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The usual two-component Normal mixture model is instructed in these slides http://www2.stat.duke.edu/~rcs46/modern_bayes17/lecturesModernBayes17/lecture-8/08-data-augment.pdf.

We define Latent allocation variables $Z_i$. But how to implement the R code for full conditions of $\mu_0$ and $\mu_1$?

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Numerous discussions on the Gibbs sampler for mixtures can be found on this forum:

R packages are available as well

Our book Bayesian Essentials with R (new edition of Bayesian Core) with Jean-Michel Marin contains a whole chapter on mixture models, with R codes available on my webpage. The first edition of our Monte Carlo Statistical Methods with George Casella also enjoys a full chapter on this topic.

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