I am trying to understand how to interpret Bayesian prior and posterior distributions in situations where there is believed to be variability in model parameters (due to variation in the population under study) as well as uncertainty. My understanding is that, typically, the prior distribution for a model parameter represents uncertainty in our knowledge of that parameter. My question is, would it be statistically valid to proceed with the computation of the posterior using priors that incorporate both uncertainty and variability? And would the interpretation of the posterior need to change if this was done?
- I am aware that, if a parameter is believed to vary, then it would be possible to specify a hierarchical model, with variability in the parameter quantified by a hyperparameter (and uncertainty in this hyperparameter represented by an associated hyperprior). So I guess this is one approach.
- In the context of quantitative risk analysis (QRA e.g. in books by Vose) parameter distributions can be chosen to represent uncertainty and variability. These distributions seem very similar to prior distributions. The difference being that, usually, no posterior distribution is computed - instead, something akin to a prior predictive distribution would typically be computed.