maximum likelihood learning is possible. Another option is to use mean-field inference to estimate data-dependent expectations and approximate the expected sufficient statistics by using Markov chain Monte Carlo (MCMC). This approximate inference, which must be done for each test input, is about 25 to 50 times slower than a single bottom-up pass in DBMs. This makes joint optimization impractical for large data sets, and restricts the use of DBMs for tasks such as feature representation. Spike-and-slab RBMs The need for deep learning with real-valued inputs, as in Gaussian RBMs, led to the spike-and-slab RBM (ssRBM), which models continuous-valued inputs with binary latent variables. Similar to basic RBMs and its variants, a spike-and-slab RBM is a bipartite graph, while like GRBMs, the visible units (input) are real-valued. The difference is in the hidden layer, where each hidden unit has a binary spike variable and a real-valued slab variable. A spike is a discrete probability mass at zero, while a slab is a density over continuous domain; their mixture forms a prior. An extension of ssRBM called µ-ssRBM provides extra modeling capacity using additional terms in the energy function. One of these terms enables the model to form a conditional distribution of the spike variables by marginalizing out the slab variables given an observation. In Mathematics Main articles: Gibbs measure and Log-linear model In more general mathematical setting, the Boltzmann distribution is also known as the Gibbs measure. In statistics and machine learning it is called a log-linear model. In deep learning the Boltzmann distribution is used in the sampling distribution of stochastic neural networks such as the Boltzmann machine. History The Boltzmann machine is based on a spin-glass model of Sherrington-Kirkpatrick's stochastic Ising Model. The original contribution in applying such energy based models in cognitive science appeared in papers by Hinton and Sejnowski. The seminal publication by John Hopfield connected physics and statistical mechanics, mentioning spin glasses. The idea of applying the Ising model with annealed Gibbs sampling is present in Douglas Hofstadter's Copycat project. Similar ideas (with a change of sign in the energy function) are found in Paul Smolensky's "Harmony Theory". The explicit analogy drawn with statistical mechanics in the Boltzmann Machine formulation led to the use of terminology borrowed from physics (e.g., "energy" rather than "harmony"), which became standard in the field. The widespread adoption of this terminology may have been encouraged by the fact that its use