Boltzmann machine
= Boltzmann machine
{c}
{title2=$p_\beta(x)=Z^{-1}e^{-\beta E(x)}$}
{wiki}
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> $E(x)=-\sum_{i<j}w_{ij}x_ix_j+\sum_i\theta_ix_i$. Here $\beta>0$ controls the preference for low <energy>, and the <partition function> $Z=\sum_xe^{-\beta E(x)}$ normalizes all states. In a computational model, $\beta$ is an algorithmic inverse <temperature>; interpreting it as $1/(k_BT)$ additionally requires physical <energy> units.
A random single-unit <Gibbs sampling> update has <conditional probability>
$$
p_\beta(x_k=1\mid x_{-k})=\frac1{1+e^{-2\beta h_k}},\qquad h_k=\sum_{j\ne k}w_{kj}x_j-\theta_k.
$$
Indeed, the two possible <energies> differ by $-2h_k$; 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 $\beta$, 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 $p_\beta$. A short run need not yet sample that distribution accurately.
As $\beta\to\infty$, 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>.