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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 $z_n$ ?

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