Keynes (1911) derived general forms of probability density functions for which the "most probable value" is given by the arithmetic mean, the geometric mean, the harmonic mean, or the median. His approach was based on indirect (i.e., posterior) distributions and used a constant prior distribution for the parameter of interest. It was therefore equivalent to maximum likelihood (ML) estimation, the technique later introduced by Fisher (1912). Keynes' results suffer from the fact that he did not discuss the supports of the distributions, the sets of possible parameter values, and the normalising constants required to make sure that the derived functions are indeed densities. Taking these aspects into account, we show that several of the distributions proposed by Keynes reduce to well-known ones, like the exponential, the Pareto, and a special case of the generalised inverse Gaussian distribution. Keynes' approach based on the arithmetic, the geometric, and the harmonic mean can be generalised to the class of quasi-arithmetic means. This generalisation allows us to derive further results. For example, assuming that the ML estimator of the parameter of interest is the exponential mean of the observations leads to the most general form of an exponential family with location parameter introduced by Dynkin (1961) and Ferguson (1962, 1963).
ML estimator criterion function median quasi-arithmetic mean exponential family