Change of notation

Note the change to a vector notation. A variable like $\boldsymbol{x}$ represents now a specific configuration. We can generate an infinity of such configurations. The final partition function is then the sum over all such possible configurations, that is

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

changes to

$$Z(\boldsymbol{\Theta})=\sum_{\boldsymbol{x}}\sum_{\boldsymbol{h}} f(\boldsymbol{x},\boldsymbol{h};\boldsymbol{\Theta}).$$

If we have a binary set of variable $x_i$ and $h_j$ and $M$ values of $x_i$ and $N$ values of $h_j$ we have in total $2^M$ and $2^N$ possible $\boldsymbol{x}$ and $\boldsymbol{h}$ configurations, respectively.

We see that even for the modest binary case, we can easily approach a number of configuration which is not possible to deal with.

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