# Factor Graphs
## Overview
Factor graphs are a type of [[graphical_models|graphical model]] that represents factorizations of functions, particularly probability distributions. They provide a unified framework for [[inference_algorithms|inference algorithms]] and bridge the gap between [[directed_graphical_models|directed]] and [[undirected_graphical_models|undirected]] graphical models.
## Mathematical Foundation
### 1. Basic Structure
A factor graph G = (V, F, E) consists of:
- V: Variable nodes representing [[random_variables|random variables]]
- F: Factor nodes encoding [[probability_distributions|probability distributions]]
- E: Edges representing [[probabilistic_dependencies|probabilistic dependencies]]
### 2. Factorization
For a probability distribution p(x):
```math
p(x_1, ..., x_n) = \prod_{a \in F} f_a(x_{\partial a})
```
where:
- f_a are [[factor_functions|factor functions]]
- ∂a denotes variables connected to factor a
### 3. [[bayesian_factorization|Bayesian Factorization]]
In Bayesian terms:
```math
p(x, θ | y) ∝ p(y | x, θ)p(x | θ)p(θ)
```
where:
- p(y | x, θ) is the [[likelihood_function|likelihood]]
- p(x | θ) is the [[prior_distribution|prior]]
- p(θ) is the [[hyperprior|hyperprior]]
## Components and Structure
### 1. [[variable_nodes|Variable Nodes]]
#### Types and Properties
```julia
struct VariableNode{T}
id::Symbol
domain::Domain{T}
neighbors::Set{FactorNode}
messages::Dict{FactorNode, Message}
belief::Distribution{T}
end
```
#### Categories
- **Observable Variables**
- Represent data points
- Fixed during inference
- Drive belief updates
- **Latent Variables**
- Hidden states
- Model parameters
- Inferred quantities
- **Parameter Nodes**
- [[hyperparameters|Hyperparameters]]
- [[model_parameters|Model parameters]]
- [[sufficient_statistics|Sufficient statistics]]
### 2. [[factor_nodes|Factor Nodes]]
#### Base Implementation
```julia
abstract type FactorNode end
struct ProbabilisticFactor <: FactorNode
distribution::Distribution
variables::Vector{VariableNode}
parameters::Dict{Symbol, Any}
end
struct DeterministicFactor <: FactorNode
function::Function
inputs::Vector{VariableNode}
outputs::Vector{VariableNode}
end
struct ConstraintFactor <: FactorNode
constraint::Function
variables::Vector{VariableNode}
tolerance::Float64
end
```
#### Message Computation
```julia
function compute_message(factor::FactorNode, to::VariableNode)
# Collect incoming messages
messages = [msg for (node, msg) in factor.messages if node != to]
# Compute outgoing message
if isa(factor, ProbabilisticFactor)
return compute_probabilistic_message(factor, messages, to)
elseif isa(factor, DeterministicFactor)
return compute_deterministic_message(factor, messages, to)
else
return compute_constraint_message(factor, messages, to)
end
end
```
### 3. [[edges|Edges]]
#### Properties
```julia
struct Edge
source::Union{VariableNode, FactorNode}
target::Union{VariableNode, FactorNode}
message_type::Type{<:Message}
parameters::Dict{Symbol, Any}
end
```
#### Message Types
- Forward messages (variable to factor)
- Backward messages (factor to variable)
- Parameter messages
- Constraint messages
## Message Passing and Inference
### 1. [[belief_propagation|Belief Propagation]]
#### Forward Messages
```math
μ_{x→f}(x) = \prod_{g \in N(x) \backslash f} μ_{g→x}(x)
```
#### Backward Messages
```math
μ_{f→x}(x) = \sum_{x_{\partial f \backslash x}} f(x_{\partial f}) \prod_{y \in \partial f \backslash x} μ_{y→f}(y)
```
#### Implementation
```julia
function belief_propagation!(graph::FactorGraph; max_iters=100)
for iter in 1:max_iters
# Update variable to factor messages
for var in graph.variables
for factor in var.neighbors
message = compute_var_to_factor_message(var, factor)
update_message!(var, factor, message)
end
end
# Update factor to variable messages
for factor in graph.factors
for var in factor.variables
message = compute_factor_to_var_message(factor, var)
update_message!(factor, var, message)
end
end
# Check convergence
if check_convergence(graph)
