ELICITATION OF PRIOR DISTRIBUTIONS TO ESTIMATE VARIANCE COMPONENTS

Abstract

In the Bayesian context, when variance components are modeled in normal hierarchical models, the inverted gamma distribution (IG) is typically used as the prior density for each component. However, the literature indicates that this prior density is highly informative, and thus the Half Cauchy distribution (HC) is recommended. The aim of this study was to evaluate, using simulation (in the context of high-dimensional data such as in the case of genomic selection applications), the suitability of the scaled inverse chi-squared (X-2v,S) distribution, which belongs to the family of scaled inverse gamma distributions, and HC as prior densities for the variance components in the Bayesian Ridge regression model. The evaluation was carried out when the number of observations in the response variable is greater than the number of predictor variables (n > p) as well as in high dimensions (n << p). The Bayesian learning of the posterior distribution was evaluated using the Hellinger distance (HD). The results of the Bayesian analysis were also compared with those obtained with the restricted maximum likelihood (REML). Results indicate that when n > p, the REML method underestimates the variance of the random effect, whereas in scenarios in which n << p, the method overestimates the same parameter when the variance of the error is large (greater than or equal to 6.0) and gives consistent estimations when the error variance is moderate (equal to 1.0). On the other hand, under prior distribution (X-2v,S) and in both scenarios (n > p) and n << p, it was observed that the parameters can be overestimated or underestimated, depending on the fixed values used to simulate the data. For the case of the HC prior distribution, the credibility intervals for both the variance of the effects of the predictor variables and the variance of the error contain the true values of the parameters, and their precisions increase with the sample size.In the Bayesian context, when variance components are modeled in normal hierarchical models, the inverted gamma distribution (IG) is typically used as the prior density for each component. However, the literature indicates that this prior density is highly informative, and thus the Half Cauchy distribution (HC) is recommended. The aim of this study was to evaluate, using simulation (in the context of high-dimensional data such as in the case of genomic selection applications), the suitability of the scaled inverse chi-squared (X-2v,S) distribution, which belongs to the family of scaled inverse gamma distributions, and HC as prior densities for the variance components in the Bayesian Ridge regression model. The evaluation was carried out when the number of observations in the response variable is greater than the number of predictor variables (n > p) as well as in high dimensions (n << p). The Bayesian learning of the posterior distribution was evaluated using the Hellinger distance (HD). The results of the Bayesian analysis were also compared with those obtained with the restricted maximum likelihood (REML). Results indicate that when n > p, the REML method underestimates the variance of the random effect, whereas in scenarios in which n << p, the method overestimates the same parameter when the variance of the error is large (greater than or equal to 6.0) and gives consistent estimations when the error variance is moderate (equal to 1.0). On the other hand, under prior distribution (X-2v,S) and in both scenarios (n > p) and n << p, it was observed that the parameters can be overestimated or underestimated, depending on the fixed values used to simulate the data. For the case of the HC prior distribution, the credibility intervals for both the variance of the effects of the predictor variables and the variance of the error contain the true values of the parameters, and their precisions increase with the sample size