Comming from this and this example it seems wise to parameterise Gamma as

But I’m straggling to think of priors for the inverse_phi. In the linked examples the exponential(1) is the one suggested but looking into it such prior returns really high variance.

In my case I could make the assumption that the variance could be ~10% on the mean (the data spread from (0, 1.2))

inv_phi <- 40;
mu <- 0.5;
rgamma(1e5, inv_phi, rate = inv_phi/mu) %>% as_tibble %>% summarise_all(list(m=mean, s=sd));
curve(dgamma(x, inv_phi, rate = inv_phi/mu), xlim = c(0, 2))
# A tibble: 1 × 2
m s
<dbl> <dbl>
1 0.500 0.0790

Looking at plausible values for inverse_phi seems that higher values should be encouraged (>1) - so the opposite of what an exp(1) would do.

I would argue that your choice of prior for the shape parameter of a gamma distribution (which you call inverse phi) depends entirely on what you are modeling and what you expect the variance to be relative to the mean. To get a sense of this relative variation implied by your prior choice for shape, either:

check the 95 central confidence interval of a gamma distribution with shape and rate equal to the shape of your “inverse phi”. This distribution will have mean 1.0 by default. A value of the rate parameter != shape would simply scale the distribution. As such, the 95CI is a measure of the relative variability of the gamma distribution of interest. You can check this for a range of values of the shape parameter to get a sense of the range of relative variatiability that a choice of prior for the shape parameter would imply.

analytically calculate the coefficient of variation (CV) for your gamma, which is a simple function of the shape parameter: 1/sqrt(shape). And if you have a prior belief about the CV, you could even specify a prior for that and derive the shape parameter as 1/CV^2.