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Cramer-Rao lower bound

As explained in Machine Learning Algorithms in Depth

The Cramer-Rao lower bound is a lower bound on the variance of an unbiased estimator under regularity conditions.

The Cramer–Rao lower bound limits the variance of any unbiased estimator when p(x|θ) satisfies the regularity condition. It equals the inverse Fisher information matrix, so greater information permits lower achievable variance. section 7.3.1

For active learning, variance reduction can optimize the inverse information matrix by:

  • A-optimality: minimizing Tr(I⁻¹(θ)). section 7.3.1
  • D-optimality: minimizing |I⁻¹(θ)|. section 7.3.1
  • E-optimality: minimizing λmax[I⁻¹(θ)]. section 7.3.1
  • For K parameters, Fisher information is a K × K matrix; evaluating candidates may require matrix inversion, costing O(UK³) across U unlabeled instances. section 7.3.1

How Machine Learning Algorithms in Depth covers it

How deep it goes
Defines the bound, states its regularity condition, connects it to Fisher information, and compares optimization criteria.
Examples
Uses multivariate K-parameter information matrices and active-learning query selection as computational examples.
What it assumes
Assumes probability, parameter estimation, unbiasedness, regularity conditions, variance, and Fisher information.
What you can do after
Evaluate variance-reduction query strategies using A-, D-, or E-optimality and account for their matrix-inversion cost.
How it connects
Connects unbiased estimation, bias–variance tradeoff, Fisher information, active learning, and optimal experimental design.
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