# Marginal Cluster assignments for Dirichlet Process mixture model

I am watching Tamara' Broderick video on Dirichlet Process mixture models where she talks about computing $$p(z_n = k | z_1,z_2,..z_{n-1})$$ at ardoun 16:06. The z's are drawn from $$\rho_1 \sim beta(a_1,a_2)\\z_i \sim Categorical(\rho_1,1-\rho_1)$$

The posterior cluster assignment $$p(z_n=k|z_1,z_2,...z_n)$$ works out to be $$\frac{a_{1,n}}{a_{1,n} + a_{2,n}}$$ where $$a_{1,n}$$ represents the initial starting parameter $$a_1$$ + total number of z's that are equal to 1. $$a_{2,n}$$ represents the initial starting parameter $$a_2$$ + total number of z's that are equal to 2.

Tamara Broderick has proved that this is the case. Intuitively however, I feel that since $$z_i$$ are all drawn i.i.d, the cluster assignments of the past $$n-1$$ draws should not affect the probability of the next draw $$z_n$$. Hence the probability should still be $$\rho_1$$ for $$k=1$$ and $$1 - \rho_1$$ for $$k=2$$. Why should the past $$n-1$$ draws of $$z$$ have an effect on the next draw $$z_n$$ ?