How to model the sum of two random variable?

Modeling
1

I’m trying to find latent variable from observed data produced by the sum of 2 random variables.

This is the python code producing the simulated data :

data = []
for p in range(1000): #paniers
    s = np.random.exponential(scale=10,size=2).sum() 
    data.append(s)

This builds 1000 data points, each of them is the sum of two random draws from the same exponential random variable.

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Below is my incomplete model. I’m trying to find the latent variable beta, which is the parameter of the exponential random variable used to create the data.

data {
    int<lower=0> N;
    real observed[N];
}
parameters {
    real<lower=0,upper=100> beta;
}
model {
    real e;
    real s;
    s = 0;
    for (i in 1:2)
       e ~ exponential(beta);
       s += e;
    observed ~  ????
}

How do I replace the ??? to ‘connect’ my observed data to my model ?

Any help appreciated.

2

It’d be helpful if you could write out the maths of your model. If you’re trying to model Y = X_0 + X_1 where X_i \sim \text{exponential}(\lambda) then the probability density function of Y can be written exactly.

1 Like
3

Yes it’s that model. but it is important to notice that X0 & X1 are not observed.

Your statement about the exponential distribution is interesting but I’d like a more general answer since I used exponential just as an example. I’d like to have a genera answer in term of the distribution from wich X0 & X1 are drawn.

4

I suspected this was the case. In general I think what you need to do is work out the maths of what the pdf of the actual observed random variable is conditional on the generating parameter. Stan takes a log-density up to normalisation, so that’s what you need to provide.

1 Like