What does concurve or convex means in terms of Log likelihood function?

Question

As a non native english speaker and a naive student, I found difficulties in understanding what this means.

While studying a paper, I found that some log likelihood functions showed convexity property in that study. There are nothing more about this property in that paper.

I spent some time to understand what this property is. However, I think I just wasted my time.

Is it something about maximizing log likelihood so that we can find better estimate from a regression model?

Any suggestions would be much appreciated. Thank you.

Answer 1

A rough analogy: a convex function is shaped like a valley, a concave function is shaped like a hill. Regression uses maximum likelihood estimation, which is about maximising the log-likelihood function. In some cases, this maximum can be found with an analytic formula. In other cases, it needs to be estimated iteratively. The computer guesses a location and calculates the log-likelihood, then tries another location and sees whether the new log-likelihood is higher. If the computer is on a hill, it can find the summit by walking in whatever direction is uphill, but if it's in a valley, the best direction to walk in is less clear, and sometimes it may fail to find a summit if it gets stuck in a valley or on a flat plain.

Answer 2

https://en.wikipedia.org/wiki/Convex_optimization

Properties

The following are useful properties of convex optimization problems:[16][12]

every local minimum is a global minimum;

the optimal set is convex;

if the objective function is strictly convex, then the problem has at most one optimal point.