## 1 Introduction  
### Motivation  
Seven million people die each year due to the adverse health effects of air pollution, with 4.2 million deaths attributed to exposure to ambient (outdoor) air pollution [1]. Approximately 91% of the world's population lives in areas where air pollution exceeds guideline limits established by the World Health Organization (WHO) [2]. Several health-relevant ambient air pollutants include carbon monoxide (CO), ozone (O\({}_{3}\)), nitrogen oxides (NO and NO\({}_{2}\)), sulfur dioxide (SO\({}_{2}\)),and particulate matter (PM) [3]. Atmospheric PM has become the subject of extensive research due to its impact on human health, ecosystems, and the climate [4]. A critically important class of atmospheric PM is ultrafine particles (UFPs), which are particles smaller than 100 nm in size. UFPs have very high surface area to mass ratios and preferentially deposit in the tracheebronchial and alveolar regions of the human respiratory system. A large fraction of UFPs are derived from emissions associated with traffic, industrial activities, and domestic heating [5]. UFPs tend to dominate atmospheric PM number size distributions and contribute little to PM mass concentrations that are presently used as air quality indicators (e.g., PM\({}_{2.5}\) and PM\({}_{10}\)). Thus, particle number (PN) concentrations are more useful in describing the abundance of UFPs [6].  
In addition to UFPs, another health-relevant component of atmospheric PM is black carbon (BC). BC is emitted from the incomplete combustion of carbonaceous material and is associated with vehicle exhaust, coal-fired power plants, and biomass burning for heating and cooking [7]. BC is an important component of PM in cities, contributing 5%-15% to the total PM mass concentrations in urban air [8; 9]. BC has negative implications for human health [10], regional and global climate [11], and extreme weather events [12]. Due to its relatively short lifetime in the atmosphere [13], it has been suggested that mitigation of BC emissions may reduce global warming [10].  
Due to the aforementioned BC impacts, the United Nations Environmental Programme recommends monitoring this pollutant in cities side-by-side with other air pollution indicators (such as PM\({}_{2.5}\) and CO) for the purpose of mitigation and adaptation policy making and planning. In practice, BC measurement is not as trivial as other pollutant indicators [14; 15] and it is expensive (a proper BC measurement setup is on the order of approximately $50,000) to plan in some situations. Complications with data acquisition, such as instrument failure, data corruption, and calibration frequency, can lead to missing BC data during research campaigns or continuous measurements [16]. Standard interpolation methods are ineffective to fill such data gaps if they are relatively long in duration. BC levels have been shown to vary proportionally with those of traffic-related gaseous pollutants, such as CO, NO, and NO\({}_{2}\)[6]. Therefore, it may be possible to estimate BC mass concentrations using other air pollution metrics and indicators. Such an approach can help fill critical gaps in the measurement of BC that has long been hindered by instrumentation cost and measurement complexity. In order to address the aforementioned challenges, we propose the development of BC proxies through a data-driven model with a Bayesian framework. A BC proxy can be defined as a mathematical model that estimates BC mass concentrations using other available measured variables, such as levels of PM or gaseous pollutants. Deriving a proxy for BC is in many cases cost efficient and accurate. BC, PM\({}_{2.5}\), CO, and NO\({}_{x}\) originate from similar urban air pollution sources; thus, presently existing monitoring stations and their databases can be used to estimate BC concentrations not only in the present/future, but also historically. In this study, we introduce BC proxies that are based upon PM measurements by establishing relationships between BC and size-fractionated PM concentrations.