# Variational Inference in RxInfer ## Overview Variational Inference (VI) in RxInfer provides a powerful framework for approximate Bayesian inference. It transforms the inference problem into an optimization problem by finding the best approximation to the true posterior distribution. ```mermaid graph TD subgraph True Posterior T1[Complex Distribution] T2[Intractable Computation] end subgraph Variational Approximation V1[Simple Distribution] V2[Tractable Computation] end subgraph Optimization O1[KL Divergence] O2[Free Energy] end T1 --> O1 V1 --> O1 O1 --> O2 style T1 fill:#f9f style T2 fill:#f9f style V1 fill:#bbf style V2 fill:#bbf style O1 fill:#bfb style O2 fill:#bfb ``` ## Core Concepts ### 1. Variational Distribution The approximating distribution \(q(z)\): ```julia @model function variational_example() # Prior z ~ Normal(0, 1) # Likelihood x ~ Normal(z, 1) # Variational approximation specified in constraints return z end @constraints function model_constraints() # Specify variational family q(z) :: NormalMeanPrecision end ``` ### 2. Evidence Lower Bound (ELBO) ```mermaid graph LR subgraph ELBO Components E1[Expected Log-Likelihood] E2[KL Divergence] end subgraph Optimization O1[Maximize ELBO] O2[Update Parameters] end E1 --> O1 E2 --> O1 O1 --> O2 style E1 fill:#f9f style E2 fill:#f9f style O1 fill:#bbf style O2 fill:#bbf ``` ### 3. Factorization Different factorization patterns: ```julia # Mean-field factorization @constraints function mean_field() q(x, y, z) = q(x)q(y)q(z) end # Structured factorization @constraints function structured() q(x, y, z) = q(x)q(y, z) end ``` ## Inference Methods ### 1. Mean-Field VI ```julia @model function mean_field_example() # Parameters μ ~ Normal(0, 1) σ ~ Gamma(1, 1) # Observations x ~ Normal(μ, σ) end @constraints function mean_field_constraints() # Mean-field factorization q(μ, σ) = q(μ)q(σ) # Distribution families q(μ) :: NormalMeanPrecision q(σ) :: GammaShapeRate end ``` ### 2. Structured VI ```julia @model function structured_example() # State space model x = Vector{Random}(undef, T) x[1] ~ Normal(0, 1) for t in 2:T x[t] ~ Normal(x[t-1], 1) end end @constraints function structured_constraints() # Keep temporal structure q(x[1:T]) :: MultivariateNormal end ``` ### 3. Amortized VI ```julia @model function amortized_example() # Encoder network ϕ = encoder_params() # Amortized variational distribution z ~ encoder_network(x, ϕ) # Decoder network θ = decoder_params() x ~ decoder_network(z, θ) end ``` ## Optimization Techniques ### 1. Natural Gradient Descent ```julia # Configure natural gradient optimization result = infer( model = my_model(), data = my_data, optimizer = NaturalGradient( learning_rate = 0.1, momentum = 0.9 ) ) ``` ### Optimization Flow ```mermaid graph TD subgraph Initialization I1[Initialize Parameters] I2[Set Learning Rate] end subgraph Updates U1[Compute Gradient] U2[Natural Gradient] U3[Parameter Update] end subgraph Monitoring M1[ELBO] M2[Convergence] end I1 --> U1 I2 --> U2 U1 --> U2 U2 --> U3 U3 --> M1 M1 --> M2 M2 -->|Not Converged| U1 style I1 fill:#f9f style I2 fill:#f9f style U1 fill:#bbf style U2 fill:#bbf style U3 fill:#bbf style M1 fill:#bfb style M2 fill:#bfb ``` ### 2. Stochastic VI ```julia # Stochastic optimization with mini-batches result = infer( model = my_model(), data = my_data, batch_size = 32, optimizer = StochasticVI( learning_rate = 0.01 ) ) ``` ### 3. Fixed-Point Updates ```julia # Use fixed-point updates when available result = infer( model = my_model(), data = my_data, update_method = FixedPoint() ) ``` ## Advanced Features ### 1. Custom Variational Families ```julia # Define custom variational distribution struct CustomVariational <: VariationalDistribution parameters::NamedTuple end # Implement required methods function compute_natural_params(d::CustomVariational) # Convert to natural parameters end function update_variational_params(d::CustomVariational, nat_params) # Update from natural parameters end ``` ### 2. Hybrid Inference ```julia @model function hybrid_model() # VI for some variables z ~ Normal(0, 1) # Exact inference for others x ~ conjugate_prior() end @constraints function hybrid_constraints() # Mixed inference strategy q(z) :: NormalMeanPrecision # VI q(x) :: ExactPosterior # Exact end ``` ### 3. Convergence Monitoring ```julia # Track ELBO and parameters subscribe!(result.elbo) do elbo println("ELBO: ", elbo) end subscribe!(result.parameters) do params println("Parameters: ", params) end ``` ## Best Practices ### 1. Model Design ```mermaid mindmap root((VI Design)) Model Structure Factorization Dependencies Conjugacy Optimization Learning Rate Batch Size Momentum Monitoring ELBO Gradients Parameters ``` ### 2. Distribution Selection - Choose appropriate variational families - Consider trade-offs between flexibility and tractability - Use conjugate relationships when possible ### 3. Optimization Strategy - Start with simpler factorizations - Monitor convergence carefully - Adjust optimization parameters based on performance ## Debugging and Diagnostics ### 1. ELBO Monitoring ```julia using Plots function plot_elbo(result) plot( result.elbo_history, label = "ELBO", xlabel = "Iteration", ylabel = "ELBO Value" ) end ``` ### 2. Parameter Tracking ```julia function monitor_parameters(result) for (param, value) in result.variational_params println("$param: mean = $(mean(value)), std = $(std(value))") end end ``` ### 3. Convergence Checks ```julia function check_convergence(elbo_history; threshold = 1e-6) if length(elbo_history) < 2 return false end return abs(elbo_history[end] - elbo_history[end-1]) < threshold end ``` ## References - [[inference_algorithms|Inference Algorithms]] - [[message_passing|Message Passing]] - [[factor_graphs|Factor Graphs]] - [[model_specification|Model Specification]] - [[optimization_techniques|Optimization Techniques]]