In a basic BYM model may be written as
sometimes with covariates but that doesn't matter much here. Where s are the spatially structured effects and u the unstructured effects over units.
In Congdon (2020) they refer to the fair prior on these as one in which
where 
is the average number of neighbors in the adjacency matrix.
It is defined similarly (in terms of precision, I think) in Bernardinelli et al. (1995).
However, for the gamma distribution, scaling appears to only impact the scale term
I haven't been able to find a worked example for this, and don't understand how the priors are arrived at, for example, in the well-known lip cancer data
I am hoping someone could help me understand how these are reached in this setting, even in the simple case of two gamma hyperpriors.
References
Congdon, P. D. (2019). Bayesian Hierarchical Models: With Applications Using R, Second Edition (2nd edition). Chapman and Hall/CRC.
Bernardinelli, L., Clayton, D. and Montomoli, C. (1995). Bayesian estimates of disease maps: How important are priors? Statistics in Medicine 14 2411–2431.
