# Variational Methods in Cognitive Modeling --- type: mathematical_concept id: variational_methods_001 created: 2024-02-05 modified: 2024-03-15 tags: [mathematics, variational-methods, optimization, inference, variational-inference] aliases: [variational-calculus, variational-inference, variational-bayes] semantic_relations: - type: implements links: - [[active_inference]] - [[free_energy_principle]] - [[bayesian_inference]] - [[belief_updating]] - type: mathematical_basis links: - [[information_theory]] - [[probability_theory]] - [[optimization_theory]] - [[functional_analysis]] - [[differential_geometry]] - type: relates links: - [[belief_updating]] - [[expectation_maximization]] - [[monte_carlo_methods]] - [[path_integral_free_energy]] - [[stochastic_optimization]] - [[optimal_transport]] - type: applications links: - [[deep_learning]] - [[probabilistic_programming]] - [[active_inference]] - [[state_estimation]] - [[dynamical_systems]] - type: documented_by links: - [[../../docs/guides/implementation_guides_index|Implementation Guides]] - [[../../docs/api/api_documentation_index|API Documentation]] --- ## Overview Variational methods provide the mathematical foundation for approximating complex probability distributions and optimizing free energy in cognitive modeling. This document outlines key mathematical principles, implementation approaches, and applications, with a particular focus on variational inference. For foundational mathematical concepts, see [[variational_calculus]], and for physical applications, see [[path_integral_free_energy]]. ## Theoretical Foundations ### Variational Inference Framework The core idea of variational inference (see [[bayesian_inference]], [[information_theory]]) is to approximate complex posterior distributions $p(z|x)$ with simpler variational distributions $q(z)$ by minimizing the KL divergence: ```math q^*(z) = \arg\min_{q \in \mathcal{Q}} \text{KL}(q(z) || p(z|x)) ``` This optimization is equivalent to maximizing the Evidence Lower BOund (ELBO) (see [[free_energy]], [[information_theory]]): ```math \text{ELBO}(q) = \mathbb{E}_{q(z)}[\ln p(x,z) - \ln q(z)] ``` ### Mean Field Approximation Under the mean field assumption (see [[statistical_physics]], [[information_geometry]]), the variational distribution factorizes as: ```math q(z) = \prod_{i=1}^M q_i(z_i) ``` This leads to the coordinate ascent updates (see [[optimization_theory]], [[natural_gradients]]): ```math \ln q_j^*(z_j) = \mathbb{E}_{q_{-j}}[\ln p(x,z)] + \text{const} ``` ### Stochastic Variational Inference For large-scale problems (see [[stochastic_optimization]], [[monte_carlo_methods]]), stochastic optimization of the ELBO: ```math \nabla_{\phi} \text{ELBO} = \mathbb{E}_{q(z;\phi)}[\nabla_{\phi} \ln q(z;\phi)(\ln p(x,z) - \ln q(z;\phi))] ``` ## Advanced Implementation ### 1. Variational Autoencoder ```python class VariationalAutoencoder: def __init__(self): self.components = { 'encoder': ProbabilisticEncoder( architecture='hierarchical', distribution='gaussian' ), 'decoder': ProbabilisticDecoder( architecture='hierarchical', distribution='bernoulli' ), 'prior': LatentPrior( type='standard_normal', learnable=True ) } def compute_elbo( self, x: torch.Tensor, n_samples: int = 1 ) -> torch.Tensor: """Compute ELBO using reparameterization trick""" # Encode input mu, log_var = self.components['encoder'](x) # Sample latent variables z = self.reparameterize(mu, log_var, n_samples) # Decode samples x_recon = self.components['decoder'](z) # Compute ELBO terms recon_loss = self.reconstruction_loss(x_recon, x) kl_loss = self.kl_divergence(mu, log_var) return recon_loss - kl_loss ``` ### 2. Normalizing Flow ```python class NormalizingFlow: def __init__(self): self.components = { 'base': BaseDensity( type='gaussian', learnable=True ), 'transforms': TransformSequence( architectures=['planar', 'radial'], n_layers=10 ), 'optimizer': FlowOptimizer( method='adam', learning_rate='adaptive' ) } def