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 $f\colon K \to \mathbb{R}$ 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 strongly generalizing Jensen's inequality. Establishing these bounds requires the investigation of the convex hull of the graph of $f$, which can be difficult for complicated $f$. On the other hand, once these inequalities are established, they hold, just like Jensen's inequality, for any random variable $X$. Hence, these bounds are of particular interest in cases where $f$ is fairly 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不等式在测度与概率论、统计学、机器学习、信息论以及数学与数据科学的许多其他领域中普遍存在。它指出，对于凸域$K \subseteq \mathbb{R}^{d}$上的任意凸函数$f\colon K \to \mathbb{R}$和任意取值为$K$的随机变量$X$，有$\mathbb{E}[f(X)] \geq f(\mathbb{E}[X])$。本文推导了任意定义域$K$上任意函数$f$的$\mathbb{E}[f(X)]$的严格上界和下界——称为“图凸包界”，从而强烈推广了Jensen不等式。建立这些界需要研究函数$f$的图的凸包，这对于复杂的$f$而言可能较为困难。然而，一旦这些不等式成立，它们便与Jensen不等式一样，适用于任意随机变量$X$。因此，这些界在函数$f$相对简单而$X$复杂或未知的情形下尤为有价值。本文覆盖了$f$的定义域与值域处于有限维和无限维空间的情况，同时包含了条件期望和马尔可夫算子的类似界。