Lele Subhash R
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2R3, Canada.
Entropy (Basel). 2023 Aug 25;25(9):1262. doi: 10.3390/e25091262.
In scientific problems, an appropriate statistical model often involves a large number of canonical parameters. Often times, the quantities of scientific interest are real-valued functions of these canonical parameters. Statistical inference for a specified function of the canonical parameters can be carried out via the Bayesian approach by simply using the posterior distribution of the specified function of the parameter of interest. Frequentist inference is usually based on the profile likelihood for the parameter of interest. When the likelihood function is analytical, computing the profile likelihood is simply a constrained optimization problem with many numerical algorithms available. However, for hierarchical models, computing the likelihood function and hence the profile likelihood function is difficult because of the high-dimensional integration involved. We describe a simple computational method to compute profile likelihood for any specified function of the parameters of a general hierarchical model using data doubling. We provide a mathematical proof for the validity of the method under regularity conditions that assure that the distribution of the maximum likelihood estimator of the canonical parameters is non-singular, multivariate, and Gaussian.
在科学问题中，一个合适的统计模型通常涉及大量的规范参数。通常，科学关注的量是这些规范参数的实值函数。对于规范参数的指定函数进行统计推断，可以通过贝叶斯方法简单地使用感兴趣参数的指定函数的后验分布来进行。频率主义推断通常基于感兴趣参数的轮廓似然。当似然函数是解析的时，计算轮廓似然仅仅是一个有许多可用数值算法的约束优化问题。然而，对于层次模型，由于涉及高维积分，计算似然函数以及因此的轮廓似然函数是困难的。我们描述一种简单的计算方法，使用数据加倍来计算一般层次模型参数的任何指定函数的轮廓似然。我们在正则条件下为该方法的有效性提供了数学证明，这些条件确保规范参数的最大似然估计量的分布是非奇异的、多元的和高斯的。
