Abstract
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.
Highlights
This tutorial describes a parameter estimation technique that is little-known in social sciences, namely maximum a posteriori estimation
We reviewed a seldom-used technique for parameter estimation in social sciences, namely Maximum A Posteriori (MAP) estimation
MAP estimation is an extension of the regular MLE technique which can be used in conjunction with priors of any types
Summary
This tutorial describes a parameter estimation technique that is little-known in social sciences, namely maximum a posteriori estimation. In MLE, the likelihood of a set of parameters given the data is computed and a search for the parameters that maximize the likelihood is performed This technique is very general and can be applied to any population distribution (e.g., ex-Gaussian, lognormal, Weibull, etc.; see Luce, 1986, Appendix A, for a review of some distribution functions). The γ parameter of the Weibull distribution applied to response time data is always found to be in the vicinity of 2 plus or minus 1 (Logan, 1992, Cousineau and Shiffrin, 2004, Huber and Cousineau, 2003, Rouder, Lu, Speckman, Sun, & Jiang, 2005) This prior knowledge could be entered in a BE analysis using a normal distribution with mean 2 and standard deviation 0.5. The probability density function (pdf) of the Weibull distribution is given by:
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