# Research Documentation

---
title: Research Documentation
type: index
status: stable
created: 2024-02-06
tags:
  - research
  - theory
  - mathematics
  - statistics
semantic_relations:
  - type: implements
    links: [[../concepts/cognitive_modeling_concepts]]
  - type: relates
    links:
      - [[../concepts/cognitive_modeling_concepts]]
      - [[knowledge_base/mathematics/active_inference_pomdp]]
---

## Overview
This directory contains research documentation focusing on theoretical foundations, mathematical frameworks, and statistical methods in cognitive modeling.

## Theoretical Foundations

### POMDP Framework
- [[active_inference_pomdp]] - Active Inference POMDP formulation
- [[belief_updating]] - Belief update mechanisms
- [[state_estimation]] - State estimation methods
- [[policy_selection]] - Policy selection theory

### Statistical Theory
- [[variational_methods]] - Variational inference methods
- [[information_theory]] - Information theoretic foundations
- [[game_theory]] - Game theoretic principles
- [[statistical_foundations]] - Core statistical concepts

## Mathematical Components

### Matrix Operations
- [[A_matrix]] - Observation model
- [[B_matrix]] - Transition dynamics
- [[C_matrix]] - Preferences
- [[D_matrix]] - Prior beliefs
- [[E_matrix]] - Action distribution

### Free Energy Formulations
- [[variational_free_energy]] - VFE computation
- [[expected_free_energy]] - EFE formulation
- [[efe_components]] - EFE decomposition
- [[free_energy_principle]] - Theoretical foundation

### Information Measures
- [[epistemic_value]] - Information gain
- [[pragmatic_value]] - Goal-directed value
- [[expected_free_energy_update]] - EFE updates
- [[compute_efe]] - EFE computation

## Implementation Theory

### Core Algorithms
- [[belief_propagation]] - Message passing algorithms
- [[variational_inference]] - Variational methods
- [[monte_carlo_methods]] - Sampling approaches
- [[gradient_descent]] - Optimization techniques

### State Space Models
- [[state_spaces]] - State space theory
- [[markov_models]] - Markov processes
- [[hidden_states]] - Hidden state inference
- [[observation_models]] - Observation modeling

### Policy Optimization
- [[policy_optimization]] - Policy improvement
- [[action_selection]] - Action selection methods
- [[exploration_exploitation]] - Exploration strategies
- [[temperature_parameter]] - Temperature scaling

## Validation Framework

### Statistical Tests
- [[hypothesis_testing]] - Statistical testing
- [[model_comparison]] - Model evaluation
- [[parameter_estimation]] - Parameter inference
- [[goodness_of_fit]] - Fit measures

### Performance Metrics
- [[information_metrics]] - Information measures
- [[prediction_error]] - Error metrics
- [[convergence_analysis]] - Convergence studies
- [[stability_analysis]] - Stability measures

## Research Methods

### Experimental Design
- [[experiment_design]] - Design principles
- [[control_variables]] - Control methods
- [[sampling_strategies]] - Sampling approaches
- [[power_analysis]] - Statistical power

### Data Analysis
- [[statistical_analysis]] - Analysis methods
- [[bayesian_analysis]] - Bayesian approaches
- [[frequentist_analysis]] - Frequentist methods
- [[causal_analysis]] - Causality studies

### Visualization
- [[data_visualization]] - Visualization methods
- [[belief_evolution]] - Belief trajectories
- [[free_energy_landscape]] - Energy landscapes
- [[policy_visualization]] - Policy analysis

## Integration Points

### Theory Integration
- [[active_inference_theory]] - Active inference
- [[predictive_coding]] - Predictive processing
- [[optimal_control]] - Control theory
- [[reinforcement_learning]] - RL connections

### Implementation Links
- [[model_implementation]] - Implementation guides
- [[numerical_methods]] - Numerical approaches
- [[optimization_methods]] - Optimization techniques
- [[validation_methods]] - Validation approaches

## Related Documentation
- [[../concepts/cognitive_modeling_concepts|Core Concepts]]
- [[../guides/implementation_guides|Implementation Guides]]
- [[../api/api_documentation|API Documentation]]
- [[../examples/usage_examples|Usage Examples]]

## References
- [[friston_2017]] - Active Inference
- [[da_costa_2020]] - Active Inference POMDP
- [[parr_2019]] - Free Energy Theory
- [[buckley_2017]] - Tutorial Paper