# Self-designed objective for multiple linear regression

#### baiyang11

A multiple linear regression is to use several predictor variables to predict the outcome of a response variable, like the following relationship:

$y_{i}=\beta_{1}x_{i1}+...+\beta_{p}x_{ip}+$ $\epsilon_{i}$, $i=1,...,n$

I understand the typical objective to learn the $\beta$ paramters is least-squares, which means to minimize the sum of the sqaure of $\epsilon_{i}$. Now I want other kinds of objective, for example to maximize the Shannon entropy of the sequences of $\epsilon$ (or other self-specified objective). I googled towards this direction but no luck. I am wondering if there is any problem (and tool to solve it if possible) I can look into to do that?

#### chiro

MHF Helper
Hey baiyang11.

Are you familiar with expectation maximization? Also you will need to be familiar with multi-variable optimization techniques and constructing an iteration system (of matrices) to converge to the true value under these conditions.

When looking at GLM's you will find that they use matrix calculus to iterate towards a solution (also in EM) and you can combine that style of thinking with optimization and iterate towards a solution.

The matrix approach will require a bit more linear algebra than what is normally touched on in statistics.