Probability model

We define a probability

$$p(x_i,h_j;\boldsymbol{\Theta}) = \frac{f(x_i,h_j;\boldsymbol{\Theta})}{Z(\boldsymbol{\Theta})},$$

where $f(x_i,h_j;\boldsymbol{\Theta})$ is a function which we assume is larger or equal than zero and obeys all properties required for a probability distribution and $Z(\boldsymbol{\Theta})$ is a normalization constant. Inspired by statistical mechanics, we call it often for the partition function. It is defined as (assuming that we have discrete probability distributions)

$$Z(\boldsymbol{\Theta})=\sum_{x_i\in \boldsymbol{X}}\sum_{h_j\in \boldsymbol{H}} f(x_i,h_j;\boldsymbol{\Theta}).$$

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