What's the best way of predicting variable $z$ given variables ${a_1, a_2, ..., a_n}$?

I'm using a Bayesian Model to predict variable $z$. I have parameters $a_i$ that I have inferred a prior distribution:

$$a_1 = a_2 = a_3 \sim Normal(0,100)$$ $$a_4 = a_5 \sim Uniform(100,200)$$ $$a_6 = a_7 \sim Beta(1,1)$$ $$a_6 = a_8 \sim Gamma(1,1)$$

But I'm having 2 problems:

1. I can't figure out a way of finding the likelihood function. The output $z$ consists of data the are very concentrated around zero (some negative values too) and have long tail for positive values. Should I just say it looks like an exponential distribution and assume that distribution as my likelihood function?
2. How should I relate all those 8 parameters to predict $z$? Is it like try-and-error? Like, I guess that it could be linear relationship so I say $z = a_1 + b_1*a_2 + ... + b_8*a_8$ and predict $z$, and if that is not good I try another model?