Suppose we have a continuous probability distribution with density function $f$ whose support is $[a,b]$ and we know that for some finite set of values $\{ v_i \}_{i=1}^n$ between $a$ and $b$ that the $\operatorname{CDF}[f,v_i]=q_i$ where $$
a < v_1 \le \cdots \le v_n < b
$$ Essentially, we know the CDF of the distribution at various points in the interval, but we don't know it over the entire interval $[a,b]$.  

> How do we figure out the [maximal entropy probability distribution][1] in this context?


  [1]: https://en.wikipedia.org/wiki/Maximum_entropy_probability_distribution