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Statistical Foundations
Overview
Core statistical foundations for cognitive modeling, focusing on probabilistic inference, information theory, and optimization methods.
Probability Theory
Fundamentals
- probability_axioms - Basic probability laws
- random_variables - Random variable theory
- probability_distributions - Distribution types
- conditional_probability - Conditional laws
Advanced Topics
- measure_theory - Measure-theoretic probability
- stochastic_processes - Random processes
- martingales - Martingale theory
- ergodic_theory - Ergodicity concepts
Statistical Inference
Classical Methods
- maximum_likelihood - ML estimation
- hypothesis_testing - Statistical tests
- confidence_intervals - Interval estimation
- regression_analysis - Regression methods
Bayesian Methods
- bayesian_inference - Bayesian approach
- prior_distributions - Prior specification
- posterior_computation - Posterior analysis
- model_selection - Bayesian model choice
Information Theory
Core Concepts
- entropy - Information content
- mutual_information - Information sharing
- kl_divergence - Distribution divergence
- fisher_information - Information geometry
Applications
- information_gain - Active learning
- channel_capacity - Communication limits
- rate_distortion - Compression theory
- information_bottleneck - Information constraints
Optimization Methods
Gradient-Based
- gradient_descent - First-order methods
- natural_gradients - Information geometry
- conjugate_gradients - Second-order methods
- stochastic_optimization - Stochastic methods
Variational Methods
- variational_inference - VI algorithms
- expectation_maximization - EM algorithm
- variational_bayes - VB methods
- message_passing - Message algorithms
Implementation
Numerical Methods
- monte_carlo - MC methods
- importance_sampling - IS techniques
- mcmc - MCMC algorithms
- particle_methods - Particle filters
Software Tools
- statistical_computing - Computing tools
- probabilistic_programming - PPL frameworks
- inference_engines - Inference libraries
- visualization_tools - Plotting utilities
Applications
Model Validation
- cross_validation - CV methods
- bootstrapping - Bootstrap techniques
- model_diagnostics - Diagnostic tools
- residual_analysis - Residual checks
Performance Analysis
- convergence_analysis - Convergence study
- complexity_analysis - Computational cost
- stability_analysis - Numerical stability
- sensitivity_analysis - Parameter sensitivity
Documentation Links
- ../../docs/research/research_documentation_index
- ../../docs/guides/implementation_guides_index
- ../../docs/api/api_documentation_index
- ../../docs/examples/usage_examples_index
References
- casella_berger - Statistical Inference
- mackay - Information Theory
- robert_casella - Monte Carlo Methods
- bishop - Pattern Recognition