Joint distribution

The restricted Boltzmann machine is described by a Boltzmann distribution

$$ \begin{align*} P_{rbm}(\boldsymbol{x},\boldsymbol{h},\boldsymbol{\Theta}) = \frac{1}{Z(\boldsymbol{\Theta})} \exp{-(E(\boldsymbol{x},\boldsymbol{h},\boldsymbol{\Theta}))}, \end{align*} $$

where \( Z \) is the normalization constant or partition function discussed earlier and defined as

$$ \begin{align*} Z(\boldsymbol{\Theta}) = \int \int \exp{-E(\boldsymbol{x},\boldsymbol{h},\boldsymbol{\Theta})} d\boldsymbol{x} d\boldsymbol{h}. \end{align*} $$

It is common to set the temperature \( T \) to one. It is omitted in the equations above. The energy is thus a dimensionless function.