation] --> E[RL: Value Function] D --> F[AI: Belief Distribution] G[Action Selection] --> H[RL: Policy Gradient] G --> I[AI: Free Energy Minimization] J[Exploration] --> K[RL: Explicit Strategies] J --> L[AI: Epistemic Value] end style B fill:#bbf,stroke:#333 style C fill:#f99,stroke:#333 style E fill:#bbf,stroke:#333 style F fill:#f99,stroke:#333 style H fill:#bbf,stroke:#333 style I fill:#f99,stroke:#333 style K fill:#bbf,stroke:#333 style L fill:#f99,stroke:#333 ``` Key mappings between frameworks: 1. **Value Function** ↔ **Negative Expected Free Energy** 2. **Reward** ↔ **Log Prior Probability** 3. **Policy Gradient** ↔ **Free Energy Gradient** 4. **Exploration Bonus** ↔ **Epistemic Value** ### Optimal Control Theory Active inference can be related to optimal control theory: ```math \begin{aligned} \text{LQG Control:} \quad J &= \mathbb{E}\left[\sum_t (x_t^T Q x_t + u_t^T R u_t)\right] \\ \text{Active Inference:} \quad G &= \mathbb{E}\left[\sum_t (-\log p(o_t) + \mathbb{I}[q(s_t|\pi)||q(s_t|o_t,\pi)])\right] \end{aligned} ``` Where: - $J$ is the cost functional in optimal control - $G$ is the expected free energy in active inference - $Q, R$ are cost matrices - $\mathbb{I}$ is mutual information ### Planning as Inference Active inference instantiates planning as probabilistic inference: ```math \begin{aligned} p(a|o) &\propto p(o|a)p(a) \\ P(\pi) &\propto \exp(-\gamma G(\pi)) \end{aligned} ``` Where planning is transformed into: 1. Inferring the posterior over policies 2. Selecting actions according to this posterior 3. Updating the model based on new observations ## Computational Complexity Analysis ### Time Complexity The computational complexity of active inference scales with: 1. **State Dimension**: $O(d_s^3)$ for matrix operations in belief updates 2. **Policy Depth**: $O(T)$ for T-step policies 3. **Policy Breadth**: $O(|\Pi|)$ for number of considered policies For standard implementations: - State inference: $O(n_{\text{iter}} \cdot d_s^3)$ - Policy evaluation: $O(|\Pi| \cdot T \cdot d_s^3)$ - Total per step: $O(n_{\text{iter}} \cdot d_s^3 + |\Pi| \cdot T \cdot d_s^3)$ ### Space Complexity Space requirements scale with: 1. **Model Parameters**: $O(d_s^2 \cdot d_a + d_s \cdot d_o)$ 2. **Belief States**: $O(d_s)$ 3. **Policy Storage**: $O(|\Pi| \cdot T)$ ### Approximation Methods To address computational complexity: 1. **Amortized Inference**: Neural networks to approximate inference 2. **Sparse Precision Matrices**: Exploit structure in covariance 3. **Monte Carlo Methods**: Sample-based approximations of expected free energy 4. **Hierarchical Policies**: Decompose policies into manageable sub-policies ## Best Practices ### Model Design 1. Choose appropriate dimensions 2. Initialize transition models 3. Set precision parameters 4. Design reward function (prior preferences) 5. Structure hierarchical models for efficiency ### Implementation 1. Monitor convergence of inference 2. Handle numerical stability issues 3. Validate inference quality 4. Test policy selection 5. Implement efficient matrix operations ### Training 1. Tune learning rates 2. Adjust temperature (precision) parameters 3. Balance exploration-exploitation 4. Validate performance metrics 5. Implement adaptive precision ## Common Issues ### Technical Challenges 1. State inference instability 2. Policy divergence 3. Reward (preference) specification 4. Local optima 5. Computational scaling ### Solutions 1. Careful initialization and regularization 2. Gradient clipping for stability 3. Preference learning from demonstrations 4. Multiple restarts or