Log concave functions have some very interesting and useful properties. I’ll list some of these shortly after a three definitions.
A function is convex if the line segment joining two points on the graph lies above the graph. In symbols, f(x) is convex if for every t between 0 and 1, and for every x and y,
f(tx + (1-t)y) ≤ t f(x) + (1-t) f(y).
A function f(x) is concave if –f(x) is convex. Equivalently, the line connecting two points on the graph lies below the graph. In symbols, reverse the direction of the inequality for convexity.
A function f(x) is log concave if log( f(x) ) is concave.
The basic properties of convex functions are obvious. It’s easy to show that the sum of two convex functions is convex, the maximum of two convex functions is convex, etc. It would seem that log concave functions would be unremarkable because they are so simply related to convex functions. But they have some surprising properties.
- The Laplace transform of a non-negative function is a log convex.
- The running average of a log concave function is also log concave.
- If a random variable has a log concave PDF (probability density function), then the CDF (cumulative distribution function) is also log concave.
- The convolution of two log concave functions is log concave.
- If two independent random variables have log-concave PDFs, then their sum also has a log concave PDF.
A few properties of log concave functions do follow easily from analogous properties of convex and concave functions.
- The product of log concave functions is log concave.
- A function f(x) is log concave if and only if for every t between 0 and 1 and for every x and y, f(tx + (1-t)y) ≥ f(x)t f(y)1-t
- A smooth function f(x) is log concave if and only if the inequality f ∇2 f ≤ ∇ f ⋅ ∇ f holds everywhere.
Here are a few specific functions that are log concave.
- The exponential function eax
- The inverse logit function ex/(ex + 1)
- The function ∏ xi / ∑ xi
- The function xa for a ≥ 0
- Determinant is a log concave function on positive definite matrices
For more, see Stephen Boyd’s book Convex Optimization.