| type |
id |
matrix_type |
created |
modified |
complexity |
tags |
related_spaces |
semantic_relations |
| matrix_spec |
B_matrix_001 |
transition |
2024-03-15 |
2024-03-15 |
advanced |
| matrix |
| transition |
| active-inference |
| dynamics |
| probability |
| control |
|
|
|
Overview
The B-matrix is a fundamental component in POMDPs and active inference frameworks, representing state transition probabilities under different actions. It encodes the dynamics of the environment and how actions influence state changes, forming the basis for prediction, planning, and control.
Core Concepts
Fundamental Definition
- transition_probability - Basic concept
- Conditional probability P(s'|s,π)
- State transitions
- Action influence
- Temporal dynamics
Key Properties
- markov_property - Memory independence
- History independence
- Current state sufficiency
- Future prediction
Structural Characteristics
- matrix_structure - Organization
- Dimensionality
- Sparsity patterns
- Symmetry properties
- Conservation laws
Mathematical Framework
Formal Definition
B_{ijk} = P(s'_i|s_j,π_k)
# Constraints
∑_i B_{ijk} = 1 ∀j,k
B_{ijk} ≥ 0 ∀i,j,k
Matrix Structure
dimensions:
rows: num_states # Next state (s')
cols: num_states # Current state (s)
depth: num_actions # Actions/policies (π)
constraints:
probability:
- sum(axis=0) == 1.0 # Column-wise normalization
- all_values >= 0 # Non-negative probabilities
structure:
- rows == cols # Square matrix per action
- depth == num_policies
Probabilistic Properties
Implementation Details
Data Structures
Basic Structure
class BMatrix:
def __init__(self, num_states: int, num_actions: int):
self.B = np.zeros((num_states, num_states, num_actions))
self.initialize_transitions()
def initialize_transitions(self):
"""Initialize with identity or prior knowledge"""
for a in range(self.num_actions):
self.B[:,:,a] = np.eye(self.num_states) # Start with self-transitions
Advanced Features
def get_transition_distribution(self, state: int, action: int) -> Distribution:
"""Get probability distribution over next states"""
return Distribution(self.B[:, state, action])
def sample_next_state(self, state: int, action: int) -> int:
"""Sample next state from transition distribution"""
return np.random.choice(
self.num_states,
p=self.B[:, state, action]
)
Storage Formats
- matrix_storage - Data management
- Dense arrays
- Sparse representations
- Compressed formats
- Memory mapping
Computational Methods
- transition_computation - Processing
- Matrix operations
- Parallel computation
- GPU acceleration
- Distributed processing
Learning and Adaptation
Learning Methods
Maximum Likelihood
def update_transitions_ml(self,
state: int,
action: int,
next_state: int,
learning_rate: float):
"""Update transitions using maximum likelihood"""
target = np.zeros(self.num_states)
target[next_state] = 1
self.B[:, state, action] = (1 - learning_rate) * self.B[:, state, action] + \
learning_rate * target
Bayesian Updates
def update_transitions_bayes(self,
state: int,
action: int,
next_state: int,
prior_strength: float):
"""Update transitions using Bayesian inference"""
self.counts[next_state, state, action] += 1
alpha = self.counts[:, state, action]
self.B[:, state, action] = dirichlet.mean(alpha + prior_strength)
Structure Learning
- causal_discovery - Structure identification
- Sparsity patterns
- Invariant relationships
- Causal mechanisms
- Independence testing
Online Adaptation
- dynamic_learning - Real-time updates
- Incremental learning
- Adaptive rates
- Forgetting factors
- Confidence tracking
Applications
Planning and Control
Policy Evaluation
def evaluate_policy(self, policy: np.ndarray, horizon: int) -> np.ndarray:
"""Evaluate state occupancy under policy"""
state_dist = initial_distribution
for t in range(horizon):
action = policy[t]
state_dist = self.B[:,:,action] @ state_dist
return state_dist
Optimal Control
- optimal_control - Control methods
- LQR formulation
- Model predictive control
- Stochastic optimal control
- Risk-sensitive control
Prediction and Simulation
Forward Simulation
def simulate_trajectory(self,
initial_state: int,
policy: List[int],
num_samples: int) -> np.ndarray:
"""Simulate multiple trajectories under policy"""
trajectories = np.zeros((num_samples, len(policy) + 1))
trajectories[:,0] = initial_state
for t, action in enumerate(policy):
for n in range(num_samples):
current_state = int(trajectories[n,t])
trajectories[n,t+1] = self.sample_next_state(current_state, action)
return trajectories
State Prediction
- state_prediction - Future states
- Expected states
- Uncertainty propagation
- Confidence bounds
- Risk assessment
Integration with Other Components
With State Space
With Action Space
With Observation Model
Advanced Topics
Information Theory
- transition_information - Information measures
- Entropy rate
- Channel capacity
- Information flow
- Predictive information
Geometric Properties
- transition_geometry - Geometric aspects
- Manifold structure
- Geodesics
- Parallel transport
- Curvature
Stability Analysis
- transition_stability - Stability properties
- Fixed points
- Attractors
- Lyapunov stability
- Structural stability
Optimization and Efficiency
Computational Optimization
Numerical Stability
- numerical_methods - Numerical issues
- Conditioning
- Error propagation
- Precision control
- Stability preservation
Resource Management
- resource_optimization - Resources
- Memory allocation
- Computation scheduling
- Load balancing
- Power efficiency
Best Practices
Implementation Guidelines
Validation Methods
- validation_methods - Quality assurance
- Unit testing
- Integration testing
- Performance testing
- Validation metrics
Maintenance Procedures
References
See Also