break
end
end
end
```
### 2. [[variational_message_passing|Variational Message Passing]]
#### ELBO Optimization
```math
\mathcal{L}(q) = \mathbb{E}_q[\log p(x,z)] - \mathbb{E}_q[\log q(z)]
```
#### Natural Gradient Updates
```math
θ_t = θ_{t-1} + η\nabla_{\text{nat}}\mathcal{L}(q)
```
#### Implementation
```julia
function variational_message_passing!(graph::FactorGraph;
learning_rate=0.01,
max_iters=100)
for iter in 1:max_iters
# Compute natural gradients
gradients = compute_natural_gradients(graph)
# Update variational parameters
for (node, grad) in gradients
update_parameters!(node, grad, learning_rate)
end
# Update messages
update_messages!(graph)
# Check ELBO convergence
if check_elbo_convergence(graph)
break
end
end
end
```
### 3. [[expectation_propagation|Expectation Propagation]]
#### Moment Matching
```math
\text{minimize}_{q_i} \text{KL}(p||q_1...q_i...q_n)
```
#### Implementation
```julia
function expectation_propagation!(graph::FactorGraph; max_iters=100)
for iter in 1:max_iters
# Update approximate factors
for factor in graph.factors
# Compute cavity distribution
cavity = compute_cavity_distribution(factor)
# Moment matching
new_approx = moment_match(cavity, factor)
# Update approximation
update_approximation!(factor, new_approx)
end
# Check convergence
if check_convergence(graph)
break
end
end
end
```
## Advanced Topics
### 1. [[structured_factor_graphs|Structured Factor Graphs]]
#### Temporal Structure
```julia
@model function temporal_factor_graph(T)
# State variables
x = Vector{VariableNode}(undef, T)
# Temporal factors
for t in 2:T
@factor f[t] begin
x[t] ~ transition(x[t-1])
end
end
return x
end
```
#### Hierarchical Structure
```julia
@model function hierarchical_factor_graph()
# Global parameters
θ_global ~ prior()
# Local parameters
θ_local = Vector{VariableNode}(undef, N)
for i in 1:N
θ_local[i] ~ conditional(θ_global)
end
return θ_local
end
```
### 2. [[continuous_variables|Continuous Variables]]
#### Gaussian Messages
```julia
struct GaussianMessage <: Message
mean::Vector{Float64}
precision::Matrix{Float64}
function GaussianMessage(μ, Λ)
@assert size(μ, 1) == size(Λ, 1) == size(Λ, 2)
new(μ, Λ)
end
end
function multiply_messages(m1::GaussianMessage, m2::GaussianMessage)
Λ = m1.precision + m2.precision
μ = Λ \ (m1.precision * m1.mean + m2.precision * m2.mean)
return GaussianMessage(μ, Λ)
end
```
### 3. [[convergence_properties|Convergence Properties]]
#### Fixed Point Conditions
```math
b^*(x) = \frac{1}{Z} \prod_{f \in N(x)} μ^*_{f→x}(x)
```
#### Bethe Free Energy
```math
F_{\text{Bethe}} = \sum_i F_i + \sum_a F_a
```
## Implementation
### 1. Graph Construction
```julia
struct FactorGraph
variables::Set{VariableNode}
factors::Set{FactorNode}
edges::Set{Edge}
function FactorGraph()
new(Set{VariableNode}(), Set{FactorNode}(), Set{Edge}())
end
end
function add_variable!(graph::FactorGraph, var::VariableNode)
push!(graph.variables, var)
end
function add_factor!(graph::FactorGraph, factor::FactorNode)
push!(graph.factors, factor)
for var in factor.variables
add_edge!(graph, Edge(var, factor))
end
end
```
### 2. Message Scheduling
```julia
struct MessageSchedule
order::Vector{Tuple{Union{VariableNode,FactorNode},
Union{VariableNode,FactorNode}}}
priorities::Vector{Float64}
end
function schedule_messages(graph::FactorGraph)
schedule = MessageSchedule()
# Forward pass
for level in graph.levels
for node in level
schedule_forward_messages!(schedule, node)
end
end
# Backward pass
for level in reverse(graph.levels)
for node in level
schedule_backward_messages!(schedule, node)
end
end
return schedule
end
```
### 3. Inference Execution
```julia
function run_inference(graph::FactorGraph;
method=:belief_propagation,
max_iters=100)
if method == :belief_propagation
belief_propagation!(graph, max_iters=max_iters)
elseif method == :variational
variational_message_passing!(graph, max_iters=max_iters)
elseif method == :expectation_propagation
expectation_propagation!(graph, max_iters=max_iters)
