ABSTRACT
Again, let Yobs denote the observed (or observable) data and θ the parameter vector of interest. A central subject of statistics is to make inference on θ based on Yobs. Frequentist/classical method arrives its inferential statements by combining point estimators of parameters with their standard errors. In a parametric inference problem, the observed vector Yobs is viewed as a realized value of a random vector whose distribution is f(Yobs|θ), which is the likelihood (function), usually denoted by L(θ|Yobs). Among all the estimation approaches, the maximum likelihood estimation (MLE) is the most popular one, where θ is estimated by θˆ that maximizes f(Yobs|θ). When sample sizes are small to moderate, a useful alternative to MLE is to utilize prior knowledge on the parameter and thus incorporate a prior distribution for the parameter into the likelihood and then compute the observed posterior distribution f(θ|Yobs). The posterior mode is defined as an argument θ˜ that maximizes f(θ|Yobs) or equivalently maximizes its logarithm log f(θ|Yobs). Computationally, finding the MLE or the mode becomes an optimization problem.
