I'm trying to understand Boltzmann machines. Tutorials explain it with two formulas.

Logistic function for the probability of single units:

$$p(unit=1)=\frac{1}{1+e^{-\sum_{x}wx } }$$

and, when the machine is running, every state of the machine goes to the probability:

$$p(State= state\ with\ energy\ E_i )=\frac{e^{-E_i}}{\sum_i e^{-E_i}}$$

So, the state depends on the units, and then, if I understand correctly, the second formula is a consequence of the first.

So, how can it be the proof that the distribution of $p(state)$ is a consequence of $p(unit)$?

  • $\begingroup$ Although you have asked this question more than 3 years ago, I think it would be nice if you could clarify what those symbols in the equations mean. More precisely, what is $w$ and $x$, and what does "$State= state\ with\ energy\ E_i$" mean? $\endgroup$
    – nbro
    May 18, 2020 at 12:15


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