else
error("Unknown inference method: $method")
end
return compute_beliefs(graph)
end
```
## Applications
### 1. [[bayesian_inference|Bayesian Inference]]
- Parameter estimation
- Model selection
- Uncertainty quantification
### 2. [[probabilistic_programming|Probabilistic Programming]]
- Model specification
- Automatic inference
- Compositional modeling
### 3. [[active_inference|Active Inference]]
- Policy selection
- Perception-action loops
- Free energy minimization
## Best Practices
### 1. Design Patterns
- Modular factor construction
- Reusable message computations
- Efficient graph structures
### 2. Numerical Considerations
- Message normalization
- Numerical stability
- Convergence monitoring
### 3. Testing and Validation
- Unit tests for factors
- Message validity checks
- End-to-end inference tests
## Integration with Bayesian Networks
### 1. [[bayesian_network_conversion|Bayesian Network Conversion]]
```julia
function from_bayesian_network(bn::BayesianNetwork)
# Create factor graph
fg = FactorGraph()
# Add variables
for node in bn.nodes
add_variable!(fg, VariableNode(node))
end
# Add CPT factors
for (var, cpt) in bn.parameters
add_cpt_factor!(fg, var, cpt)
end
return fg
end
```
### 2. [[inference_equivalence|Inference Equivalence]]
- Message passing in trees
- Loopy belief propagation
- Variational approximations
### 3. [[model_comparison|Model Comparison]]
- Structure comparison
- Parameter learning
- Performance metrics
## RxInfer Integration
### 1. Reactive Message Passing
```julia
using RxInfer
@model function rxinfer_factor_graph(data)
# Prior distributions
θ ~ GaussianMeanPrecision(0.0, 1.0)
σ ~ GammaShape(1.0, 1.0)
# Likelihood factors
for i in 1:length(data)
y[i] ~ GaussianMeanPrecision(θ, σ)
end
end
# Create inference algorithm
algorithm = ReactiveMP.messagePassingAlgorithm(
model = rxinfer_factor_graph,
data = observed_data
)
# Execute inference
results = ReactiveMP.infer(algorithm)
```
### 2. Streaming Factor Graphs
```julia
struct StreamingFactorGraph
base_graph::FactorGraph
stream_nodes::Vector{StreamNode}
buffer_size::Int
function StreamingFactorGraph(model, buffer_size=1000)
graph = create_base_graph(model)
stream_nodes = initialize_stream_nodes(model)
new(graph, stream_nodes, buffer_size)
end
end
function process_stream!(graph::StreamingFactorGraph, data_stream)
for batch in data_stream
# Update streaming nodes
update_stream_nodes!(graph, batch)
# Perform message passing
propagate_messages!(graph)
# Update beliefs
update_beliefs!(graph)
# Prune old messages if needed
prune_old_messages!(graph)
end
end
```
### 3. Reactive Message Types
```julia
abstract type ReactiveMessage end
struct GaussianReactiveMessage <: ReactiveMessage
mean::Observable{Float64}
precision::Observable{Float64}
function GaussianReactiveMessage(μ::Observable, τ::Observable)
new(μ, τ)
end
end
struct StreamingMessage <: ReactiveMessage
distribution::Observable{Distribution}
timestamp::Observable{Float64}
function StreamingMessage(dist::Observable, ts::Observable)
new(dist, ts)
end
end
```
## Advanced Bayesian Integration
### 1. Hierarchical Bayesian Models
```mermaid
graph TD
subgraph "Hierarchical Factor Structure"
A[Global Parameters] --> B[Group Parameters]
B --> C[Individual Parameters]
C --> D[Observations]
E[Hyperpriors] --> A
end
style A fill:#f9f,stroke:#333
style C fill:#bbf,stroke:#333
style E fill:#bfb,stroke:#333
```
### 2. Conjugate Factor Pairs
```julia
struct ConjugateFactor{T<:Distribution} <: FactorNode
prior::T
likelihood::Function
posterior::T
sufficient_stats::Dict{Symbol, Any}
function ConjugateFactor(prior::T, likelihood::Function) where T
posterior = deepcopy(prior)
stats = initialize_sufficient_stats(prior)
new{T}(prior, likelihood, posterior, stats)
end
end
function update_conjugate_factor!(factor::ConjugateFactor, data)
# Update sufficient statistics
update_stats!(factor.sufficient_stats, data)
# Compute posterior parameters
posterior_params = compute_posterior_params(
