Why is the Fisher information matrix so important, and why do we need to calculate it?

Question

I am currently taking a module in predictive analytics and I have come across the Fisher Information Matrix.

Can somebody explain why this is so important, its use and why we need to calculate it.

Thanks in advance

Answer 1

The Fisher Information matrix is extremely important. It tells how much information one (input) parameter carries about another (output) value. So if you had a complete model of human physiology, you could use the Fisher information to tell how knowledge about 1) eating habits, 2) exercise habits, 3) sleep time, and 4) lipstick color affected a person's body mass. You'd find that the entries corresponding to the first three variables would be large but that the last would be zero.

Answer 2

Another perspective, the Fisher information matrix is very important because from its inverse we can estimate the variance and covariance of the parameter estimators of a likelihood function. $Var\left(\hat{\beta_j}\right)=I^{-1}\left(\hat{\beta}_j\right)$