  Data-driven forecasts of air quality have recently achieved more accurate
short-term predictions. Despite their success, most of the current data-driven
solutions lack proper quantifications of model uncertainty that communicate how
much to trust the forecasts. Recently, several practical tools to estimate
uncertainty have been developed in probabilistic deep learning. However, there
have not been empirical applications and extensive comparisons of these tools
in the domain of air quality forecasts. Therefore, this work applies
state-of-the-art techniques of uncertainty quantification in a real-world
setting of air quality forecasts. Through extensive experiments, we describe
training probabilistic models and evaluate their predictive uncertainties based
on empirical performance, reliability of confidence estimate, and practical
applicability. We also propose improving these models using "free" adversarial
training and exploiting temporal and spatial correlation inherent in air
quality data. Our experiments demonstrate that the proposed models perform
better than previous works in quantifying uncertainty in data-driven air
quality forecasts. Overall, Bayesian neural networks provide a more reliable
uncertainty estimate but can be challenging to implement and scale. Other
scalable methods, such as deep ensemble, Monte Carlo (MC) dropout, and
stochastic weight averaging-Gaussian (SWAG), can perform well if applied
correctly but with different tradeoffs and slight variations in performance
metrics. Finally, our results show the practical impact of uncertainty
estimation and demonstrate that, indeed, probabilistic models are more suitable
for making informed decisions. Code and dataset are available at
\url{https://github.com/Abdulmajid-Murad/deep_probabilistic_forecast}