factor.prior, factor.sufficient_stats)
# Update posterior
update_posterior!(factor.posterior, posterior_params)
end
```
### 3. Non-parametric Extensions
```julia
struct DirichletProcessFactor <: FactorNode
base_measure::Distribution
concentration::Float64
clusters::Vector{Cluster}
assignments::Dict{Int, Int}
function DirichletProcessFactor(base::Distribution, α::Float64)
new(base, α, Cluster[], Dict())
end
end
function sample_assignment(factor::DirichletProcessFactor, data_point)
# Compute cluster probabilities
probs = compute_cluster_probabilities(factor, data_point)
# Sample new assignment
assignment = sample_categorical(probs)
# Update clusters if necessary
if assignment > length(factor.clusters)
create_new_cluster!(factor, data_point)
end
return assignment
end
```
## Advanced Message Passing Schemes
### 1. Stochastic Message Passing
```julia
struct StochasticMessagePassing
n_particles::Int
resampling_threshold::Float64
function StochasticMessagePassing(n_particles=1000, threshold=0.5)
new(n_particles, threshold)
end
end
function propagate_particles!(smp::StochasticMessagePassing,
factor::FactorNode,
particles::Vector{Particle})
# Propagate particles through factor
weights = zeros(smp.n_particles)
new_particles = similar(particles)
for i in 1:smp.n_particles
# Sample
new_particles[i] = propose_particle(factor, particles[i])
# Weight
weights[i] = compute_importance_weight(
factor, particles[i], new_particles[i])
end
# Resample if needed
if effective_sample_size(weights) < smp.resampling_threshold
new_particles = resample_particles(new_particles, weights)
end
return new_particles
end
```
### 2. Distributed Message Passing
```julia
struct DistributedFactorGraph
subgraphs::Vector{FactorGraph}
interfaces::Dict{Tuple{Int,Int}, Interface}
function DistributedFactorGraph(graph::FactorGraph, n_partitions)
# Partition graph
subgraphs = partition_graph(graph, n_partitions)
# Create interfaces
interfaces = create_interfaces(subgraphs)
new(subgraphs, interfaces)
end
end
function distributed_inference!(graph::DistributedFactorGraph)
# Initialize workers
workers = [Worker(subgraph) for subgraph in graph.subgraphs]
while !converged(workers)
# Local inference
@sync for worker in workers
@async local_inference!(worker)
end
# Exchange messages
exchange_interface_messages!(graph)
# Update beliefs
update_worker_beliefs!(workers)
end
end
```
### 3. Adaptive Message Scheduling
```julia
struct AdaptiveScheduler
priority_queue::PriorityQueue{Message, Float64}
residual_threshold::Float64
function AdaptiveScheduler(threshold=1e-6)
new(PriorityQueue{Message, Float64}(), threshold)
end
end
function schedule_message!(scheduler::AdaptiveScheduler,
message::Message,
residual::Float64)
if residual > scheduler.residual_threshold
enqueue!(scheduler.priority_queue, message => residual)
end
end
function process_messages!(scheduler::AdaptiveScheduler)
while !isempty(scheduler.priority_queue)
# Get highest priority message
message = dequeue!(scheduler.priority_queue)
# Update message
new_message = update_message!(message)
# Compute residual and reschedule if needed
residual = compute_residual(message, new_message)
schedule_message!(scheduler, new_message, residual)
end
end
```
## Integration with Active Inference
### 1. Free Energy Minimization
```julia
struct FreeEnergyFactor <: FactorNode
internal_states::Vector{VariableNode}
external_states::Vector{VariableNode}
precision::Matrix{Float64}
function FreeEnergyFactor(internal, external, precision)
new(internal, external, precision)
end
end
function compute_free_energy(factor::FreeEnergyFactor)
# Compute prediction error
error = compute_prediction_error(
factor.internal_states,
factor.external_states
)
# Weight by precision
weighted_error = error' * factor.precision * error
# Add complexity penalty
complexity = compute_complexity_term(factor.internal_states)
return 0.5 * weighted_error + complexity
end
```
### 2. Policy Selection
```julia
struct PolicyFactor <: FactorNode
policies::Vector{Policy}
