Maximum entropy distribution with constrained quantiles 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?

 A: Let $X \sim f$ which implies that $X \in [a,b]$ almost surely.  In terms of the CDF of $X$ at the points $\{ v_i \}_{i=1}^n$, define
$$
p_i := \mathbb{P}( v_{i-1} < X \le v_i ) = q_i - q_{i-1} \;, \quad 1 \le i \le n+1
$$
where we have introduced $v_0 = a$, $v_{n+1} = b$, $q_{0}=0$ and $q_{n+1} = 1$.  Note that $\sum_{1 \le i \le n+1}  p_i = 1$ since the sum telescopes.
More to the point, we wish to maximize the entropy $h(f)$ subject to the constraints that:
$\int_a^b f(x) dx = 1$ and $\int_{v_{i-1}}^{v_i} f(x) dx = p_i$ for $1 \le i \le n+1$.  As discussed in the reference below, the density of the maximum entropy distribution which satisfies these constraints is given by:
$$
f(x) =  Z^{-1} \exp\left( \sum_{i=1}^{n+1} \lambda_i 1_{(v_{i-1}, v_i]}(x) \right) 1_{[a,b]}(x)
$$
where $Z$ is a normalization constant chosen such that $\int_a^b f(x) =1$ and where $\{ \lambda_i \}_{i=1}^{n+1}$ are Lagrange multipliers chosen such that $\int_{v_i}^{v_{i+1}} f(x) dx = p_i$.  Eliminating these Lagrange multipliers and writing the density in terms of the given quantiles yields
$$
f(x) =  \sum_i\frac{q_i - q_{i-1}}{v_i - v_{i-1}} 1_{(v_{i-1}, v_i]}(x) \;.
$$
Reference
Cover, T. M., and J. A. Thomas. "Chapter 12, Maximum Entropy." Elements of Information Theory.
John Wiley & Sons, 2012.
