  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, leading to an underdetermined system with
an infinite number of possible stocks and flows values. Bayesian statistics is
an effective way to address these challenges by principally incorporating
domain knowledge, and quantifying uncertainty in the data and providing
probabilities associated with model solutions.
  This paper presents a novel MFA methodology under the Bayesian framework. By
relaxing the mass balance constraints, we improve the computational scalability
and reliability of the posterior samples compared to existing Bayesian MFA
methods. We propose a mass based, child and parent process framework to model
systems with disaggregated processes and flows. We show posterior predictive
checks can be used to identify inconsistencies in the data and aid noise and
hyperparameter selection. The proposed approach is demonstrated on case
studies, including a global aluminium cycle with significant disaggregation,
under weakly informative priors and significant data gaps to investigate the
feasibility of Bayesian MFA. We illustrate just a weakly informative prior can
greatly improve the performance of Bayesian methods, for both estimation
accuracy and uncertainty quantification.
