Multiple prioir/posterior distributions, with Data that depends on 2 of the variables


I have 10 prior distributions one for each of theta1, theta2..... . The prior distribution for theta1 is Gamma(a1,b1), and the others are described similarly.

Now I have an observation x, where X~Poisson(theta1 +theta2). I would like to get separate posterior distribution for theta1 and theta2. I don't think there is a way of doing this because when I try and get the posterior for theta1, I use proportionality to get rid of the exp(-theta2) term, but I can't get rid of theta 2 from the (theta1 +theta2)^x term.

Could anyone shed some light on whether this is possible or not?

Thanks in advance