cognitive/knowledge_base/mathematics/path_integral_implementation.md

---
type: implementation
id: path_integral_implementation_001
created: 2024-02-05
modified: 2024-02-05
tags: [implementation, numerical-methods, path-integrals, active-inference]
aliases: [path-implementation, numerical-path-methods]
---

# Implementation Guide for Path Integral Methods

## Numerical Foundations

### Discretization Framework
```python
class PathDiscretizer:
    """Framework for path discretization schemes."""
    
    def __init__(self, dt: float, scheme: str = 'symplectic'):
        self.dt = dt
        self.scheme = self._initialize_scheme(scheme)
        
    def discretize_action(self, 
                         lagrangian: Callable,
                         path: np.ndarray) -> float:
        """Discretize continuous action functional."""
        discrete_points = self.scheme(path, self.dt)
        return self.compute_discrete_action(lagrangian, discrete_points)
```

Links to:
- [[numerical_discretization]]
- [[finite_elements]]
- [[discrete_mechanics]]

### Stability Analysis
```python
def analyze_stability(scheme: Callable,
                     test_system: Callable,
                     params: Dict) -> Dict[str, float]:
    """Analyze numerical stability of discretization."""
    return {
        'local_error': compute_local_error(scheme, test_system),
        'global_error': compute_global_error(scheme, test_system),
        'energy_drift': compute_energy_conservation(scheme, test_system)
    }
```

Links to:
- [[numerical_stability]]
- [[error_analysis]]
- [[conservation_laws]]

## Path Sampling Methods

### MCMC Implementation
```python
class PathMCMC:
    """Markov Chain Monte Carlo for path sampling."""
    
    def __init__(self, action: Callable, temperature: float):
        self.action = action
        self.beta = 1.0 / temperature
        
    def hamiltonian_mc(self,
                      initial_path: np.ndarray,
                      num_steps: int) -> List[np.ndarray]:
        """Hamiltonian Monte Carlo sampling."""
        paths = []
        current = initial_path
        momentum = self.sample_momentum(current.shape)
        
        for _ in range(num_steps):
            # Leapfrog integration
            next_path, next_momentum = self.leapfrog_step(
                current, momentum)
            
            # Metropolis acceptance
            if self.accept_state(current, next_path,
                               momentum, next_momentum):
                current = next_path
                momentum = next_momentum
            
            paths.append(current.copy())
        
        return paths
```

Links to:
- [[hamiltonian_monte_carlo]]
- [[leapfrog_integration]]
- [[metropolis_hastings]]

### Variational Methods
```python
class VariationalPathSampler:
    """Variational inference for path distributions."""
    
    def optimize_variational_parameters(self,
                                     target_density: Callable,
                                     variational_family: str,
                                     **params) -> Dict:
        """Optimize variational approximation."""
        elbo_history = []
        current_params = self.initialize_parameters(variational_family)
        
        while not self.converged(elbo_history):
            # Compute ELBO gradient
            grad = self.compute_elbo_gradient(
                current_params, target_density)
            
            # Update parameters
            current_params = self.update_parameters(
                current_params, grad, **params)
            
            # Track progress
            elbo = self.compute_elbo(
                current_params, target_density)
            elbo_history.append(elbo)
        
        return current_params
```

Links to:
- [[variational_inference]]
- [[elbo_optimization]]
- [[natural_gradients]]

## Active Inference Specifics

### Free Energy Computation
```python
class PathFreeEnergy:
    """Compute free energy for path ensembles."""
    
    def __init__(self, model_density: Callable):
        self.model = model_density
        
    def compute_path_free_energy(self,
                               paths: np.ndarray,
                               variational_density: Callable) -> float:
        """Compute free energy for path ensemble."""
        # Energy term
        energy = self.compute_path_energy(paths)
        
        # Entropy term
        entropy = self.compute_path_entropy(
            paths, variational_density)
        
        return energy - entropy
```

Links to:
- [[free_energy_principle]]
- [[path_entropy]]
- [[energy_functionals]]

### Policy Optimization
```python
class PathBasedPolicyOptimizer:
    """Optimize policies using path integral methods."""
    
    def optimize_policy(self,
                       initial_state: np.ndarray,
                       horizon: int,
                       num_samples: int) -> Callable:
        """Optimize policy using path sampling."""
        # Generate path ensemble
        paths = self.sample_paths(
            initial_state, horizon, num_samples)
        
        # Compute path weights
        weights = self.compute_path_weights(paths)
        
        # Update policy
        return self.update_policy_parameters(paths, weights)
```

Links to:
- [[policy_gradients]]
- [[path_integral_control]]
- [[importance_sampling]]

## Advanced Implementation Topics

### Parallel Computing
```python
class ParallelPathSampler:
    """Parallel implementation of path sampling."""
    
    def parallel_sample_paths(self,
                            num_paths: int,
                            num_workers: int) -> np.ndarray:
        """Generate paths in parallel."""
        with ProcessPoolExecutor(max_workers=num_workers) as executor:
            paths = list(executor.map(
                self.generate_single_path,
                range(num_paths)))
        return np.array(paths)
```

Links to:
- [[parallel_processing]]
- [[distributed_computing]]
- [[gpu_acceleration]]

### Adaptive Methods
```python
class AdaptivePathSampling:
    """Adaptive sampling methods for paths."""
    
    def adapt_step_size(self,
                       acceptance_rate: float,
                       current_step: float) -> float:
        """Adapt integration step size."""
        target_rate = 0.65  # Optimal acceptance rate
        log_step = np.log(current_step)
        return np.exp(log_step + 
                     self.learning_rate * 
                     (acceptance_rate - target_rate))
```

Links to:
- [[adaptive_mcmc]]
- [[dual_averaging]]
- [[step_size_adaptation]]

## Validation and Testing

### Unit Tests
```python
class PathIntegralTests:
    """Test suite for path integral implementations."""
    
    def test_energy_conservation(self):
        """Test energy conservation in sampling."""
        sampler = PathSampler(self.test_system)
        paths = sampler.generate_paths(num_paths=1000)
        energy_fluctuation = compute_energy_fluctuation(paths)
        assert energy_fluctuation < self.tolerance
```

Links to:
- [[unit_testing]]
- [[integration_testing]]
- [[test_driven_development]]

### Performance Benchmarks
```python
class PathBenchmarks:
    """Benchmark suite for path methods."""
    
    def benchmark_sampling(self,
                         methods: List[Callable],
                         system: Callable) -> Dict[str, float]:
        """Benchmark different sampling methods."""
        results = {}
        for method in methods:
            start_time = time.time()
            paths = method(system)
            end_time = time.time()
            
            results[method.__name__] = {
                'time': end_time - start_time,
                'efficiency': compute_sampling_efficiency(paths)
            }
        return results
```

Links to:
- [[performance_testing]]
- [[benchmarking]]
- [[profiling]]

## References
- [[neal_2011]] - MCMC Using Hamiltonian Dynamics
- [[girolami_2011]] - Riemann Manifold Langevin and Hamiltonian Monte Carlo
- [[hoffman_2014]] - The No-U-Turn Sampler
- [[betancourt_2017]] - A Conceptual Introduction to Hamiltonian Monte Carlo