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Path Integral Connections

This article maps the deep connections between path integral formulations and other mathematical frameworks in cognitive science, particularly focusing on information theory, variational methods, and active inference.

Information-Theoretic Connections

Entropy and Path Integrals

Path Entropy Mapping
```
S[P] = -∫ 𝒟x P[x(t)]log P[x(t)] ↔ H(X) = -∑ P(x)log P(x)
```
where:
- Left side: Path entropy (path_integral_information)
- Right side: Shannon entropy (information_theory)
- Connection: Functional generalization
Dynamic Information
```
I[x(t)] = ∫ dt L(x(t), ẋ(t)) ↔ I(X;Y) = ∑ P(x,y)log(P(x,y)/P(x)P(y))
```
where:
- Left side: Path information (path_integral_theory)
- Right side: Mutual information (information_theory)
- Connection: Temporal integration

Free Energy Principles

Variational Mapping
```
F[q] = ∫ 𝒟x q[x(t)]log(q[x(t)]/p[x(t)]) ↔ F = KL[q(s)||p(s|o)]
```
where:
- Left side: Path free energy (path_integral_information)
- Right side: Variational free energy (free_energy_principle)
- Connection: Path space generalization
Expected Free Energy
```
G[π] = ∫ dt E_{q(o,s|π)}[log q(s|π) - log p(o,s)] ↔ G = KL[q(o|π)||p(o)] + H(o|s,π)
```
where:
- Left side: Path expected free energy (path_integral_information)
- Right side: Expected free energy (active_inference)
- Connection: Temporal integration

Geometric Connections

Information Geometry

Metric Structure
```
g_μν[x(t)] = E[∂_μlog p(x(t))∂_νlog p(x(t))] ↔ g_ij = E[∂_ilog p(x)∂_jlog p(x)]
```
where:
- Left side: Path Fisher metric (path_integral_information)
- Right side: Fisher information metric (information_geometry)
- Connection: Functional extension
Natural Gradients
```
ẋ = -g^{μν}∂_νF[x(t)] ↔ θ̇ = -g^{ij}∂_jF(θ)
```
where:
- Left side: Path gradient flow (path_integral_information)
- Right side: Natural gradient (information_geometry)
- Connection: Infinite-dimensional generalization

Variational Methods

Action Principles

Information Action
```
A[x(t)] = ∫ dt [D(ẋ - f(x))² + Q(x)] ↔ S[φ] = ∫ dt L(φ, ∂φ)
```
where:
- Left side: Information action (path_integral_information)
- Right side: Classical action (variational_calculus)
- Connection: Information-theoretic interpretation
Optimization Structure
```
δA[x(t)] = 0 ↔ δF[q] = 0
```
where:
- Left side: Action principle (path_integral_theory)
- Right side: Variational principle (variational_methods)
- Connection: Extremal principles

Active Inference Implementation

Belief Propagation

Path Space Beliefs
```
q[x(t)] = Z⁻¹exp(-A[x(t)]/σ²) ↔ q(s) = σ(-∂F/∂s)
```
where:
- Left side: Path distribution (path_integral_information)
- Right side: Belief update (active_inference)
- Connection: Dynamical generalization
Policy Selection
```
π* = argmin_π ∫ dt G[π(t)] ↔ π* = argmin_π G(π)
```
where:
- Left side: Path policy optimization (path_integral_information)
- Right side: Policy selection (active_inference)
- Connection: Temporal integration

Computational Frameworks

Sampling Methods

Path Space MCMC

def path_mcmc(
    action: Callable,
    initial_path: np.ndarray,
    num_samples: int
) -> List[np.ndarray]:
    """Implements path space MCMC sampling."""
    paths = []
    current = initial_path

    for _ in range(num_samples):
        proposed = propose_path(current)
        if accept_path(current, proposed, action):
            current = proposed
        paths.append(current.copy())

    return paths

Connection to:

Variational Optimization

def optimize_path_density(
    variational_family: Callable,
    target_density: Callable,
    num_iterations: int
) -> Callable:
    """Optimizes variational path density."""
    current_density = initialize_density(variational_family)

    for _ in range(num_iterations):
        gradient = compute_path_gradient(
            current_density, target_density)
        current_density = update_density(
            current_density, gradient)

    return current_density