Parameterization of inverse gamma prior in Bayesian methods For a prior of $\sigma^2 \sim IG(0.01, 0.01)$, often recommended as an uninformative prior for the variance parameter in MCMC approaches and other Bayesian methods, which parameterization does this recommendation correspond to?
Is it shape/scale or shape/rate? That is, in R, would it be invgamma::dinvgamma(sigma, shape = 0.01, scale = 0.01) or invgamma::dinvgamma(sigma, shape = 0.01, rate = 0.01)?

The inverse gamma is a conjugate prior for $\sigma^2$ in typical Gaussian-data models. The underlying idea is to have a convenient proper-prior approximation to an improper $\sigma^{-2}$ prior, for a weakly informative/low information prior. In the limit as $\epsilon=\beta=\alpha$ goes to $0$, the $\text{inverse gamma}(\epsilon, \epsilon)$ distribution goes to that improper prior. This occurs when $\beta$ is a scale parameter, not a rate parameter.
This use goes back at least to about 1993 give or take, that was when I first encountered it - I believe in conversation - and indeed subsequently used it in an illustrative example. Gelman [1] credits Spiegelhalter et al (1994), so that's possibly its first appearance, though I would not be in the least surprised to discover it's a good deal older than that.
A warning: according to Gelman ([1], again) the $\text{inverse gamma}(\epsilon, \epsilon)$ for small $\epsilon$ (where 0.01 or 0.001 are typical choices of small) distribution is not as weakly informative a prior as was hoped for; he says that the results depend on which $\epsilon$ is chosen.
[1]: Gelman, A., (2006),
"Prior distributions for variance parameters in hierarchical models,"
Bayesian Analysis 1, No. 3, pp. 515–533
http://www.stat.columbia.edu/%7Egelman/research/published/taumain.pdf
