*Bounty: 100*

*Bounty: 100*

I have an unknown (discrete) probability distribution $p={p_s}$, where $p_s$ is the probability of observing configuration $s$. To each configuration is associated an energy that I can compute $E_s$.

If I want to estimate the mean

$$

langle E rangle_p := sum_s E_s p_s (1)

$$

by drawing $N$ samples, I will clearly use the sample mean estimator:

$$

M_E := frac{1}{N} sum_j E_j (2)

$$

where $E_j$ is the energy that I obtained at the $j-$th draw.

At this point something happens and my distribution $p_s$ is changed just a bit, but in an unknown way, i.e., the new distribution is

$$

p_s to p’_s = p_s +delta p_s.

$$

The energy of each configuration is unchanged.

Now I would like to estimate the new average

$$

langle E rangle_{p’} := sum_s E_s p’_s

$$

Is there a way to do that, that takes into account that I have some knowledge of $p’$ [namely, I already estimated (1) via (2)]? The goal is to minimize the number of samples $N$ that need to be taken.

**EDIT**

Let me add something as to why this may not be hopeless.

On the one hand, once my distribution changed, *without taking any additional sample* I can simply guess the average with the previous estimation. The question is essentially if I can do better than that?

On the other hand I can assume that my perturbation is of order $epsilon$, can I obtain an estimation of the new mean up to the same order (at least approximately)?

I’d be interested in any reference or even a no-go theorem or no-go argument.

**EDIT 2**

I was hoping that something like Kalman filtering or Bayesian inference could do the trick but I know too little in that field.