How to specify a flat improper prior for intercepts

[blokeman]

All my experimentation concerns qualitative outcomes (binary to quaternary) with a logit link function.

I don’t intend to actually use flat priors in the final analysis, but at this point I want to gain a better understanding of how different types of weak prior influence the logit coefficients especially of parameters on which the data itself has fairly little to say (the likelihood is weak).

As for priors yielding U-shaped distributions in the probability space, I’ve had the same argument on this forum before. Then, as now, I appeal to an authority:

@Alan Agresti (2013: 258)

Using large \sigma in normal priors for \beta implies priors on the probability scale that are highly U-shaped, with about half the probability very close to 0 and half very close to 1. This seems intuitively to be rather informative, but in fact such priors have little influence, with the posterior looking much like the likelihood function.

Agresti, Alan (2013). Categorical Data Analysis. 3rd ed. Hoboken, New Jersey: John Wiley & Sons.