From https://en.wikipedia.org/wiki/Observed_information the observed Fisher information matrix is just the negative Hessian of the log likelihood function. If your log likelihood function is not convex then the hessian will not be positive-definite (and thus indefinite).

From https://en.wikipedia.org/wiki/Observed_information the observed Fisher information matrix is just the negative Hessian of the log likelihood function. If your log likelihood function is not convex then the hessian will not be positive-definite (and thus indefinite).

This is the case if you are using the "Observed Fisher Information Matrix" from https://en.wikipedia.org/wiki/Scoring_algorithm. If you are using the canonical notion of the Fisher Information Matrix (https://en.wikipedia.org/wiki/Fisher_information#Matrix_form) then it must be positive semi-definite.