A Bayesian approach for the modelling of material stocks and flows with incomplete data

Material Flow Analysis (MFA) is used to quantify and understand the life cycles of materials from production to end of use, which enables environmental, social and economic impacts and interventions. MFA is challenging as available data is often limited and uncertain, giving rise to an underdetermined system with an infinite number of solutions when attempting to calculate the values of all stocks and flows in the system. Bayesian statistics is an effective way to address these challenges as it rigorously quantifies uncertainty in the data and propagates it in a system flow model to provide the probabilities associated with model solutions. Furthermore, the Bayesian approach provides a natural way to incorporate useful domain knowledge about the system through the elicitation of the prior distribution. This paper presents a novel Bayesian approach to MFA. We propose a mass based framework that directly models the flow and change in stock variables in the system, including systems with simultaneous presence of stocks and disaggregation of processes. The proposed approach is demonstrated on a global aluminium cycle, under a scenario where there is a shortage of data, coupled with weakly informative priors that only require basic information on flows and change in stocks. Bayesian model checking helps to identify inconsistencies in the data, and the posterior distribution is used to identify the variables in the system with the most uncertainty, which can aid data collection. We numerically investigate the properties of our method in simulations, and show that in limited data settings, the elicitation of an informative prior can greatly improve the performance of Bayesian methods, including for both estimation accuracy and uncertainty quantification.