# How is the distribution of the state related to the distribution of the units in Boltzmann machines?

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 $$p(unit)$$?

• 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?
– nbro
May 18 '20 at 12:15