As we will see below, a typical Boltzmann machines employs a probability distribution
$$ p(\boldsymbol{x},\boldsymbol{h};\boldsymbol{\Theta}) = \frac{f(\boldsymbol{x},\boldsymbol{h};\boldsymbol{\Theta})}{Z(\boldsymbol{\Theta})}, $$where \( f(\boldsymbol{x},\boldsymbol{h};\boldsymbol{\Theta}) \) is given by a so-called energy model. If we assume that the random variables \( x_i \) and \( h_j \) take binary values only, for example \( x_i,h_j=\{0,1\} \), we have a so-called binary-binary model where
$$ f(\boldsymbol{x},\boldsymbol{h};\boldsymbol{\Theta})=-E(\boldsymbol{x}, \boldsymbol{h};\boldsymbol{\Theta}) = \sum_{x_i\in \boldsymbol{X}} x_i a_i+\sum_{h_j\in \boldsymbol{H}} b_j h_j + \sum_{x_i\in \boldsymbol{X},h_j\in\boldsymbol{H}} x_i w_{ij} h_j, $$where the set of parameters are given by the biases and weights \( \boldsymbol{\Theta}=\{\boldsymbol{a},\boldsymbol{b},\boldsymbol{W}\} \). Note the vector notation instead of \( x_i \) and \( h_j \) for \( f \). The vectors \( \boldsymbol{x} \) and \( \boldsymbol{h} \) represent a specific instance of stochastic variables \( x_i \) and \( h_j \). These arrangements of \( \boldsymbol{x} \) and \( \boldsymbol{h} \) lead to a specific energy configuration.