"Mixing" poisson and normal distributions

I am running an experiment using the Inference Compilation algorithm, and I’m currently stuck in a problem. My model samples a variable that is basically a counter that represents the number of objects in a scene (that is further rendered given the needed sampled properties). I have n_objects = pyro.sample("N", dist.Poisson(torch.tensor(poisson_param))) in my model, and my issue is in building the inference network: I have a proposal feedforward nn that outputs the parameters of the distribution of "N", but choosing the distribution is the real challenge… I think the best option is some discrete distribution similar to poisson but with some tunable parameter to regulate the sampling variance, because I noted that using a simple poisson distribution to infer values for "N" gives a range that is too large for this application. Any idea of how to implement such distribution? Thank you for any help in advance and let me know if something is missing in this description!