forward( self, x: torch.Tensor, return_logdet: bool = True ) -> Tuple[torch.Tensor, torch.Tensor]: """Forward pass through flow""" z = x log_det = 0.0 for transform in self.components['transforms']: z, ldj = transform(z) log_det += ldj if return_logdet: return z, log_det return z ``` ### 3. Amortized Inference ```python class AmortizedInference: def __init__(self): self.components = { 'inference_network': InferenceNetwork( architecture='residual', uncertainty='learnable' ), 'generative_model': GenerativeModel( type='hierarchical', latent_dims=[64, 32, 16] ), 'training': AmortizedTrainer( method='importance_weighted', n_particles=10 ) } def infer( self, x: torch.Tensor, n_samples: int = 1 ) -> Distribution: """Perform amortized inference""" # Get variational parameters params = self.components['inference_network'](x) # Sample from variational distribution q = self.construct_distribution(params) z = q.rsample(n_samples) # Compute importance weights log_weights = ( self.components['generative_model'].log_prob(x, z) - q.log_prob(z) ) return self.reweight_distribution(q, log_weights) ``` ## Advanced Methods ### 1. Structured Inference - [[graphical_models]] (see also [[belief_networks]], [[markov_random_fields]]) - Factor graphs - Message passing (see [[belief_propagation]]) - Structured approximations - [[copula_inference]] (see also [[multivariate_statistics]]) - Dependency modeling - Multivariate coupling - Vine copulas ### 2. Implicit Models - [[adversarial_variational_bayes]] - GAN-based inference - Density ratio estimation - Implicit distributions - [[flow_based_models]] - Invertible networks - Change of variables - Density estimation ### 3. Sequential Methods - [[particle_filtering]] - Sequential importance sampling - Resampling strategies - Particle smoothing - [[variational_sequential_monte_carlo]] - Amortized proposals - Structured resampling - Flow transport ## Applications ### 1. Probabilistic Programming - [[automatic_differentiation]] - Reverse mode - Forward mode - Mixed mode - [[program_synthesis]] - Grammar induction - Program inversion - Symbolic abstraction ### 2. Deep Learning - [[deep_generative_models]] - VAEs - Flows - Diffusion models - [[bayesian_neural_networks]] - Weight uncertainty - Function-space inference - Ensemble methods ### 3. State Space Models - [[dynamical_systems]] - Continuous dynamics - Jump processes - Hybrid systems - [[time_series_models]] - State estimation - Parameter learning - Structure discovery ## Research Directions ### 1. Theoretical Extensions - [[optimal_transport]] - Wasserstein inference - Gradient flows - Metric learning - [[information_geometry]] - Natural gradients - Statistical manifolds - Divergence measures ### 2. Scalable Methods - [[distributed_inference]] - Parallel algorithms - Communication efficiency - Consensus methods - [[neural_inference]] - Learned optimizers - Meta-learning - Neural architectures ### 3. Applications - [[scientific_computing]] - Uncertainty quantification - Inverse problems - Model selection - [[decision_making]] - Policy learning - Risk assessment - Active learning ## References - [[blei_2017]] - "Variational Inference: A Review for Statisticians" - [[kingma_2014]] - "Auto-Encoding Variational Bayes" - [[rezende_2015]] - "Variational Inference with Normalizing Flows" - [[hoffman_2013]] - "Stochastic Variational Inference" ## See Also - [[variational_calculus]] - [[bayesian_inference]] - [[monte_carlo_methods]] - [[optimization_theory]] - [[information_theory]] - [[probabilistic_programming]] - [[deep_learning]] ## Numerical Methods ### Optimization Algorithms - [[gradient_descent]] - First-order methods - [[conjugate_gradient]] - Second-order methods - [[quasi_newton]] - Approximate Newton - [[trust_region]] - Trust region methods ### Sampling Methods - [[importance_sampling]] - IS techniques - [[hamiltonian_mc]] - HMC sampling - [[sequential_mc]] - SMC