MLE of Integer Valued Normal Distribution

Question

If Z is a normal random variable on $\mathbb{R}^d$ with parameters $(\mu,\Sigma)$ and we know that $\mu\in \mathbb{Z}^d$ and $\Sigma \in \mathbb{Z}^{d+}$; then how can we solve this MLE problem for the MLES of $\mu$ and $\Sigma$?

Answer 1

Solve this as an integer constrained nonlinear programming (optimization) problem. Maximize likelihood (or minimize log likelihood or whatever) subject to the constraints that mu and sigma^2 are integers, and sigma^2 is positive. Use mixed integer nonlinear optimization software to do so (technically, all your variables are integer, so it's not really mixed, nevertheless software for mixed will work). You could use KNITRO for finding a local optimum, or use BARON to try to find a global optimum. Depending on the size and difficulty of the problem, there is no guarantee that BARON will find the global optimum (but if it tells you it has, you can trust it). There are probably other nonlinear optimizers as well which can handle integer variables. Unless you use a global optimizer, there is no guarantee you will be finding the global optimum, which is what you really want. Even if you have found a global optimum, for instance in BARON, I don't think you are able to get a guarantee that there aren't other points which achieve the same global optimum objective value, which in the case of MLE, would be very relevant to know.

One thing you could try is using YALMIP (free) under MATLAB. Without having to acquire even a trial license to KNITRO or BARON, you can use YALMIP"s own BMIBNB global branch and bound optimizer, which in principle will try to do the same thing as BARON, but is not as sophisticated, so its chances of success aren't as good, and chances of success may depend on the quality of Linear Programming and continuous variable local nonlinear optimizer available to solve subproblems).