BOLTZMANN MACHINE

Energy-based probabilistic model with Gibbs sampling

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A Restricted Boltzmann Machine (RBM) is a two-layer energy-based model with visible units v and hidden units h. The energy function is E(v,h) = −∑ᵢⱼ wᵢⱼvᵢhⱼ − ∑ᵢ bᵢvᵢ − ∑ⱼ cⱼhⱼ, and the joint probability is P(v,h) ∝ exp(−E(v,h)). Training uses contrastive divergence (CD-k): sample a positive phase from data (v⁺,h⁺), then run k steps of Gibbs sampling for the negative phase (v⁻,h⁻). The weight update ΔW = α(⟨vh⊤⟩_data − ⟨vh⊤⟩_model) maximizes the log-likelihood. "Restricted" means no intra-layer connections, making the conditional distributions P(h|v) and P(v|h) factorize — enabling efficient block Gibbs sampling. The animation shows activation propagation and weight strengths.