A Boltzmann machine is a stochastic neural network that learns a probability distribution specified by pairwise energies. To compare it directly with a Hopfield network, use the bipolar energy . Here controls the preference for low energy, and the partition function normalizes all states. In a computational model, is an algorithmic inverse temperature; interpreting it as additionally requires physical energy units.
A random single-unit Gibbs sampling update has conditional probabilityIndeed, the two possible energies differ by ; their normalized Boltzmann weights give this logistic function. Random selection of one unit at a time satisfies detailed balance. For finite weights and finite positive , all conditional probabilities lie strictly between zero and one, so the finite Markov chain can reach every state and can remain in place. Its unique stationary distribution is . A short run need not yet sample that distribution accurately.
As , a nonzero field selects its sign, approaching asynchronous Hopfield energy descent. At zero field the stochastic rule remains a fair coin, whereas the deterministic convention above retains the old value. At finite inverse temperature, uphill moves permit exploration beyond one local minimum. Recall and learning are different operations: sampling changes the units at fixed weights; learning changes the weights to fit the training data.
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