# Truncated count model -- including information about the number of unobserved realisations

## Background

Suppose we have a model such that $$Y \sim \mathcal{M}(\theta)$$ is a discrete random variable taking values in $$[0, 1, \ldots]$$. We would like to make inference about $$\theta$$ from a collection of observations $$\boldsymbol y = \{y_1, y_2, \ldots, y_J\},\: y_i >0$$, i.e., we only observe realisations of $$Y$$ if they are non-zero. There is some literature from the sixties on performing inference when $$\mathcal{M}$$ is a Poisson distribution, for instance. I have a question for which I haven't seen a Bayesian treatment, which is most likely due to my own ignorance and/or poor Googling skills.

Suppose I have some (imperfect) knowledge about the size of the "population", $$N = J + n_0$$ (see below). First question is (i) should I include this information into the model? and (ii) how should I do that? Below I discuss my partial answers to these. What I am asking for is: feedback as to whether these are correct and where in the literature I can find more information.

## An attempt at a solution

Let $$\pi(\theta)$$ be a joint prior on the parameters and let $$L(\boldsymbol y | \theta)$$ be the likelihood, such that the joint posterior is $$p(\theta | \boldsymbol y) \propto L(\boldsymbol y | \theta)\pi(\theta)$$. We can use a "compressed" likelihood of the form $$\prod_{i = 0}^U \text{Pr}(i | \theta)^{n_i}$$, where $$n_i$$ is number of occurrences of $$i$$ in the sample $$\boldsymbol y$$ and $$U$$ is the maximum such value. It seems to me that we can see $$n_0$$ as an extra parameter in the model and assign it a prior $$\pi_K(n_0)$$. I wonder if we can then use $$L^\prime(\boldsymbol y | \theta) := \prod_{i = 1}^U \text{Pr}(i | \theta)^{n_i}\sum_{j=J+1}^\infty \text{Pr}(0 |\theta)^j \pi_K(j)$$ as the new likelihood. Summation could also be from $$J + 1$$ to $$N$$ (instead of $$\infty$$). I choose to marginalise over $$n_0$$ to get around having to simulate the discrete parameter $$n_0$$, which is important for fitting the model in Stan, for instance.