One of the major paradigms for statistical inference, Bayesian inference allows one to assign prior distributions to model parameters of a statistical model. Common examples in parametric models include the mean and variance parameters of the normal distribution, regression coefficients in regression models, and parameters controlling hazard functions in survival analyses. There are many good books (e.g. Gelman et al. 2013) and also resources in the internet on Bayesian inference. See, for example, links and references in https://en.wikipedia.org/wiki/Bayesian_inference.

The prior distributions not only allow researchers to incorporate expert knowledge in analyses but also help to avoid identification problems and numerical instabilities in the estimation. In addition to parametric models, prior distributions have been introduced for nonparametric models (Walker et al. 1999; Green 1995; Arjas and Gasbarra 1994). Also hierarchical models can be constructed in cases where the data are nested e.g. children within families, families within municipalities etc. In survival analysis various frailty models have been introduced to account for the dependencies between observations within groups.