Application of a Modified g -Parameter Prior in Bayesian Model Averaging to Water Pollution in Ibadan (Published)
A special technique that measures the uncertainties embedded in model selection processes is Bayesian Model Averaging (BMA) which depends on the appropriate choices of model and parameter priors. Inspite the importance of the parameter priors’ specification in BMA, the existing parameter priors give exitremely low Posterior Model Probability (PMP). Therefore, this paper elicits modified g-parameter priors to improve the performance of the PMP and predictive ability of the model with an application to the Water Pollution of Asejire in Ibadan. The modified g-parameter priors gj = ,established the consistency conditions and asymptotic properties using the models in the literature. The results show that the PMP with the best prior (gj=) had the least standard deviations (0.0411 at n=100,000 and 0:000 at n=1000) for models 1 & 2 respectively; and had the highest posterior means (0.9577 at n=100,000 and 1.000 at n=1000) for models 1 & 2 respectively. The point and overall predictive performances for the best prior were 2.357 at n=50 and 2.335 at n=100,000 when compared with the BMA Log Predictive Score threshold of 2.335. Applying this best g-parameter prior in modeling the Asejire river, it indicates that the dissolved solids (mg/l) and total solids (mg/l) are the most important pollutants in the river model with their PIP of 6.14% and 6.1% respectively.
Keywords: Posterior Inclusion Probability (PIP), dissolved solids, log-predictive score, model uncertainty
On a Modified g-Parameter Prior in Bayesian Model Averaging (Published)
A special technique that measures the uncertainties embedded in model selection processes is Bayesian Model Averaging (BMA) which depend on the appropriate choices of model and parameter priors. As important as parameter priors’ specification in BMA, the existing parameter priors based on fast increasing sample sizes compared to the number of regressors in a model give low Posterior Model Probability (PMP). Therefore, this research aimed at eliciting a modified g-parameter priors to improve the performance of the PMP and predictive ability of the model. From the functional form of the g-priors used; gj = / where and are functions of regressors per model j and sample size n with , the tools of BMA like Bayes Theorem, Bayes Factor (BF), Posterior Model Probability (PMP), Prior Inclusion Probability (PIP) and Shrinkage Factor (SF) through the modified g-parameter priors gj = established the superiority of the consistency’s conditions and asymptotic properties of the prior(s) using the Fernandez, Ley and Steel (FLS) models (1 & 2); and respectively with as sample sizes. The result from the analysis revealed that the performance of PMP was reliable with the least standard deviations (0.1994 SD 0.0411) and (0.1086SD0.000) for model 1 and model 2 respectively; and it was convergent with the highest means (0.5378Mean0.9577) and (0.8342Mean1.000) for model 1 and model 2 respectively. For the three modified g-parameter priors, the best reliability occurred when n = 100; 000 for Model 1 and Model 2 with (0.0631, 0.0521 and 0.0411) and (0.00, 0.00 and 0.00) respectively; also, the best convergence occurred with (0.9343, 0.9460 and 0.9577) and (1.00, 1.00 and 1.00) for Model 1 and Model 2 respectively when n = 100; 000. The predictive performance affirmed the goodness of the elicited g-parameter priors when n = 50 for Point prediction with (2.302, 2.357, 2.357); and when n = 100; 000 for Overall prediction with (2.332, 2.334, 2.335) which were all closed to the LPS threshold 2.335 according to BMA specification.
Keywords: Dissolved Solids (DS), Log-Predictive Score (LPS), Model Uncertainty (MU), Posterior Inclusion Probability (PIP)