Improving maximum likelihood estimation using prior probabilities: A tutorial on maximum a posteriori estimation and an examination of the weibull distribution

This tutorial describes a parameter estimation technique that is little-known in social sciences, namely maximum a posteriori estimation. This technique can be used in conjunction with prior knowledge to improve maximum likelihood estimation of the best-fitting parameters of a data set. The estimates are based on the mode of the posterior distribution of a Bayesian analysis. The relationship between maximum a posteriori estimation, maximum likelihood estimation, and Bayesian estimation is discussed, and example simulations are presented using the Weibull distribution. We show that, for the Weibull distribution, the mode produces a less biased and more reliable point estimate of the parameters than the mean or the median of the posterior distribution. When Gaussian priors are used, it is recommended to underestimate the shape and scale parameters of the Weibull distribution to compensate for the inherent bias of the maximum likelihood and Bayesian methods which tend to overestimate these parameters. We conclude with a discussion of advantages and limitations of maximum a posteriori estimation.