TITLE: When do people actually use the maximum entropy distribution?
QUESTION [18 upvotes]: One of the standard problems in convex optimization is the calculation of the maximum entropy distribution that satisfies some set of criteria.  For example, if $\mathbf{x} \in \mathbb R^n$ is an unknown vector of probabilities, one might solve a problem like $$\mathrm{maximize}_\mathbf{x\geq0} -\sum_{i=1}^n x_i \log x_i ~~~s.t. \\ A\mathbf{x}\leq \mathbf b \\ \sum_{i=1}^n x_i = 1~,$$which would of course find the probability distribution that satisfies a certain set of inequalities whose entropy is as large as possible.  My question is:  where do people actually use this (maybe not just using inequality constraints, other kinds of constraints would be possible as well of course)?  I see problems of this kind arising in textbooks very often, but I don't see research papers in which one really makes use of these distributions to some specific aim.

REPLY [5 votes]: Thermodynamics can be derived from this principle:
\begin{align*}
  0 &\overset!=\delta \Big[\underbrace{-k_B\langle\ln p\rangle}_{S} + \alpha\Big(\langle1\rangle - 1\Big) + \beta\Big(\langle E\rangle - U\Big) + \mu\Big(\langle N\rangle - N\Big)\Big]
\\\text{where}\ \langle x\rangle :&= \sum_i p_i x_i \ \text{denotes the expectation value of $x$}
\end{align*}
$k_B$ is the Boltzmann constant, which makes the entropy $S$ equivalent to the physical one, while the Lagrangian multiplicators $\beta$ and $\mu$ which conserve (internal) energy and particle count respectively turn out to be equivalent to the thermodynamical quantities $\beta^{-1}=k_B T$ (i.e. $\propto$ inverse temperature) and chemical potential $\mu$.
For the canonical ensemble (i.e. no particle exchange, $N_i\equiv N$) this yields the Boltzmann distribution
$$p_i = \frac{e^{-\beta E_i}}{\sum\limits_j e^{-\beta E_j}}.$$