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.