Bayesian analog to Generalized Estimating Equations?

#1
GEE2 (second order generalized estimating equations, see Prentice and Zhao 1991 Biometrics) can be used to model the mean and covariance of a response without having to specify a likelihood. For example, suppose I have a continuous response and I want to model the mean with parameters beta and the covariance with parameters alpha. Under certain regularity conditions, GEE2 can provide consistent estimators for both beta and alpha without having to specify any likelihood. I think the main benefit of GEE2 is that it does not rely on any strong parametric assumptions. For example, if you have a continuous response of interest that is not really normally distirubted, GEE2 may be more robust than fitting a normal model with MLE.

I know that fundamentally bayesian statistics requires a likelihood. Nontheless, I'm wondering if there is any bayesian analog to GEE2 that can let me estimate mean and covariance parameters for a continuous response without having to assume the response is normally distributed (for example). I.e. I want to obtain more robust estimates of mean and covariance parameters (beta and alpha) that do not rely on strong parametric assumptions of the data.

The reason I'd prefer to do something like GEE2 within a Bayesian context is because I want to use MCMC to do the estimation (if possible), since I trust MCMC a lot more than the modified newton-raphson algorithm proposed for GEE2 (too sensitive to initial starting values IMO).