How do interpret a vague prior for hierarchical modeling?

Question

I am new to Bayesian analysis and using the following WINBUGS example to understand Bayesian hierarchical modeling:

I have 2 questions:

1) For the fixed effects terms, i.e., the beta0 and beta1 terms, I would like to know why is the values of (0.0, 1.0E-5) is a vague prior, as opposed to (0.0, 10,000) for example.

Does the setting of these vague prior hyperparameters depend on whether or not the covariate data is standardized/normalized (For example, 1.0E-5 is used as the vague prior when the data is normalized between 0~1 and 10,000 would be the vague prior for non-normalized data) ?

2) For the tau.h and tau.c in the model, I am seeing this used a 'fair' prior. What difference/effect would it make if both of these were set to: (0.5,0.0005). I have seen it used here for instance. Should I use the prior with lowest DIC? And is (0.5,0.0005) a 'fair' prior?

any insights are appreciated.

Answer 1

A few quick answers:

• JAGS parameterizes the Normal distribution in terms of mean and precision (precision=1/variance), so a precision of 1e-5 means a variance of 1e5 or a standard deviation of 316. That this is "vague" or "weak" does depend on the scale of the covariate data. "Weak" essentially means that the standard deviation $\gg$ the scale of the data.
• I haven't read Best et al 1999 (as cited in your code), but Gamma(eps,eps) where eps << 1 is a typical weak prior for precisions: it gives a positive distribution with a large coefficient of variance (i.e., "vague") and a mean of 1 (JAGS parameterizes Gamma with shape and rate, so the mean is shape/rate = eps/eps = 1. This is again slightly sensitive to the scaling of the relevant covariate.
• You should be aware that the Gamma(eps,eps) (which is used in part because it's a conjugate prior for the precision of a Normal distribution, thus mathematically/computationally convenient) prior has been shown to have some bad properties in cases where the data is not very strong (and thus the prior has an effect); it often gives unrealistically large peak densities near zero, see e.g. Gelman 2006.

Gelman, Andrew. “Prior Distributions for Variance Parameters in Hierarchical Models.” Bayesian Analysis 1, no. 3 (xx xx 2006): 515–33.