methods - [[variational_sampling]] - Variational approaches ### Implementation Considerations - [[numerical_stability]] - Stability issues - [[convergence_criteria]] - Convergence checks - [[hyperparameter_tuning]] - Parameter selection - [[computational_efficiency]] - Efficiency concerns ## Validation Framework ### Quality Metrics ```python class VariationalMetrics: """Quality metrics for variational methods.""" @staticmethod def compute_kl_divergence(p: np.ndarray, q: np.ndarray) -> float: """Compute KL divergence between distributions.""" return np.sum(p * (np.log(p + 1e-10) - np.log(q + 1e-10))) @staticmethod def compute_elbo(model: GenerativeModel, variational_dist: Distribution, data: np.ndarray) -> float: """Compute Evidence Lower BOund.""" return model.expected_log_likelihood(data, variational_dist) - \ model.kl_divergence(variational_dist) ``` ### Performance Analysis - [[convergence_analysis]] - Convergence properties - [[complexity_analysis]] - Computational complexity - [[accuracy_metrics]] - Approximation quality - [[robustness_tests]] - Stability testing ## Integration Points ### Theory Integration - [[active_inference]] - Active inference framework (see also [[free_energy_principle]]) - [[predictive_coding]] - Predictive processing (see also [[hierarchical_inference]]) - [[message_passing]] - Belief propagation (see also [[factor_graphs]]) - [[probabilistic_inference]] - Probabilistic methods (see also [[bayesian_statistics]]) ### Implementation Links - [[optimization_methods]] - Optimization techniques (see also [[natural_gradients]]) - [[inference_algorithms]] - Inference methods (see also [[monte_carlo_methods]]) - [[sampling_approaches]] - Sampling strategies (see also [[mcmc_methods]]) - [[numerical_implementations]] - Numerical methods (see also [[numerical_optimization]]) ## Documentation Links - [[../../docs/research/research_documentation_index|Research Documentation]] - [[../../docs/guides/implementation_guides_index|Implementation Guides]] - [[../../docs/api/api_documentation_index|API Documentation]] - [[../../docs/examples/usage_examples_index|Usage Examples]] ## References - [[jordan_1999]] - Introduction to Variational Methods - [[wainwright_2008]] - Graphical Models - [[zhang_2018]] - Natural Gradient Methods --- title: Variational Methods type: concept status: stable created: 2024-02-12 tags: - mathematics - optimization - inference semantic_relations: - type: foundation links: - [[calculus_of_variations]] - [[optimization_theory]] - type: relates links: - [[variational_inference]] - [[optimal_control]] - [[machine_learning]] --- # Variational Methods ## Core Concepts ### Calculus of Variations 1. **Euler-Lagrange Equation** ```math \frac{d}{dx}\frac{∂L}{∂y'} - \frac{∂L}{∂y} = 0 ``` where: - L is Lagrangian - y is function - y' is derivative 2. **Hamilton's Principle** ```math δS = δ\int_{t_1}^{t_2} L(q,\dot{q},t)dt = 0 ``` where: - S is action - L is Lagrangian - q is generalized coordinate ### Variational Optimization 1. **Functional Gradient** ```math \frac{δF}{δf} = \lim_{ε→0} \frac{F[f + εη] - F[f]}{ε} ``` where: - F is functional - f is function - η is test function 2. **Natural Gradient** ```math \nabla_F f = G^{-1}\nabla f ``` where: - G is Fisher information matrix - ∇f is Euclidean gradient ## Advanced Methods ### Variational Inference 1. **Evidence Lower Bound** ```math ELBO(q) = E_q[log p(x,z)] - E_q[log q(z)] ``` where: - q(z) is variational distribution - p(x,z) is joint distribution 2. **Reparameterization Trick** ```math z = g_φ(ε,x), ε ~ p(ε) ``` where: - g_φ is transformation - ε is noise variable - φ are parameters ### Optimal Transport 1. **Wasserstein Distance** ```math W_p(μ,ν) = (\inf_γ \int ||x-y||^p dγ(x,y))^{1/p} ``` where: - μ,ν are distributions - γ is transport plan 2. **Kantorovich Duality** ```math W_1(μ,ν) = \sup_{||f||_L≤1} \int f d(μ-ν) ``` where: - f is potential function - ||f||_L is Lipschitz