I have a discrete random variable $X$, with possible values $\{E_k\}$ , following a Boltzmann distribution, with probability mass function given by:

$$ p_k = p(X=E_k) = e^{-\beta E_k}/Z $$

where $Z = \Sigma_k \ e^{-\beta E_k}$ is a normalization factor, and $\beta > 0$.

I have $N$ independent samples $x_i \sim X$. What is the best estimator of $\beta$?

I would like to know what would be the maximum-likelihood estimator, or the minimum variance unbiased estimator, of the parameter of the Boltzmann distribution behind the samples $x_i$.

A) Intuitively I think of:

  1. approximate the probability mass with $\hat p_k =$ (counts of $x_i = E_k)/N$
  2. fit a logarithmic regression $\log \hat p_k(x_i) = -\hat \beta x_i + \hat \alpha$
  3. use $\hat \beta$ as an estimator of $\beta$.

B) Using maximum-likelihood I get:

  1. likelihood function: $L(\beta, {\bf x}) = \Pi_{i=1}^N \ e^{-\beta x_i}/Z $
  2. log-likelihood: $l(\beta, {\bf x}) = - N\log Z - \Sigma_{i=1}^N \ \beta x_i$
  3. maximum-likelihood: $$\frac{\partial l}{\partial \beta} = N \ \Sigma_{k} \ E_k \frac{e^{-\beta E_k}}{Z} - \Sigma_{i=1}^N \ x_i = 0 \ \Leftrightarrow \ \bar X (\beta) = \frac{1}{N}\Sigma_{i=1}^N \ x_i $$

I guess this means I should choose $\beta$ such that the expected value of $X$ matches the sample average.

How can I compare estimators A and B? What is the estimators' expected value and variance? Is a better estimator known?



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