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.



When do people actually use the maximum entropy distribution?
In finance, finding risk neutral probabilities can be done via max-entropy methods.


In short, you observe prices $p\_i$ of a finite number of instruments $\phi\_i$, and you seek a probability measure $P$ such that $p\_i = \int\phi\_i\,\mathrm{d}P$. There is in general no uniqueness of $P$, and finding such a $P$ can be challenging. Finding $P$ by a max-entropy method is one very practical way to do this.


There are quite a lot of papers which do exactly this, to name only two:


<http://papers.ssrn.com/sol3/papers.cfm?abstract_id=648>


<http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1968344>