norm ### Stochastic Methods 1. **Stochastic Gradient Descent** ```math θ_{t+1} = θ_t - α_t\nabla_θ L(θ_t,x_t) ``` where: - θ are parameters - α is learning rate - L is loss function 2. **Stochastic Variational Inference** ```math λ_{t+1} = λ_t + ρ_t\nabla_λ L_t(λ_t) ``` where: - λ are variational parameters - ρ is step size - L is local ELBO ## Applications ### Machine Learning 1. **Variational Autoencoders** ```math L(θ,φ;x) = E_{q_φ(z|x)}[log p_θ(x|z)] - KL(q_φ(z|x)||p(z)) ``` where: - θ,φ are parameters - q_φ is encoder - p_θ is decoder 2. **Normalizing Flows** ```math log p_K(x) = log p_0(f^{-1}_K ∘...∘ f^{-1}_1(x)) + \sum_{k=1}^K log|det \frac{∂f^{-1}_k}{∂x}| ``` where: - p_K is transformed density - f_k are invertible maps ### Physics 1. **Quantum Mechanics** ```math δ\int ψ^*[-\frac{ℏ^2}{2m}\nabla^2 + V]ψ dx = 0 ``` where: - ψ is wavefunction - V is potential - ℏ is Planck constant 2. **Field Theory** ```math S[φ] = \int d^4x \mathcal{L}(φ,∂_μφ) ``` where: - S is action - φ is field - L is Lagrangian density ### Control Theory 1. **Linear Quadratic Regulator** ```math J = \int_0^T (x^TQx + u^TRu)dt ``` where: - Q,R are cost matrices - x is state - u is control 2. **Model Predictive Control** ```math min_u \sum_{k=0}^{N-1} l(x_k,u_k) + V_f(x_N) ``` where: - l is stage cost - V_f is terminal cost - N is horizon ## Implementation ### Optimization Algorithms ```python class VariationalOptimizer: def __init__(self, objective: Callable, method: str = 'natural'): """Initialize variational optimizer. Args: objective: Objective functional method: Optimization method """ self.objective = objective self.method = method def optimize(self, initial_params: np.ndarray, n_steps: int) -> np.ndarray: """Optimize variational parameters. Args: initial_params: Starting parameters n_steps: Number of optimization steps Returns: optimal_params: Optimized parameters """ params = initial_params.copy() for _ in range(n_steps): if self.method == 'natural': grad = self.natural_gradient(params) else: grad = self.euclidean_gradient(params) params = self.update_step(params, grad) return params ``` ### Variational Inference ```python class VariationalInference: def __init__(self, model: ProbabilisticModel, guide: VariationalGuide): """Initialize variational inference. Args: model: Probabilistic model guide: Variational guide """ self.model = model self.guide = guide def elbo(self, x: torch.Tensor) -> torch.Tensor: """Compute ELBO. Args: x: Observed data Returns: elbo: Evidence lower bound """ # Sample from guide z = self.guide.sample(x) # Compute log probabilities log_p = self.model.log_prob(x, z) log_q = self.guide.log_prob(z, x) return log_p - log_q ``` ## Advanced Topics ### Information Geometry 1. **Statistical Manifolds** ```math ds² = g_{ij}(θ)dθ^idθ^j ``` where: - g_{ij} is Fisher metric - θ are statistical parameters 2. **Natural Gradient Flow** ```math \dot{θ} = -g^{ij}∂_jF ``` where: - g^{ij} is inverse metric - F is free energy ### Quantum Variational Methods 1. **Variational Quantum Eigensolver** ```math E(θ) = ⟨ψ(θ)|H|ψ(θ)⟩ ``` where: - ψ(θ) is parameterized state - H is Hamiltonian 2. **Quantum Approximate Optimization** ```math |ψ(β,γ)⟩ = e^{-iβ_pH_B}e^{-iγ_pH_C}...e^{-iβ_1H_B}e^{-iγ_1H_C}|s⟩ ``` where: - H_B,H_C are Hamiltonians - β,γ are parameters ## Future Directions ### Emerging Areas 1. **Deep Variational Methods** - Neural ODEs - Continuous normalizing flows - Variational transformers 2. **Quantum Applications** - Quantum machine learning - Quantum simulation - Quantum control ### Open Problems 1. **Theoretical Challenges** - Non-convex optimization - Convergence guarantees - Sample complexity 2. **Practical Challenges** - Scalability - Robustness - Model selection ## Related Topics 1. [[optimization_theory|Optimization Theory]] 2. [[information_geometry|Information Geometry]] 3. [[quantum_computing|Quantum Computing]] 4. [[machine_learning|Machine Learning]]