How can I write the density function of the dirichlet process?

The dirichlet process has two representations commonly used: the chinese restaurant process and the stick-breaking representation. I understand how they work, and how to do sampling on both representations, which is essentially how to write out their marginal representations, or the probability of the next sample given the others. However, say we have a standard DP mixture model:

G   ~  DP(G0, alpha)
z_i ~  G
x_i ~  Normal(z_i, sigma)

If we want to derive a variational algorithm for this model we'll have to define an approximating distribution (usually either a truncated stick-breaking distribution or a finite symmetrical dirichlet distribution) and compute E_q[log P(G)], where G is the DP prior, and I'm not sure how to do this without actually writing down the density of the DP, except I can't find it in most references. Another problem is computing E_q[log P(z|G)], but the papers I know of doing variational inference for DP mixture models ( Blei and Jordan. Variational methods for the dirichlet process and Kurihara, Welling, and Teh. Collapsed variational dirichlet process mixture models) seem to think it's ok to compute E_q[log P(z_i|z-)] (where z- is the set of all zs other than z_i) and use the marginal form of the DP for that. Looking at the resulting update equations it seems as if only a small change is involved between DP mixtures and standard finite mixtures, but I can't wrap my head around:

1. How can I actually compute the value of the lower bound?

2. How to write down P(G) and P(z|G) so that I can write down the lower bound, derive it, and get to the update equations? Why is it ok to use the marginal equation P(z_i|z-) in this case but not necessarily so in the finite case?

3. Am I just supposed to not bound P(G)?