Is it possible to have rate predictions which are greater than 1 with Poisson regression in r?

I was trying to fit the Poisson rate regression and predict the rate by using StanMoMo in R. However, when I tried to extract mufor(The predictions were saved in this variable) from the result. Few values were greater than zero. It was very strange. Since I tried to use the sample dataset that the author provided, there was no such problem. So I am wondering why is this the case. Please give me some suggestions, Thank you!

I may not be available to provide the original dataset. I am still trying my best to generate a reproducible example.

The code in stanMoMo package:
StanMoMo/APCmodel.stan at master ยท kabarigou/StanMoMo (github.com)

 gf=append_row(g,g_p);
  if (family==0){
    for (t in 1:Tfor) for (x in 1:J) {
    mufor[pos] = a[x] + k_p[t]+gf[T+t-x+J];
    pos += 1;
  }
  mufor = exp(mufor);