Câu trả lời:
Hãy xem cái này: http://en.wikipedia.org/wiki/Fisher_inif#Matrix_form
Từ định nghĩa, chúng ta có
If this component wise notation is too ugly, note that the Fisher Information matrix can be written as , in which the scores vector is defined as
Hence, we have the one-liner
WARNING: not a general answer!
If corresponds to a full-rank exponential family, then the negative Hessian of the log-likelihood is the covariance matrix of the sufficient statistic. Covariance matrices are always positive semi-definite. Since the Fisher information is a convex combination of positive semi-definite matrices, so it must also be positive semi-definite.