# 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]]