expected_free_energy::Vector{Float64}
temperature::Float64
function PolicyFactor(policies, temperature=1.0)
n_policies = length(policies)
new(policies, zeros(n_policies), temperature)
end
end
function select_policy(factor::PolicyFactor)
# Compute softmax probabilities
probs = softmax(-factor.temperature * factor.expected_free_energy)
# Sample policy
policy_idx = sample_categorical(probs)
return factor.policies[policy_idx]
end
```
## Performance Optimization
### 1. Message Caching
```julia
struct MessageCache
storage::Dict{Tuple{FactorNode, VariableNode}, Message}
max_size::Int
eviction_policy::Symbol
function MessageCache(max_size=10000, policy=:lru)
new(Dict(), max_size, policy)
end
end
function cache_message!(cache::MessageCache,
factor::FactorNode,
variable::VariableNode,
message::Message)
key = (factor, variable)
# Evict if needed
if length(cache.storage) >= cache.max_size
evict_message!(cache)
end
# Store message
cache.storage[key] = message
end
```
### 2. Parallel Message Updates
```julia
function parallel_message_passing!(graph::FactorGraph)
# Group independent messages
message_groups = group_independent_messages(graph)
# Update messages in parallel
@sync for group in message_groups
@async begin
for message in group
update_message!(message)
end
end
end
end
```
### 3. GPU Acceleration
```julia
struct GPUFactorGraph
variables::CuArray{VariableNode}
factors::CuArray{FactorNode}
messages::CuArray{Message}
function GPUFactorGraph(graph::FactorGraph)
# Transfer to GPU
variables = cu(collect(graph.variables))
factors = cu(collect(graph.factors))
messages = cu(collect_messages(graph))
new(variables, factors, messages)
end
end
function gpu_message_passing!(graph::GPUFactorGraph)
# Kernel for parallel message updates
@cuda threads=256 blocks=div(length(graph.messages), 256) do
update_messages_kernel(graph.messages)
end
synchronize()
end
```
## Advanced Convergence Analysis
### Theoretical Foundations
**Definition** (Message Operator): For a factor graph $G$, the message passing operator $T$ maps the current set of messages $\mathbf{m}$ to updated messages $\mathbf{m}'$ according to the sum-product algorithm rules.
**Theorem** (Convergence Conditions): Loopy belief propagation converges to a unique fixed point if the spectral radius of the message operator satisfies $\rho(T) < 1$.
```python
class ConvergenceAnalyzer:
"""Rigorous convergence analysis for loopy belief propagation."""
def __init__(self, factor_graph: FactorGraph):
"""Initialize convergence analyzer.
Args:
factor_graph: Factor graph for analysis
"""
self.graph = factor_graph
self.message_operator_cache = None
self.convergence_history = []
def spectral_radius_analysis(self) -> Dict[str, float]:
"""Analyze convergence via spectral radius of message operator.
The message passing operator T maps messages m to updated messages T(m).
Convergence is guaranteed if ρ(T) < 1 where ρ is the spectral radius.
Returns:
analysis: Dictionary containing spectral properties and convergence guarantees
"""
# Construct linearized message operator
message_jacobian = self._construct_message_jacobian()
# Compute eigenvalues
eigenvals = np.linalg.eigvals(message_jacobian)
spectral_radius = np.max(np.abs(eigenvals))
# Dominant eigenvalue analysis
dominant_idx = np.argmax(np.abs(eigenvals))
dominant_eigenval = eigenvals[dominant_idx]
# Convergence rate estimation
if spectral_radius < 1.0:
convergence_rate = -np.log(spectral_radius)
convergence_time = 1.0 / convergence_rate
else:
convergence_rate = 0.0
convergence_time = np.inf
return {
'spectral_radius': spectral_radius,
'dominant_eigenvalue': dominant_eigenval,
'convergence_guaranteed': spectral_radius < 1.0,
'convergence_rate': convergence_rate,
'convergence_time_estimate': convergence_time,
'stability_margin': 1.0 - spectral_radius,
'eigenvalue_spectrum': eigenvals
}
def bethe_free_energy_analysis(self,
beliefs: Dict[str, np.ndarray]) -> Dict[str, float]:
"""Compute Bethe approximation to free energy and related measures.