annealing 5. Hierarchical decomposition ## Future Directions ### Theoretical Developments 1. **Information Geometry**: Formal connections to differential geometry 2. **Thermodynamic Interpretations**: Links to non-equilibrium statistical physics 3. **Quantum Active Inference**: Extension to quantum probability ### Practical Advances 1. **Large-Scale Models**: Scaling to high-dimensional state spaces 2. **Hardware Implementations**: Neuromorphic computing architectures 3. **Hybrid Systems**: Integration with deep learning and other frameworks ### Emerging Applications 1. **Artificial General Intelligence**: Integrated perception-action systems 2. **Human-AI Interaction**: Modeling and predicting human behavior 3. **Social Systems**: Extending to social cognition and cultural dynamics ## Quantum Active Inference ### Quantum Formulation The quantum extension of active inference uses quantum probability theory: ```math \begin{aligned} \text{Quantum State:} \quad |\psi\rangle &= \sum_s \sqrt{q(s)}|s\rangle \\ \text{Density Matrix:} \quad \rho &= |\psi\rangle\langle\psi| \\ \text{Quantum Free Energy:} \quad F_Q &= \text{Tr}(\rho H) + \text{Tr}(\rho \ln \rho) \end{aligned} ``` ### Quantum Policy Selection ```math \begin{aligned} \text{Policy Operator:} \quad \hat{\pi} &= \sum_\pi |\pi\rangle\langle\pi| \\ \text{Expected Free Energy:} \quad G_Q(\pi) &= \text{Tr}(\rho_\pi H) + S(\rho_\pi) \\ \text{Policy Selection:} \quad P(\pi) &= \frac{\exp(-\beta G_Q(\pi))}{Z} \end{aligned} ``` ### Implementation Framework ```python class QuantumActiveInference: def __init__(self, hilbert_dim: int, n_policies: int): """Initialize quantum active inference model.""" self.hilbert_dim = hilbert_dim self.n_policies = n_policies # Initialize quantum operators self.hamiltonian = self._init_hamiltonian() self.policy_operators = self._init_policy_operators() def compute_quantum_free_energy(self, state: np.ndarray) -> float: """Compute quantum free energy.""" # Convert state to density matrix rho = self._state_to_density_matrix(state) # Compute energy term energy = np.trace(rho @ self.hamiltonian) # Compute von Neumann entropy entropy = -np.trace(rho @ np.log(rho)) return energy + entropy ``` ## Advanced Mathematical Formulations ### Differential Geometry Framework Active inference can be formulated using differential geometry: ```math \begin{aligned} \text{Tangent Space:} \quad T_pM &= \text{span}\{\partial_\theta^i\} \\ \text{Cotangent Space:} \quad T^*_pM &= \text{span}\{d\theta_i\} \\ \text{Metric:} \quad g_{ij} &= \mathbb{E}_{p(o|\theta)}\left[\frac{\partial \ln p}{\partial \theta^i}\frac{\partial \ln p}{\partial \theta^j}\right] \end{aligned} ``` ### Symplectic Structure ```math \begin{aligned} \text{Symplectic Form:} \quad \omega &= \sum_i dp_i \wedge dq_i \\ \text{Hamiltonian Flow:} \quad X_H &= \omega^{-1}(dH) \\ \text{Action:} \quad S[q] &= \int_0^T (p\dot{q} - H(p,q))dt \end{aligned} ``` ### Category Theory Perspective ```mermaid graph TD subgraph "Categorical Active Inference" A[State Category] --> B[Observation Category] B --> C[Action Category] C --> D[Policy Category] D --> A E[Free Energy Functor] --> F[Natural Transformation] F --> G[Adjoint Functors] end style A fill:#f9f,stroke:#333 style E fill:#bbf,stroke:#333 style G fill:#bfb,stroke:#333 ``` ## Advanced Implementation Frameworks ### Deep Active Inference Networks ```python class DeepActiveInferenceNetwork(nn.Module): def __init__(self, state_dim: int, obs_dim: int, latent_dim: int): """Initialize deep active inference network.""" super().