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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.