Making a note of something that confused me at first while reading this: if \(\mathbf{X}=(X_1\dots X_n)\sim N(0,\Sigma)\), then the principal component \(v\cdot\mathbf{X}\) (where \(v\) is the principal eigenvector of \(\Sigma\)) is NOT the latent variable for \(\mathbf{X}\) – indeed, \(P(\mathbf{X}|v\cdot\mathbf{X})\) does not factor. E.g. in the two-dimensional case, knowing \(X_1+X_2\) doesn’t induce independence between \(X_1, X_2\) at all – in fact, it makes them completely determined from one another.

Instead, the latent is some \(\Lambda\sim N(0,1)\) with conditionals \(X_1=\Lambda+\varepsilon_1\),