What is the maximum entropy distribution given values for several quantiles of one sample? Suppose I know the value of n specific quantiles from a large sample. What is the maximum entropy distribution over the real line given those values? 
 A: Maximum entropy problems do not always admit a solution. The generic expression for the maximum entropy density $f(x)$ given a set of integral constraints
\begin{equation}
\int dx \, h_i(x) \, f(x) = c_i
\end{equation}
with $i=1\ldots N$ is
\begin{equation}
f(x) = e^{\mu + \sum_{i=1}^N \lambda_i h_i(x)} \;.
\end{equation}
The values of the parameter $\mu$ and $\lambda_i$ have to be found by imposing the fulfillment of the constraints and the fact the $f(x)$ integrates to one, that is it is a proper density. I left the boundaries of the integral in the constraints unspecified on purpose. The reason will become clear in a moment.
The quantile constraints $F(x_i)=q_i$ where $F$ is the distribution function (the integral of the $f(x)$), translate in having $h_i(x)=1-\theta_{x_i}(x)$ and $c_i=q_i$, where $\theta_z(x)$ is the Heaviside theta function, which is equal to $1$ if $x>z$ and zero otherwise. The problem is that now the expression for $f(x)$ given above is not an (improper) integrable function on the real line. Summarizing, your problem does not have a solution. If however you add further constraints, like a finite support or some specified moment, than the problem might become solvable.
