2018 Volume 23 Issue 1 Pages 39-56
Ronald A. Fisher pointed out that Bayes (1763) estimation, which uses flat prior distributions, contains a fatal problem: we can create any estimate we like by changing the transformation used before the analysis. Therefore, Fisher proposed the maximum likelihood estimation (ML estimation) to replace Bayes (1763) estimation. However, the ML estimation procedure is sometimes more difficult than Bayes estimation using the Markov chain Monte Carlo (MCMC) method. In such cases, we should use the adjusted Bayes (1763) estimation to obtain ML estimates; we can use the median of the posterior distribution as the ML estimate if we use an appropriate transformation (empirical Jeffreys prior) that makes the posterior distribution roughly symmetrical. We can find the appropriate transformation by finding the Box-Cox transformation that makes the skew of the posterior distribution nearly zero. In this case, we can use the 2.5% and 97.5% quantiles of the posterior distribution as the Fisherian 95% confidence interval. In this case, the posterior distribution approximates the fiducial distribution that is the probability distribution of parameters. I show an example of the calculation using a dataset for the population of sika deer, Cervus nippon, in Hokkaido, Japan.
