Question

For a convex function \(f\), \(E[f(x)] \geq f(E[x])\). Flip the sign for a concave function.

A function \(f(x)\) is convex if \(f''(x) \geq 0\) everywhere in its domain. For example, if \(f(x) = \log x\), \(f''(x) = -1/x^2\), so the log function is concave for \(x \in (0, \infty]\). A visual illustration of Jensen’s inequality is shown below.

Image(filename='figs/jensen.png')

When is Jensen’s inequality an equality? From the diagram, we can see that this only happens if the function \(f(x)\) is a constant! We will make use of this fact later on in the lecture.