"Challenges to Developing Likelihood Functions for Practical Bayesian Analysis"
Bayesian theorem has been used widely in many research fields. While the theorem is mathematically rigorous, developing the likelihood function for Bayesian analysis is always challenging in practical Bayesian analysis. This seminar will present several such challenges in environmental modeling. In particular, this study presents a systematic and comprehensive investigation on impacts of eight residual models on Bayesian inference and predictive performance of three mechanistic soil respiration models; such an investigation has not been reported in literature. The residual models are the basis of formulating likelihood functions used in Bayesian inference to estimate posterior parameter distributions. Since the residual models are intrinsically unknown, assumptions on residual probability distributions are always required. This study considers three pairs of assumptions: homoscedastic/heteroscedastic, independent/auto-correlated, and Gaussian/no-Gaussian residuals. The eight residual models are constructed by accounting for the assumptions in a stepwise manner. The three soil respiration models have different levels of complexity with respect to number of carbon pools and explicit representations of soil moisture controls on carbon degradation and microbial uptake rates. The residual models have substantial impacts on Bayesian inference and predictive performance of the soil respiration models. The level of complexity of the best model is justified by the modeling results under all the eight residual models. The residual models that assume residual independence and homoscedasticity should not be used for any models. While accounting residual heteroscedasticity in the residual models improves Bayesian inference and predictive performance, it is not the case for accounting residual autocorrelation. The residual models using the skew exponential power distribution can better characterize the residual distributions than those using the Gaussian distribution. Although the conclusions of this study are empirical, the analysis may provide insights for selecting appropriate residual models for Bayesian modeling of soil respiration models.