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I'm having trouble implementing a sum of $N$ binomials (and a poisson distribution) with Stan.

Observed data is $(y_i)$ for $1 \leq i \leq M$, and $(x_{ij})$ for $1 \leq i \leq M$, $1 \leq j \leq N$.

Assuming the following generating process :

$$ y_i = \sigma + \sum_{j=1}^N y_{ij} \\ \textrm{with } y_{ij} \sim \textrm{Binomial}(x_{ij},p_j)\\ \textrm{and } \sigma \sim \textrm{Poisson}(\lambda) $$

I would like to fit my data against the paramters $(p_j)$ and $\lambda$, using informative priors on $p_j$ (though this is not relevant here). But I don't know how to implement this with Stan. Binomials are discrete and only produce integers which are not allowed as Stan parameters. Besides, I have no idea how to model a sum of N Binomials + a Poisson variable in Stan.

Assuming the binomials are well approximated by poisson distributions would make things much easier. We then obtain :

$$y_i \sim \textrm{Poisson}(\lambda + \sum_{j=1}^N p_{j} x_{ij})$$

I wrote the following stan-model based on this approximation :

data {
    int<lower=0> M;
    int<lower=0> N;
    int<lower=0> y[M];
    int<lower=0> x[M,N];
}
parameters {
    vector <lower=0,upper=1>[N] p;
    real <lower=0> sigma;
}

transformed parameters {
    real<lower=0> yf[M];

    for (i in 1:M) {
        yf[i] = sigma;
        for (k in 1:N) {
            yf[i] += p[k]*x[i,k];
        }
    }
}

model {
    p ~ uniform(0,1); // un-informative prior for simplification
    sigma ~ uniform(0,50);
    y ~ poisson(yf);
}

In my case, $x$'s orders of magnitude range between $0$ and $10^5$, and $p$'s between $10^{-6}$ and $10^{-2}$. Usually $x_{ij}p_j \sim \mathcal{O}(1)$ but sometimes it can exceed a few hundreds. Therefore, I doubt the Poisson approximation is really appropriate.

Is Stan suited for this problem ? If it is, how to implement this model without requiring the Poisson approximation ?

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1 Answer 1

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Stan is able to fit such a model. I guess that in your model $\sum_{j=1}^{N} y_{i j} $ is there to capture the mean and $\sigma$ to capture the deviation. I would then propose the following adaptation:

$y_{i} \sim \mathcal{N}(\sum_{j=1}^{N} y_{i j}, \sigma) $

$\text { with } y_{i j} \sim \operatorname{Binomial}\left(x_{i j}, p_{j}\right)$

$\text { and } \sigma \sim \operatorname{Poisson}(\lambda)$

In the model block, you can say that the mean is the sum of $y_{ij}$ and give a binomial prior to your $y_{ij}$.

Hope it helps.

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