I have a problem whith understanding the litterature, though.
As far as I understand the Muraki/Engelhard paper and other IRT textbooks (Embertson & Reise, 2000; Ayala, 2009), they assume categorical indicators with a preestablished number of possible response categories (like dichotome data or Likert-type data). My problem is how to apply the EAP to obtain factor scores with count data (Poisson-type).
Mplus computes factor scores for count factor indicators. The general formula of (222) of Appendix 11 is used. With counts the f(y) part of the formula is a Poisson or Negative binomial distribution. The expected value of this expression is used as the estimated factor score. This is obtained via numerical integration. In other words, there is not an explicit formula that can be used. The formulas for computing the posterior on page 151, bottom two formulas, in http://statmodel.com/download/ChapmanHall06V24.pdf may also be helpful.
Poisson and negbin don't estimate residual variances, although negbin gives a scale factor which is used in the above computation.