### Reaching equilibrium state in a RBM

Chuck Norris can sample to infinity many times but we cannot. Say we want to reach the equilibrium state of an RBM, do we have a convergence criterion that tell us how “close” we are to equilibrium distribution ? In other words, how can we quantify the residual error if we stop after N steps ?

#### 1 Response to “Reaching equilibrium state in a RBM”

1. April 21, 2013 at 20:51

The problem is that only Chuck Norris knows the true distribution we are looking for.
Let say that we want to estimate $I = \int f(x)p(x)dx$ with $S_n = \frac{1}{n} \sum_t f(x_t)$ where $X_t$ comes from the MCMC. We can evaluate the variance of this estimator $Var(S_n) = \frac{Var(f(x))}{n}$ with $Var(f(x)) \approx \frac{1}{n-1} \sum_t (f(x_t) - S_n)^2$. The main problem with this method is that if your MCMC has not found all the modes of your distribution, you will not have the right estimation since you do not take all the possible data into account.