cognitive/knowledge_base/mathematics/variational_methods.md

Variational Methods in Cognitive Modeling

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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
  • ../../docs/api/api_documentation_index

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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:

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):

\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:

q(z) = \prod_{i=1}^M q_i(z_i)

This leads to the coordinate ascent updates (see optimization_theory, natural_gradients):

\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:

\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

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

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

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

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
• ../../docs/guides/implementation_guides_index
• ../../docs/api/api_documentation_index
• ../../docs/examples/usage_examples_index

References

• jordan_1999 - Introduction to Variational Methods
• wainwright_2008 - Graphical Models
• zhang_2018 - Natural Gradient Methods