The limma User's Guide says on p. 62 that the parameter (s_0)^2 is the mean of the inverse chi-squared prior for the true residual variances (sigma_g)^2. However, if I get

Smyth, G. K. (2004). Linear Models and Empirical Bayes Methods for Assessing Differential Expression in Microarray Experiments. Statistical Applications in Genetics and Molecular Biology, 3(1), 1–25. https://doi.org/10.2202/1544-6115.1027

correctly (more precisely: section 3 (p. 6 bottom)), the prior for the (sigma_g)^2 is rather a scaled inverse chi-squared distribution with scaling parameter (s_0)^2. In that case, the mean is d_0 * (s_0)^2 / (d_0 - 2) (see Wikipedia). So my question is: Am I right that the correct mean for the prior of the (sigma_g)^2 is d_0 * (s_0)^2 / (d_0 - 2) and not (s_0)^2 ?

Please see the errata to my 2004 paper in SAGMB, which is on the 3rd page of the pdf from the journal website. Or see the reprint incorporating corrections that is available from my website http://www.statsci.org/smyth/pubs/2003to2014.html

The exponential family canonical parameter here is 1/sigma_g^2, often called the precision parameter, rather than sigma^2_g itself. The expected value of the prior distribution for 1/sigma_g^2 is 1/(s^2_0). The Bayesian calculations are done for the precision rather than for the variance.

The prior distribution for sigma^2_g, on the other hand, does not necessarily have a finite mean or variance and is not very useful.