A concave function is one in which, for any two points we choose, the value of the function in between those points is larger than the value of the line segment connecting the points. Mathematically, a function $f$ is convex if, for any $\x_1, \x_2$, in the domain of $f$,
\[f(w\x_1 + (1-w)\x_2) \ge wf(\x_1) + (1-w)f(\x_2)\]for all \(w \in (0,1)\). The function \(f\) is strictly concave if
\[f(w\x_1 + (1-w)\x_2) > wf(\x_1) + (1-w)f(\x_2).\]It is worth mentioning that the domain of \(f\) must be a convex set in order for \(f\) to be a concave function. Otherwise, the idea of concavity is undefined.
Properties
- A strictly concave function has at most one unique maximum.
- If a strictly concave function has a local maximum, then that point is the global maximum.
- Concave functions are unimodal.
See here for proofs (concavity proofs are the same as convexity proofs, with the inequalities reversed).
Sufficient conditions
For any function that is twice differentiable, a sufficient condition for concavity unimodality is that its Hessian matrix \(H(\x)\) is negative definite for all \(\x\):
Theorem: Let \(f: \real^d \to \real\) be twice differentiable. If \(\nabla^2 f(x) \gle \zero\) for all \(\x\) in the domain of \(f\), then f is strictly concave.
The converse is not necessarily true.
Log-concave
A function \(f:\real^d \to \real\) is said to be log-concave if \(\log f\) is concave.
A sufficient condition for the likelihood to be log-concave is that the information matrix is positive definite for all \(\bt\) (see “Sufficient conditions” above).
Many common parametric models are log-concave, including everything in the exponential family (with respect to their natural parameters).