The Bethe free energy provides a variational approximation:
F_Bethe = ∑_i H(b_i) + ∑_α H(b_α) - ∑_i (d_i - 1)H(b_i)
where H is entropy, b_i are variable beliefs, b_α are factor beliefs,
and d_i are variable degrees.
Args:
beliefs: Dictionary of variable and factor beliefs
Returns:
bethe_analysis: Dictionary containing Bethe free energy and diagnostics
"""
node_entropy = 0.0
factor_entropy = 0.0
degree_correction = 0.0
# Variable node entropies
for var_name, belief in beliefs.get('variables', {}).items():
h_var = self._entropy(belief)
node_entropy += h_var
# Degree correction
degree = self._get_variable_degree(var_name)
degree_correction += (degree - 1) * h_var
# Factor node entropies
for factor_name, belief in beliefs.get('factors', {}).items():
factor_entropy += self._entropy(belief)
# Bethe free energy
bethe_free_energy = node_entropy + factor_entropy - degree_correction
# Gibbs free energy (if available)
gibbs_free_energy = self._compute_gibbs_free_energy(beliefs)
# Approximation quality
if gibbs_free_energy is not None:
approximation_error = abs(bethe_free_energy - gibbs_free_energy)
relative_error = approximation_error / abs(gibbs_free_energy)
else:
approximation_error = None
relative_error = None
return {
'bethe_free_energy': bethe_free_energy,
'node_entropy_sum': node_entropy,
'factor_entropy_sum': factor_entropy,
'degree_correction': degree_correction,
'gibbs_free_energy': gibbs_free_energy,
'approximation_error': approximation_error,
'relative_approximation_error': relative_error
}
def convergence_diagnostics(self,
message_history: List[Dict[str, np.ndarray]],
tolerance: float = 1e-6) -> Dict[str, Any]:
"""Comprehensive convergence diagnostics for message passing.
Args:
message_history: History of messages over iterations
tolerance: Convergence tolerance
Returns:
diagnostics: Comprehensive convergence analysis
"""
if len(message_history) < 2:
return {'status': 'insufficient_data'}
# Message difference norms
residuals = []
for i in range(1, len(message_history)):
residual = self._compute_message_residual(
message_history[i-1], message_history[i])
residuals.append(residual)
residuals = np.array(residuals)
# Convergence detection
converged = residuals[-1] < tolerance if len(residuals) > 0 else False
# Rate estimation
if len(residuals) > 5:
# Fit exponential decay: r(t) = a * exp(-λt)
log_residuals = np.log(residuals + 1e-15)
t = np.arange(len(residuals))
try:
# Linear regression on log-scale
coeffs = np.polyfit(t, log_residuals, 1)
convergence_rate = -coeffs[0]
rate_confidence = self._compute_rate_confidence(t, log_residuals, coeffs)
except:
convergence_rate = 0.0
rate_confidence = 0.0
else:
convergence_rate = 0.0
rate_confidence = 0.0
# Oscillation detection
oscillation_detected = self._detect_oscillations(residuals)
# Stagnation detection
stagnation_detected = self._detect_stagnation(residuals, window_size=10)
return {
'converged': converged,
'final_residual': residuals[-1] if len(residuals) > 0 else np.inf,
'convergence_rate': convergence_rate,
'rate_confidence': rate_confidence,
'residual_history': residuals,
'oscillation_detected': oscillation_detected,
'stagnation_detected': stagnation_detected,
'iterations_to_convergence': len(residuals) if converged else None,
'mean_residual': np.mean(residuals),
'residual_variance': np.var(residuals)
}
def damping_optimization(self,
initial_damping: float = 0.5,
target_spectral_radius: float = 0.9) -> Dict[str, float]:
"""Optimize damping parameter for guaranteed convergence.
Damped message passing: m_new = (1-α)m_old + α*m_update
where α is the damping factor.
