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 maximum entropy probability distribution that satisfies these constraints?