# Why mixture model with Gibbs sampling works?

I just have a question about why Gibbs sampling can correctly estimate parameters with random initial value? That is to say,we can sample z by: \begin{align} p(z_i=k \,|\, \cdot) &\propto \prod_i^n \prod_k^K \left( \pi_k \frac{1}{\sigma\sqrt{2\pi}} \text{exp}\left\{ \frac{-(x_i-\theta_k)^2}{2\sigma^2} \right\} \right)^{\delta(z_i, k)} \\ &\propto \pi_k \cdot \frac{1}{\sigma\sqrt{2\pi}} \text{exp}\left\{ \frac{-(x_i-\theta_k)^2}{2\sigma^2} \right\} \end{align} Then update parameters After several iteration of these two-step, it gets true distribution with random initialization. Why it converges to the true distribution? I don't understand. Which information contribute to the change of parameters to the true value?