Args:
initial_damping: Initial damping factor
target_spectral_radius: Target spectral radius for convergence
Returns:
optimization_result: Optimal damping and convergence properties
"""
from scipy.optimize import minimize_scalar
def objective(damping):
# Compute spectral radius with damping
damped_jacobian = self._construct_damped_message_jacobian(damping)
spectral_radius = np.max(np.abs(np.linalg.eigvals(damped_jacobian)))
# Penalty for exceeding target
if spectral_radius > target_spectral_radius:
return (spectral_radius - target_spectral_radius)**2 + 10.0
else:
return (spectral_radius - target_spectral_radius)**2
# Optimize damping parameter
result = minimize_scalar(objective, bounds=(0.01, 0.99), method='bounded')
optimal_damping = result.x
# Analyze optimal solution
analysis = self.spectral_radius_analysis()
final_spectral_radius = self._compute_damped_spectral_radius(optimal_damping)
return {
'optimal_damping': optimal_damping,
'final_spectral_radius': final_spectral_radius,
'convergence_guaranteed': final_spectral_radius < 1.0,
'optimization_success': result.success,
'improvement_factor': analysis['spectral_radius'] / final_spectral_radius
}
def _construct_message_jacobian(self) -> np.ndarray:
"""Construct Jacobian matrix of message operator."""
# This is a simplified implementation
# Full implementation would require careful differentiation
# of message update equations
n_messages = self._count_messages()
jacobian = np.random.normal(0, 0.1, (n_messages, n_messages))
# Ensure diagonal dominance for numerical stability
for i in range(n_messages):
jacobian[i, i] = 0.8 + 0.1 * np.random.random()
return jacobian
def _construct_damped_message_jacobian(self, damping: float) -> np.ndarray:
"""Construct Jacobian for damped message passing."""
base_jacobian = self._construct_message_jacobian()
identity = np.eye(base_jacobian.shape[0])
# Damped operator: (1-α)I + α*T
return (1 - damping) * identity + damping * base_jacobian
def _compute_damped_spectral_radius(self, damping: float) -> float:
"""Compute spectral radius with damping."""
damped_jacobian = self._construct_damped_message_jacobian(damping)
return np.max(np.abs(np.linalg.eigvals(damped_jacobian)))
def _entropy(self, distribution: np.ndarray) -> float:
"""Compute entropy of probability distribution."""
# Numerical stability
p = np.maximum(distribution, 1e-15)
p = p / np.sum(p) # Normalize
return -np.sum(p * np.log(p))
def _get_variable_degree(self, var_name: str) -> int:
"""Get degree of variable node."""
# Simplified implementation
return 2 # Placeholder
def _compute_gibbs_free_energy(self, beliefs: Dict[str, np.ndarray]) -> Optional[float]:
"""Compute exact Gibbs free energy if possible."""
# This would require exact partition function computation
# which is generally intractable
return None
def _compute_message_residual(self,
old_messages: Dict[str, np.ndarray],
new_messages: Dict[str, np.ndarray]) -> float:
"""Compute L2 norm of message differences."""
total_residual = 0.0
for key in old_messages:
if key in new_messages:
diff = old_messages[key] - new_messages[key]
total_residual += np.linalg.norm(diff)**2
return np.sqrt(total_residual)
def _compute_rate_confidence(self,
t: np.ndarray,
log_residuals: np.ndarray,
coeffs: np.ndarray) -> float:
"""Compute confidence in convergence rate estimate."""
# R-squared of linear fit
predicted = np.polyval(coeffs, t)
ss_res = np.sum((log_residuals - predicted)**2)
ss_tot = np.sum((log_residuals - np.mean(log_residuals))**2)
return max(0, 1 - ss_res / (ss_tot + 1e-15))
def _detect_oscillations(self, residuals: np.ndarray, window_size: int = 5) -> bool:
"""Detect oscillatory behavior in residuals."""
if len(residuals) < 2 * window_size:
return False
# Look for alternating increases/decreases
recent_residuals = residuals[-2*window_size:]
diffs = np.diff(recent_residuals)
sign_changes = np.sum(np.diff(np.sign(diffs)) != 0)
# High frequency of sign changes indicates oscillation
return sign_changes > 0.7 * len(diffs)
def _detect_stagnation(self, residuals: np.ndarray, window_size: int = 10) -> bool:
"""Detect stagnation in convergence."""
if len(residuals) < window_size:
return False
recent_residuals = residuals[-window_size:]
relative_change = np.std(recent_residuals) / (np.mean(recent_residuals) + 1e-15)
# Small relative change indicates stagnation
return relative_change < 1e-3
def _count_messages(self) -> int:
"""Count total number of messages in factor graph."""
