I am reading about Boltzmann machines and according the formulas the joint probability of the states of all units is $$ P(X = x) = \frac{1}{Z} e^{-\frac{1}{2T} \sum_i\sum_j {x_i x_j w_{ij}}} $$ $$ Z = \sum_x e^{-\frac{1}{2T}\sum_i\sum_j x_i x_j w_{ij}} $$ In thse formulas it is assumed that $ w_{ii} = 0$. However it is infeasible to calculate the quantity $Z$ directly. Therefore people use Gibbs sampling to sample different states. I don't understand how exactly the sampling is used to estimate probabilities.
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$\begingroup$ In this and most other cases the Markov chain obtained by Gibbs sampling has a stationary distribution which is P. Please see the Wikipedia article about Gibbs sampling. $\endgroup$– ValentasFeb 19, 2016 at 15:15
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