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This may occur either if for any unbiased estimator, there exists another with a strictly smaller variance, or if an MVU estimator exists, but its variance is strictly greater than the inverse of the Fisher information. The Cramér–Rao bound can also be used to bound the variance of biased estimators of given bias.
The Cramér–Rao bound [9] [10] states that the inverse of the Fisher information is a lower bound on the variance of any unbiased estimator of θ. Van Trees (1968) and Frieden (2004) provide the following method of deriving the Cramér–Rao bound , a result which describes use of the Fisher information.
The quantum Cramér–Rao bound is the quantum analogue of the classical Cramér–Rao bound. It bounds the achievable precision in parameter estimation with a quantum system: It bounds the achievable precision in parameter estimation with a quantum system:
The quantum Fisher information is a central quantity in quantum metrology and is the quantum analogue of the classical ... via the quantum Cramér–Rao bound as ...
In statistics, efficiency is a measure of quality of an estimator, of an experimental design, [1] or of a hypothesis testing procedure. [2] Essentially, a more efficient estimator needs fewer input data or observations than a less efficient one to achieve the Cramér–Rao bound.
In information geometry, the Fisher information metric [1] is a particular Riemannian metric which can be defined on a smooth statistical manifold, i.e., a smooth manifold whose points are probability distributions. It can be used to calculate the distance between probability distributions. [2] The metric is interesting in several aspects.
The achievable precision is bounded from below by the quantum Cramér-Rao bound as ... is the quantum Fisher information. [1] [11] Examples One example of ...
Historically, information geometry can be traced back to the work of C. R. Rao, who was the first to treat the Fisher matrix as a Riemannian metric. [2] [3] The modern theory is largely due to Shun'ichi Amari, whose work has been greatly influential on the development of the field.