Jensen's inequality is ubiquitous in measure and probability theory, statistics, machine learning, information theory and many other areas of mathematics and data science. It states that, for any convex function $f\colon K \to \mathbb{R}$ defined on a convex domain $K \subseteq \mathbb{R}^{d}$ and any random variable $X$ taking values in $K$, $\mathbb{E}[f(X)] \geq f(\mathbb{E}[X])$. In this paper, sharp upper and lower bounds on $\mathbb{E}[f(X)]$, termed ``graph convex hull bounds'', are derived for arbitrary functions $f$ on arbitrary domains $K$, thereby extensively generalizing Jensen's inequality. The derivation of these bounds necessitates the investigation of the convex hull of the graph of $f$, which can be challenging for complex functions. On the other hand, once these inequalities are established, they hold, just like Jensen's inequality, for \emph{any} $K$-valued random variable $X$. Therefore, these bounds are of particular interest in cases where $f$ is relatively simple and $X$ is complicated or unknown. Both finite- and infinite-dimensional domains and codomains of $f$ are covered as well as analogous bounds for conditional expectations and Markov operators.

翻译：Jensen不等式在测度论与概率论、统计学、机器学习、信息论以及数学与数据科学的诸多领域中无处不在。该不等式指出：对于定义在凸域$K \subseteq \mathbb{R}^{d}$上的任意凸函数$f\colon K \to \mathbb{R}$及任意取值为$K$的随机变量$X$，有$\mathbb{E}[f(X)] \geq f(\mathbb{E}[X])$。本文针对任意定义域$K$上的任意函数$f$，推导了$\mathbb{E}[f(X)]$的尖锐上界与下界（称为"图凸包界"），从而显著推广了Jensen不等式。这些界的推导需研究函数$f$的图凸包，对于复杂函数而言颇具挑战性。然而，一旦建立这些不等式，其与Jensen不等式一样对任意取值于$K$的随机变量$X$都成立。因此，当$f$相对简单而$X$复杂或未知时，这些界具有特殊价值。本文同时覆盖了$f$的定义域与值域为有限维及无限维情形，并给出了条件期望与Markov算子的类似界。