The Neyman-Scott problem concerns $M$ Gaussian distributions with unknown means and identical but unknown standard deviation. Two data are sampled from each distribution. As M tends to infinity, we see that the Maximum Likelihood (ML) estimate of sigma is inconsistent, under-estimating sigma by a factor of the square root of 2. One way around this problem is to use the marginalised ML estimate for sigma. An alternative is to use Minimum Message Length (MML) (Wallace and Boulton, 1968; Wallace and Freeman, 1987) to estimate sigma and the various distribution means. The MML estimate maximises the posterior probability contained in an uncertainty region of volume proportional to the reciprocal of the square root of the expected Fisher information. MML is a general, universally applicable, invariant Bayesian estimation method, and general theorems of Wallace and Freeman (1987) and Barron and Cover (1991) show MML to be consistent and efficient. We ideally further seek a related problem for which ML remains inconsistent but for which we can not marginalise. As there is no variable selection, we also note an inconsistency in the Akaike Information Criterion (AIC).

Some key words: Consistency; Efficiency; Maximum Likelihood; Minimum Message Length; MML; Multiple populations; Neyman-Scott problem.