# Probabilistic Models
## Overview
Probabilistic models are mathematical frameworks that describe systems and phenomena using probability distributions to represent uncertainty and variability. These models form the foundation of [[bayesian_inference|Bayesian inference]], [[statistical_learning|statistical learning]], and [[uncertainty_quantification|uncertainty quantification]].
## Core Components
### 1. Random Variables
- [[random_variables|Random variables]] represent uncertain quantities
- Can be discrete or continuous
- Characterized by probability distributions
### 2. Dependencies
- [[conditional_probability|Conditional probabilities]] between variables
- [[probabilistic_relationships|Probabilistic relationships]]
- [[causal_structure|Causal structures]]
### 3. Parameter Space
- Model parameters defining distributions
- [[parameter_estimation|Parameter estimation]] methods
- [[hyperparameters|Hyperparameters]] for hierarchical models
## Types of Models
### 1. [[directed_graphical_models|Directed Graphical Models]]
- Bayesian networks
- Hidden Markov models
- State space models
### 2. [[undirected_graphical_models|Undirected Graphical Models]]
- Markov random fields
- Conditional random fields
- Boltzmann machines
### 3. [[hierarchical_models|Hierarchical Models]]
- Multilevel modeling
- Nested structures
- Parameter sharing
## Mathematical Framework
### 1. Probabilistic Foundation
**Joint Distribution Factorization:**
For directed acyclic graphs:
```math
p(\mathbf{x}) = \prod_{i=1}^n p(x_i | \text{pa}(x_i))
```
For undirected graphs (Markov Random Fields):
```math
p(\mathbf{x}) = \frac{1}{Z} \prod_{C} \psi_C(\mathbf{x}_C)
```
where:
- $\text{pa}(x_i)$ are parents of variable $x_i$
- $\psi_C(\mathbf{x}_C)$ are clique potentials
- $Z$ is the partition function
### 2. Parametric Models
**Likelihood Function:**
```math
L(\boldsymbol{\theta}; \mathbf{x}) = p(\mathbf{x} | \boldsymbol{\theta}) = \prod_{i=1}^n p(x_i | \boldsymbol{\theta})
```
**Log-Likelihood:**
```math
\ell(\boldsymbol{\theta}) = \log L(\boldsymbol{\theta}; \mathbf{x}) = \sum_{i=1}^n \log p(x_i | \boldsymbol{\theta})
```
**Prior Distribution:**
```math
p(\boldsymbol{\theta}) = \prod_{j} p(\theta_j)
```
**Posterior Distribution:**
```math
p(\boldsymbol{\theta} | \mathbf{x}) = \frac{p(\mathbf{x} | \boldsymbol{\theta}) p(\boldsymbol{\theta})}{p(\mathbf{x})}
```
### 3. Exponential Family Models
Many probabilistic models belong to exponential families:
```math
p(x | \boldsymbol{\theta}) = h(x) \exp\left(\boldsymbol{\theta}^T \mathbf{t}(x) - A(\boldsymbol{\theta})\right)
```
where:
- $\mathbf{t}(x)$ are sufficient statistics
- $A(\boldsymbol{\theta})$ is the log-partition function
- $h(x)$ is the base measure
**Properties:**
- Natural parameters: $\boldsymbol{\theta}$
- Mean parameters: $\boldsymbol{\mu} = \nabla A(\boldsymbol{\theta})$
- Variance: $\text{Var}[\mathbf{t}(X)] = \nabla^2 A(\boldsymbol{\theta})$
Related: [[exponential_families]], [[natural_parameters]], [[sufficient_statistics]]
### 4. Hierarchical Structure
**Multi-level Models:**
```math
\begin{aligned}
\text{Level 1: } & y_i | \boldsymbol{\theta}_i \sim p(y_i | \boldsymbol{\theta}_i) \\
\text{Level 2: } & \boldsymbol{\theta}_i | \boldsymbol{\phi} \sim p(\boldsymbol{\theta}_i | \boldsymbol{\phi}) \\
\text{Level 3: } & \boldsymbol{\phi} \sim p(\boldsymbol{\phi})
\end{aligned}
```
**Advantages:**
- Parameter sharing across groups
- Regularization through hierarchical priors
- Uncertainty propagation across levels
Related: [[hierarchical_models]], [[multilevel_modeling]], [[mixed_effects_models]]
## Applications
### 1. Scientific Modeling
- Physical systems
- Biological processes
- Chemical reactions
### 2. Machine Learning
- [[bayesian_neural_networks|Bayesian neural networks]]
- [[probabilistic_programming|Probabilistic programming]]
- [[gaussian_processes|Gaussian processes]]
### 3. Decision Making
- [[active_inference|Active inference]]
- [[reinforcement_learning|Reinforcement learning]]
- [[optimal_control|Optimal control]]
## Implementation
### 1. Software Frameworks
- [[probabilistic_programming_languages|Probabilistic programming languages]]
- [[statistical_computing|Statistical computing]] packages
- [[inference_engines|Inference engines]]
### 2. Computational Methods
- [[monte_carlo_methods|Monte Carlo methods]]
- [[variational_inference|Variational inference]]
- [[message_passing|Message passing algorithms]]
## Best Practices
### 1. Model Selection
- [[model_complexity|Model complexity]] considerations
- [[cross_validation|Cross-validation]]
- [[information_criteria|Information criteria]]
### 2. Validation
- [[posterior_predictive_checks|Posterior predictive checks]]
- [[sensitivity_analysis|Sensitivity analysis]]
- [[robustness_testing|Robustness testing]]
## References
1. Bishop, C. M. (2006). Pattern Recognition and Machine Learning
1. Murphy, K. P. (2012). Machine Learning: A Probabilistic Perspective
1. Gelman, A., et al. (2013). Bayesian Data Analysis