Jensen Gap

May 22, 2024 — May 22, 2024

functional analysis
stochastic processes
Figure 1

Related to generic concentration of measure, but about expectation of functions.

This note exists simply because I had not heard about this concept before, but it ended up being really useful even to name it.

Remember the classic Jensen inequality, where for some convex function \(f\) and random variable \(X\) we have \[ f \left(\mathbb{E} [X]\right)\leq \operatorname {E} \left[f (X)\right]. \]

The Jensen Gap is the value \[ J(f,X) := f \left(\mathbb{E} [X]\right)\leq \operatorname {E} \left[f (X)\right]. \] for a given \(X\) and (not necessarily convex) \(f\).

Amazingly, we can sometimes say things about how big this gap is. For continuous, \(f\) and \(X\) with \(f\) differentiable the most impressive results are (Abramovich and Persson 2016; Gao, Sitharam, and Roitberg 2020).

If \(X\) is discrete and \(f\) may or may not be differentiable, we can still say some things — see Simic (2008).

1 References

Abramovich, and Persson. 2016. Some New Estimates of the ‘Jensen Gap’.” Journal of Inequalities and Applications.
Gao, Sitharam, and Roitberg. 2020. Bounds on the Jensen Gap, and Implications for Mean-Concentrated Distributions.”
Simic. 2008. On a Global Upper Bound for Jensen’s Inequality.” Journal of Mathematical Analysis and Applications.
Walker. 2014. On a Lower Bound for the Jensen Inequality.” SIAM Journal on Mathematical Analysis.