High level interpretation of Cramer-Rao bound and Fisher information matrix

Question

I am reviewing a manuscript and am struggling to understand why some statistical techniques were chosen, i.e. what information they can give. The paper looks at the effect of predicting a variable such as temperature based on noisy measurements from different sensors. Can you please correct any errors in the following interpretation of the paper's statistical tools:

1. For the temperature prediction, the authors obtain a Cramer Rao Bound (CRB) which defines the smallest error that the prediction might have. A small CRB is desirable, since it means the prediction has a smaller error.
2. The authors decompose the CRB into a numerator and a divisor. The divisor is the Fisher Information Matrix (FIM). A large FIM is desirable, since it means the prediction has a smaller error.
3. The authors lists several situations that would result in a singular FIM, which is undesirable because it means that the temperature prediction isn't valid.

Answer 1

I think you are correct in your interpretations. The below is an informal response to your points.

1. If an unbiased estimator achieves the CRB, then it is the optimal estimator out of the class of unbiased estimators for the parameter.

2. Because their estimator achieves the CRB, the greater the FIM then the more certain you are in the estimator's estimates.

3. If the FIM is singular, then you basically have unlimited uncertainty in the estimator's estimates.