Density Network over multiple parameters

I want to implement CNN with outs: mu, sigma, Q, rate1, rate2, rate3, rate4 and this probability model on top:

out_time – distributed like mixture of K normals or gammas (mu[K], sigma[K]) with probability Q, and zero with (1 - Q), looks like tweedie distribution, but a bit complex

out_1 – poisson with rate1 over out_time – Poisson(rate1 * out_time)
out_2 – poisson with rate2 over out_time – Poisson(rate2 * out_time)
out_3 – poisson with rate3 over out_time – Poisson(rate3 * out_time)
out_4 – poisson with rate4 over out_time – Poisson(rate4 * out_time)
out_5 = 2.3 * out_1 + 1.5 * out_2 + 0.9 * out_3 + 0.8 * out_4
Where out_* params are observed, and out_5 is most important.

I saw Mixture Density Networks example, but have some questions, because I am new to Edward and probabilistic programming:

  1. How I can optimise MAP over all out_* params, to predict their join pdf?
  2. How to build mixture model as described instead Mixture of same random variables?
  3. Can I define priors to mu, sigma, Q, rate1, … ?