Boltzmann machine ($p_{\beta }(x)=Z^{-1}{e}^{-\beta E(x)}$)

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 $E(x)=-{\sum }_{i<j}{w}_{ij}{x}_i{x}_j+{\sum }_i{\theta }_i{x}_i$. Here $\beta >0$ controls the preference for low energy, and the partition function $Z={\sum }_x{e}^{-\beta E(x)}$ normalizes all states. In a computational model, $\beta$ is an algorithmic inverse temperature; interpreting it as $1/(k_{B}T)$ additionally requires physical energy units.

A random single-unit Gibbs sampling update has conditional probabilityIndeed, 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.

Split the units into visible units $v$, whose patterns occur in the training data, and hidden units $z$. Let $q(v)$ be the empirical probability distribution and maximize the average log-likelihood $\ell ={\sum }_{v}q(v)\log p_{\beta }(v)$, where $p_{\beta }(v)={\sum }_z{p}_{\beta }(v,z)$. The positive phase clamps $v$ to data and averages over $q(v)p_{\beta }(z\mid v)$; the negative phase releases all units and averages over $p_{\beta }(v,z)$. The displayed difference uses these two expectations. When both units are visible, the positive pair product is supplied directly by the data.

For a proof, write $\log p_{\beta }(v)=\log {\sum }_z{e}^{-\beta E(v,z)}-\log Z$. Differentiating each finite sum gives, respectively, $-\beta ⟨\partial E{⟩}_{z\mid v}$ and $-\beta ⟨\partial E{⟩}_{v,z}$. Since $\partial E/\partial w_{ij}=-x_i{x}_j$, subtraction proves the rule. Similarly ${\partial }_{{\theta }_i}\ell =-\beta (⟨x_i{⟩}_{+}-⟨x_i{⟩}_{-})$. A small positive step along these gradients increases log-likelihood. The negative term suppresses associations overproduced by the model; strengthening observed pairs alone misses this correction. The identity requires equilibrium expectations; estimates from insufficiently mixed sampling can give a biased update.

A visible unit represents an observed coordinate of the data. Clamping it means holding that coordinate fixed while other units update. In a Boltzmann machine, hidden units are unobserved coordinates summed out to obtain the visible marginal distribution.

Take two visible bipolar units, no hidden units, zero thresholds, $\beta =1$, and $E_J(x)=-J{x}_1{x}_2$. As original training data, assign probabilities $3/8$ each to $(+,+)$ and $(-,-)$ and $1/8$ each to $(+,-)$ and $(-,+)$. The data pair product averages to $1/2$. Enumeration of all four states gives two weights $e^J$ and two $e^{-J}$, hence the partition function $Z(J)=4\cosh J$ andStarting at $J=0$, the positive-minus-negative difference is $1/2-0$. A gradient ascent step of size $1/2$ gives $J=1/4$. Each aligned state now has probability $0.311230$ and each opposed state $0.188770$, bringing the model toward the data. The log-likelihood increases from $-1.386294$ to $-1.292224$. Solving ${\ell }^{\prime }(J)=0$ gives $J_{\ast }=(\log 3)/2$, which reproduces the data exactly. Since ${\ell }^{\prime \prime }(J)=-1/{\cosh }^{2}J<0$, this is the unique maximum. These are exact enumerations and their decimal evaluations, with no sampling or laboratory measurements.

David H. Ackley, Geoffrey Hinton and Terrence J. Sejnowski tested their 1985 algorithm on a simulated 4–2–4 encoder. Two four-unit visible groups communicated through two hidden units, with no direct connections between the groups. Preparation set weights to zero and supplied corresponding one-active-unit patterns. Noisy clamping and annealed settling generated pair-coactivation readouts, compared with freely running readouts to choose weight changes. Learned hidden codes were the outcome. Equal sampling effort in clamped and free conditions supplied the comparison; noise limited weight growth. Finite settling was assumed to reach equilibrium, limiting the evidence for exact likelihood learning.^[ref]