Probabilistic PCA is a dimensionality reduction technique that analyzes data via a lower dimensional latent space. The PCA probability model is a slightly modified factor analysis model that uses $\textbf {{W}}$ $\textbf{{W}}^{ ~T}$ + $\sigma^{2}\textbf{{I}}$ as the covariance of $\textbf{x}$ where $\sigma^{2}$ is now a scalar:

$\textbf{x} \sim {N} (\textbf{x}; \textbf{{b}},\textbf {{W}}\textbf{{W}}^{ ~T} + \sigma^{2}\textbf{{I}})$

which can be equivalently expressed as:

$\textbf{x} = \textbf{Wh} + \textbf{b} + \sigma \textbf{z}$

where $\textbf{z}$ $\sim$ $\textbf{N( z ; 0, {I})}$ is noise, $\textbf{x}$ is a data vector, $\textbf{h}$ is a latent varibale, and $\textbf{W}$ is a set of principal axes relates the latent variables to the data represented as a matrix.