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In this paper we review the concepts of Bayesian evidence and Bayes factors, also known as log odds ratios, and their application to model selection. The theory is presented along with a discussion of analytic, approximate and numerical techniques. Specific attention is paid to the Laplace approximation, variational Bayes, importance sampling, thermodynamic integration, and nested sampling and its recent variants. Analogies to statistical physics, from which many of these techniques originate, are discussed in order to provide readers with deeper insights that may lead to new techniques. The utility of Bayesian model testing in the domain sciences is demonstrated by presenting four specific practical examples considered within the context of signal processing in the areas of signal detection, sensor characterization, scientific model selection and molecular force characterization., Comment: Arxiv version consists of 58 pages and 9 figures. Features theory, numerical methods and four applications

Air quality information is increasingly becoming a public health concern, since some of the aerosol particles pose harmful effects to peoples health. One widely available metric of aerosol abundance is the aerosol optical depth (AOD). The AOD is the integrated light extinction coefficient over a vertical atmospheric column of unit cross section, which represents the extent to which the aerosols in that vertical profile prevent the transmission of light by absorption or scattering. The comparison between the AOD measured from the ground-based Aerosol Robotic Network (AERONET) system and the satellite MODIS instruments at 550 nm shows that there is a bias between the two data products. We performed a comprehensive analysis exploring possible factors which may be contributing to the inter-instrumental bias between MODIS and AERONET. The analysis used several measured variables, including the MODIS AOD, as input in order to train a neural network in regression mode to predict the AERONET AOD values. This not only allowed us to obtain an estimate, but also allowed us to infer the optimal sets of variables that played an important role in the prediction. In addition, we applied machine learning to infer the global abundance of ground level PM2.5 from the AOD data and other ancillary satellite and meteorology products. This research is part of our goal to provide air quality information, which can also be useful for global epidemiology studies.