# How to find complete log likelihood for mixture of PPCA

In Appendix C of a paper by Michael E. Tipping and Christopher M. Bishop about mixture models for probabilistic PCA, the probability of a single data vector $$\mathbf{t}$$ is expressed as a mixture of PCA models (equation 69):

$$p(\mathbf{t}) = \sum_{i=1}^M\pi_i p(\mathbf{t}|i)$$

where $$\pi$$ is the mixing proportion and $$p(\mathbf{t}|i)$$ is a single probabilistic PCA model.

The model underlying the probabilistic PCA method is (equation 2)

$$\mathbf{t} = \mathbf{Wx} + \boldsymbol\mu + \boldsymbol\epsilon.$$ Where $$\mathbf{x}$$ is a latent variable. By introducing a new set of variables $$z_{ni}$$ "labelling which model is responsible for generating each data point $$\mathbf{t}_n$$", Bishop formulates the complete log likelihood as (equation 70):

$$\mathcal{L}_C = \sum_{n=1}^N\sum_{i=1}^Mz_{ni}ln\{\pi_ip(\mathbf{t}_n, \mathbf{x}_{ni})\}.$$ I would like to understand how he derives this expression as he doesn't provide a solution himself. How is this expression for the complete log likelihood found?