A common type of statistical challenge, widespread across many areas of research, involves the selection of a preferred model to describe the main features and trends in a particular data set. The objective of model selection is to balance the quality of fit to data against the complexity and predictive ability of the model achieving that fit. Several model selection techniques, including two information criteria, which aim to determine which set of model parameters the data best support, are reviewed here. The techniques rely on computing the probabilities of the different models, given the data, rather than considering the allowed values of the fitted parameters. Such information criteria have only been applied to the field of radiation epidemiology recently, even though they have longer traditions of application in other areas of research. The purpose of this review is to make two information criteria more accessible by fully detailing how to calculate them in a practical way and how to interpret the resulting values. This aim is supported with the aid of some examples involving the computation of risk models for radiation-induced solid cancer mortality fitted to the epidemiological data from the Japanese A-bomb survivors. These examples illustrate that the Bayesian information criterion is particularly useful in concluding that the weight of evidence is in favour of excess relative risk models that depend on age-at-exposure and excess relative risk models that depend on age-attained.