My question is complementary to a previous problem : Bayesian inference on binarized Poisson distribution. I retake the previous notations. Problem description :
I am counting the number of balls falling in T boxes.
X=[0, 2, 1, 0, 0, 2, 1, 0, 2, 1, 1, 0, 0, 2, 0, 0, 1, 1, 0, 0]
$\forall t \in [1, \dots, T], X_t$ is drawned as a superposition of N independant Poisson processes with intensities $\lambda_i$. Thus, $X_t \sim Poisson(\sum_{i=1}^{N} \lambda_i).$ So : $P(X_t = k) = \frac{(\sum_{i=1}^{N}\lambda_i)^{k}e^{-\sum_{i=1}^{N}\lambda_i}}{k!}$ since a sum of poissons is poisson.
Then,
I binarized the sequence to obtain $X_1$ like in the link example by :
$\forall t \in [1, \dots, T], X_1(t) = 1 \quad if \quad X(t) \geq 1$, 0 otherwise.
Then, X_1=[0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0]
If I am right $X_1(t) \sim Bernoulli(p = P(X_t \geq 1)) = Bern(p = 1 - P(X_t = 0)) = Bern(p=1-e^{-\sum_{i=1}^{N}\lambda_i})$
Finally, I would have a likelihood function with a Bernoulli form but a kind of Poisson parameter:
$P(X_1 | \lambda) = \displaystyle \prod_{t=1}^{T} (1-e^{-\sum_{i=1}^{N}\lambda_i})^{X_1(t)} (e^{-\sum_{i=1}^{N}\lambda_i})^{1-x_1(t)}$
My question : Is it possible to learn the $\lambda_i$ intensities of superposed Poisson Processes through Bayesian Inference and more specifically Gibbs sampling? Which kind of prior on $\lambda_i$ would you use to account for the exponential parameter of the bernoulli distribution and still be able to recognize a samplable posterior ? (Gamma prior seemed appealing to me even if beta distribution is conjugate with bernoulli...) Would you have any reference for estimating intensities of superposition of poisson processes ?
Thx for you help & time my dear statistician friends.