Cramér–Rao bound
Repeated independent measurements improve precision as the square root of sample count.
$$\mathrm{Var}(\hat\theta)\geq\frac{1}{\nu F_C(\theta)}$$
Physical intuition
Repeated independent measurements improve precision as the square root of sample count.
Deep dive: physical mechanism
Use the curvature of the log-likelihood to connect data fluctuations to achievable estimator precision. State whether repetitions are independent and whether the estimator is unbiased.
What is measured experimentally?
Plot the empirical variance of an estimator against the lower bound as the sample count grows.
Assumptions, noise and limitations
Correlations, nuisance parameters and biased estimators require modified bounds.
Research and simulation exercise
Monte Carlo test an estimator on simulated sensor readouts.
Connect the ideas
Explore the chapter overview, learning path, and references.