Making a note of something that confused me at first while reading this: if X = (X₁…X_n) ∼ N(0,Σ), then the principal component v ⋅ X (where v is the principal eigenvector of Σ) is NOT the latent variable for X – indeed, P(X|v⋅X) does not factor. E.g. in the two-dimensional case, knowing X₁ + X₂ doesn't induce independence between X₁, X₂ at all – in fact, it makes them completely determined from one another.

Instead, the latent is some Λ ∼ N(0,1) with conditionals X₁ = Λ + ε₁,