Discrete parameter in Stan

[Bob_Carpenter]

@ghjk

for (i in 0:big_integer) {
           log_weights[i] = poisson_log_lpmf(i | log_lambda);
           log_dens[i] = normal_lpdf(x[i] | i, sigma_x[i]);
      }

Your approach to marginalize is OK assuming that values outside of (0, 200) have negligible probability mass. I think in this case, you can replace that Poisson with a continuous density and not bother with the marginalization.