__init__() # Encoder network (recognition model) self.encoder = nn.Sequential( nn.Linear(obs_dim, 256), nn.ReLU(), nn.Linear(256, 2 * latent_dim) # Mean and log variance ) # Decoder network (generative model) self.decoder = nn.Sequential( nn.Linear(latent_dim, 256), nn.ReLU(), nn.Linear(256, obs_dim) ) # Policy network self.policy = nn.Sequential( nn.Linear(latent_dim, 256), nn.ReLU(), nn.Linear(256, state_dim) ) def encode(self, x: torch.Tensor) -> Tuple[torch.Tensor, torch.Tensor]: """Encode observations to latent space.""" h = self.encoder(x) mu, log_var = torch.chunk(h, 2, dim=-1) return mu, log_var def decode(self, z: torch.Tensor) -> torch.Tensor: """Decode latent states to observations.""" return self.decoder(z) def reparameterize(self, mu: torch.Tensor, log_var: torch.Tensor) -> torch.Tensor: """Reparameterization trick.""" std = torch.exp(0.5 * log_var) eps = torch.randn_like(std) return mu + eps * std def forward(self, x: torch.Tensor) -> Dict[str, torch.Tensor]: """Forward pass.""" # Encode mu, log_var = self.encode(x) # Sample latent state z = self.reparameterize(mu, log_var) # Decode x_recon = self.decode(z) # Compute policy policy = self.policy(z) return { 'mu': mu, 'log_var': log_var, 'z': z, 'x_recon': x_recon, 'policy': policy } ``` ### Hierarchical Message Passing ```python class HierarchicalMessagePassing: def __init__(self, n_levels: int, dims_per_level: List[int]): """Initialize hierarchical message passing.""" self.n_levels = n_levels self.dims = dims_per_level # Initialize level-specific networks self.level_networks = nn.ModuleList([ LevelNetwork(dim_in, dim_out) for dim_in, dim_out in zip(dims_per_level[:-1], dims_per_level[1:]) ]) def forward_pass(self, observation: torch.Tensor) -> List[torch.Tensor]: """Forward pass through hierarchy.""" messages = [] current = observation # Bottom-up pass for network in self.level_networks: message = network.forward_message(current) messages.append(message) current = message return messages def backward_pass(self, messages: List[torch.Tensor]) -> List[torch.Tensor]: """Backward pass through hierarchy.""" predictions = [] current = messages[-1] # Top-down pass for network in reversed(self.level_networks): prediction = network.backward_message(current) predictions.append(prediction) current = prediction return predictions[::-1] ``` ## Advanced Applications ### 1. Neuromorphic Implementation ```mermaid graph TD subgraph "Neuromorphic Architecture" A[Sensory Input] --> B[Silicon Neurons] B --> C[Synaptic Updates] C --> D[Membrane Potentials] D --> E[Spike Generation] E --> F[Action Selection] F --> G[Motor Output] end style A fill:#f9f,stroke:#333 style C fill:#bbf,stroke:#333 style E fill:#bfb,stroke:#333 ``` ### 2. Quantum Robotics ```mermaid graph LR subgraph "Quantum Robot Control" A[Quantum State] --> B[Quantum Measurement] B --> C[State Collapse] C --> D[Policy Selection] D --> E[Quantum Action] E --> F[Environment] F --> A end style A fill:#f9f,stroke:#333 style C fill:#bbf,stroke:#333 style E fill:#bfb,stroke:#333 ``` ### 3. Social Active Inference ```mermaid graph TD subgraph "Multi-Agent Network" A[Agent 1] --> B[Shared Beliefs] C[Agent 2] --> B D[Agent 3] --> B B --> E[Collective Action] E --> F[Environmental Change] F --> A F --> C F --> D end style A fill:#f9f,stroke:#333 style B fill:#bbf,stroke:#333 style E fill:#bfb,stroke:#333 ``` ## Related Documentation - [[free_energy_principle]] - [[predictive_coding]] - [[policy_selection]] - [[variational_inference]] - [[bayesian_brain_hypothesis]] - [[reinforcement_learning]] - [[optimal_control]] - [[planning_as_inference]] - [[precision_weighting]]