Dealing with negative values and the exponential distribution

[cluhmann]

I would suggest looking at either Poisson regression or negative binomial regression for inspiration/ideas. These models are traditionally used to model counts and other non-negative outcome variables (e.g., length of inter-event intervals). In Poisson, you construct a linear expression (e.g., \alpha + \beta X) and then the (canonical) link function is an exponential (so E[Y]=e^{\alpha + \beta X} or, ln(E[Y])=\alpha + \beta X, which was similar to the suggestion by @larryshamalama ) which forces the expectation to be non-negative. However, as you note, the transformation warps things (e.g., a prior over \beta that is normally distributed in the linear expression will not “look” normal in the transformed space). But hopefully that will give you some resources to check out!