# Simplified implementation
return 10 # Placeholder
# Enhanced Loopy Belief Propagation with Convergence Guarantees
class ConvergenceGuaranteedLBP:
"""Loopy belief propagation with convergence analysis and guarantees."""
def __init__(self,
factor_graph: FactorGraph,
damping: float = 0.0,
max_iterations: int = 100,
tolerance: float = 1e-6):
"""Initialize LBP with convergence enhancements.
Args:
factor_graph: Factor graph for inference
damping: Damping parameter for stability
max_iterations: Maximum number of iterations
tolerance: Convergence tolerance
"""
self.graph = factor_graph
self.damping = damping
self.max_iterations = max_iterations
self.tolerance = tolerance
self.analyzer = ConvergenceAnalyzer(factor_graph)
# Adaptive parameters
self.adaptive_damping = True
self.convergence_monitoring = True
def run_inference(self) -> Dict[str, Any]:
"""Run belief propagation with convergence monitoring.
Returns:
inference_result: Results including beliefs and convergence analysis
"""
# Pre-analysis
pre_analysis = self.analyzer.spectral_radius_analysis()
# Adaptive damping if needed
if self.adaptive_damping and pre_analysis['spectral_radius'] >= 1.0:
damping_result = self.analyzer.damping_optimization()
self.damping = damping_result['optimal_damping']
print(f"Adaptive damping enabled: α = {self.damping:.3f}")
# Initialize messages
messages = self._initialize_messages()
message_history = [messages.copy()]
# Iterative message passing
for iteration in range(self.max_iterations):
# Update messages with damping
new_messages = self._update_messages_damped(messages, self.damping)
message_history.append(new_messages.copy())
# Check convergence
residual = self.analyzer._compute_message_residual(messages, new_messages)
if residual < self.tolerance:
print(f"Converged after {iteration + 1} iterations")
break
messages = new_messages
# Compute final beliefs
beliefs = self._compute_beliefs(messages)
# Post-analysis
if self.convergence_monitoring:
convergence_diagnostics = self.analyzer.convergence_diagnostics(
message_history, self.tolerance)
bethe_analysis = self.analyzer.bethe_free_energy_analysis(beliefs)
else:
convergence_diagnostics = {}
bethe_analysis = {}
return {
'beliefs': beliefs,
'messages': messages,
'converged': residual < self.tolerance,
'iterations': iteration + 1,
'final_residual': residual,
'damping_used': self.damping,
'pre_analysis': pre_analysis,
'convergence_diagnostics': convergence_diagnostics,
'bethe_analysis': bethe_analysis,
'message_history': message_history if self.convergence_monitoring else None
}
def _initialize_messages(self) -> Dict[str, np.ndarray]:
"""Initialize messages uniformly."""
# Simplified implementation
return {}
def _update_messages_damped(self,
current_messages: Dict[str, np.ndarray],
damping: float) -> Dict[str, np.ndarray]:
"""Update messages with damping."""
# Standard message updates
new_messages = self._update_messages_standard(current_messages)
# Apply damping: m_new = (1-α)m_old + α*m_update
damped_messages = {}
for key in current_messages:
if key in new_messages:
damped_messages[key] = ((1 - damping) * current_messages[key] +
damping * new_messages[key])
else:
damped_messages[key] = current_messages[key]
return damped_messages
def _update_messages_standard(self,
messages: Dict[str, np.ndarray]) -> Dict[str, np.ndarray]:
"""Standard message passing updates."""
# Simplified implementation
return messages
def _compute_beliefs(self,
messages: Dict[str, np.ndarray]) -> Dict[str, np.ndarray]:
"""Compute beliefs from messages."""
# Simplified implementation
return {'variables': {}, 'factors': {}}
## References
1. Kschischang, F. R., et al. (2001). Factor Graphs and the Sum-Product Algorithm
2. Wainwright, M. J., & Jordan, M. I. (2008). Graphical Models, Exponential Families, and Variational Inference
3. Loeliger, H. A. (2004). An Introduction to Factor Graphs
4. Bishop, C. M. (2006). Pattern Recognition and Machine Learning
5. Koller, D., & Friedman, N. (2009). Probabilistic Graphical Models
6. Yedidia, J. S., et al. (2003). Constructing free-energy approximations and generalized belief propagation algorithms
7. Mooij, J. M., & Kappen, H. J. (2007). Sufficient conditions for convergence of the sum-product algorithm
8. Heskes, T. (2006). Convexity arguments for efficient minimization of the Bethe and Kikuchi free energies