In Bayesian estimation, when can regression coefficients and scale parameter be jointly identifiable? When not? Exercise 14.2 in Koop, Poirier and Tobias's book (i.e. Bayesian econometric methods) talks about the case that in probit model, the regression and scale parameter are not jointly identified. 
I wonder, in general, is there any rules or heuristics that can help us to judge when regression coefficients and scale parameter can be jointly identifiable and when not? 
 A: Here's some thoughts on the matter:
Consider the latent variable motivation for the probit regression. I'm going to be notationally sloppy here for the sake of brevity, hopefully you can see how it fits into a more rigorous derivation in the textbook. Consider the $i^{th}$ response $y_i$ and the underlying the unobserved random variable $y^*_i$ with predictor variable vector X and parameter vector $\theta$. Typically we assume $y^*_i = \theta^T*X_i + \epsilon$ with $\epsilon ~ N(0, \sigma_e$. Note the use of vector multiplication to represent the linear combination. 
Then, the likelihood is derived from the following $Pr(y_i = 1) = Pr(y^*_i + \epsilon > 0 ) = \phi((\theta^T*X_i)/\sigma_e)$, where $\phi$ is the normal CDF. I'm skipping a few steps for brevity. We can see that in this case, if we increase $\sigma_e$ by a constant we can scale down the $\theta$ accordingly and get the same value to input into $\phi$. Thus, we need to fix $\sigma_e$ for our likelihood to make sense, hence it is not jointly identifiable. If instead of a linear combination of predictors and model parameters we had a different function (i.e. affine) this label switching may not occur.  
