This may be a very daft question and show my ignorance of bayesian statistics, however, what is the mathematical relationship between the aggregate estimate of a parameter (in this case a linear term) and the average of the individual HB estimates from a group of respondents. I ask because clearly the scale of the HB estimate is larger in magnitude than the logit estimate, yet they are almost perfectly correlated. Is there a way to workout the inflation factor?

The reason I ask is that I need to generate estimates in the magnitude of a a nested logit model for further use in a strategic transport model, but I want to take advantage of the clear benefits of using HB estimation. However, the average of individual HB parameters is to large for use in the model, so I need an equivalence with Logit parameters. Is there a clear mathematical between the terms such that I can reduce the HB estimate to the same scale as the Logit estimate?

Many thanks in advance.