## 5 Conclusions  
This paper presents the development of BC proxies based on a Bayesian framework using WB and BB models. A considerable advantage of the Bayesian methods presented is the prevention of over-fitting and an explicit understanding of the uncertainty surrounding the BC mean prediction. The results demonstrate that the performance of the WB and BB proxies does not differ significantly. Both methods are evaluated on data obtained from a mobile air pollution measurement campaign in Jordan and give adequate results. Reasonable coefficients of determination (R\({}^{2}\)) are achieved: 0.77 and 0.78 for the WB and BB proxies, respectively. Both methods also estimate the BC mass concentration to within one standard deviation of the predictive distributions at 81% and 96% of data points for the WB and BB models, respectively. All estimations lie within 3\(\sigma\) standard deviation of the predictive distribution. Since both types of proxies provide similar performance, the WB proxy is relatively more convenient to be deployed in practice than the BB proxy as the WB model structure can be relatively easily built upon using known relationships in the data or using expert information. The data used in the BB proxy is also required to be normalised. This is not the case for the WB model, where the normalisation scaling factors may be difficult to determine when they are used in testing data. Nevertheless, the proposed method may not always be applicable to be deployed in practice, because PN measurement is not always performed at every air pollution measurement station. As demonstrated in the discussion, the developed proxy based on a single input PM\({}_{2.5}\) has shown a reduction in the proxy performance. As the first future effort which can be extended from this work is to carry out more experiments in the same and/or other regions with additional measured variables involved. The inclusion of measurements of trace gases, radiation, and meteorological variables is expected to improve the robustness of the BC proxies. Therefore, other variables associated with BC emissions can be investigated. For example, the proposed proxies can be improved by accommodating more input variables which are typically measured in most official air pollution stations and/or via low-cost sensors, such as NO\({}_{\rm x}\), which is known to correlate well with BC [78]. Furthermore, in order to improve the proxy performance, more advanced models will be developed that include temporal information for BC mass concentration and/or spatial information. For example, AutoRegressive (AR) type models representing WB, such as AutoRegressive with eXogenous inputs (ARX) models, can be developed further within Bayesian frameworks. This type of model is also equivalent to dynamic neural-networks, such as recurrent neural networks representing BB models. Finally, in order to scale-up the usage of the proposed proxy, PM\({}_{2.5}\) and other variable measurements using low-cost sensors will be carried out together with the existing instruments for validation. Then, the developed proxy will also be deployed by taking input from low-cost sensor measurements. In this way, low-cost estimation of BC mass concentrations can be realised. Conceptualisation, M.A.Z., T.H. and D.W.; methodology, M.A.Z., T.H. and D.W.; validation, M.A.Z. ; formal analysis, M.A.Z. ; investigation, M.A.Z. and T.H. ; resources, M.A.Z., T.H. and B.E.B. ; data curation, T.H. ; writing--original draft preparation, M.A.Z. ; writing--review and editing, M.A.Z., T.H., D.W., B.E.B. ; visualisation, M.A.Z. ; supervision, T.H. ; project administration, M.A.Z. and T.H. ; funding acquisition, M.A.Z. and T.H. This research was funded by the Deanship of Academic Research (DAR, project number 1516) at the University of Jordan.