How to determine the riskparamter PD, EaD, LGD for the IRB-Ansatz | Basel II
Exact models are proprietary but they usually do that using statistical analysis. For example, in this paper the authors estimate the probability of default of various banks using the probit model. Or another example is in this paper.
Models with a limited dependent variable like probit or logit and others are quite common because these models let you estimate the probability of some event occurring. However, these models can be set up in various ways depending on data.