Minimax confidence intervals in geomagnetism

SUMMARY Backus (1989) uses the prior information that the rest mass of Earth's magnetic field is less than the mass of Earth, or that the Ohmic heat liberated in the core by the currents giving rise to the main magnetic field is less than the surface heat flow, to compute the lengths of confidence intervals for low‐degree Gauss coefficients of the magnetic field. His technique for producing confidence intervals yields intervals that are longer than necessary to have probability 1 ‐ α of containing the true values. The present paper uses theory of Donoho (1989) to find lower bounds on the lengths of optimally short fixed‐length confidence intervals ( minimax confidence intervals ) for Gauss coefficients of the field of degree 1 ≥ l ≥ 12 using the heat flow constraint. The bounds on optimal minimax intervals are about 40 per cent shorter than Backus’ intervals: no procedure for producing fixed‐length confidence intervals, linear or non‐linear, can give intervals shorter than about 60 per cent the length of Backus’ in this problem. While both methods rigorously account for the fact that core field models are infinite‐dimensional, the application of the techniques to the geomagnetic problem involves approximations and counterfactual assumptions about the data errors, and so these results are likely to be extremely optimistic estimates of the actual uncertainty in Gauss coefficients.