Chris' Questions and Answers

Contents

• Unanswered Questions
• Should we put a prior on the loadings?
• Should we keep calling this "Bayesian PCA"?

Unanswered Questions

• Why do eigenvectors only scale, and not rotate?
• Do the parameters of distributions have intuitive meaning?
• Why use spot rates rather than forward rates?
  Possible answer: spot (zero-coupon) yields are the primitive curve actually published in the source data (the Bank of England's GLC nominal spot curve) — forward rates would have to be bootstrapped from it, an extra derived step. Forwards are effectively a derivative of the spot/discount curve, so they amplify noise in the underlying curve, which is undesirable for a covariance-based technique like PCA. Spot yields also keep results comparable to the standard yield-curve PCA literature (e.g. Litterman & Scheinkman), which is established on zero/spot yields, not forwards.

Should we put a prior on the loadings?

Question: Not putting a prior on the loadings feels like a significant decision. Part of me thinks we should — I'd like to keep this open rather than settle it now.

Answer: Open question, not yet decided. Bishop's Bayesian PCA paper puts the prior specifically on the loadings — an ARD prior per column of $\mathbf{W}$, which automatically shrinks unneeded components to zero and so decides the number of retained components as part of inference. What's been built so far in this project deliberately does the opposite: $\mathbf{W}$ is kept fixed at its classical value, and the prior sits on the score distribution's $\mu_i, \sigma_i$ instead — on the reasoning that a prior on individual loading entries isn't obviously interpretable as a real-world view on yield curve dynamics (a loading vector spans many tenors at once). Revisiting that and putting a prior on $\mathbf{W}$ would be a considerably bigger undertaking, closer to reimplementing Bishop's model than extending the current one — worth keeping on the table rather than closing off.

Update: this is now being explored as a second track, Bayesian PCA (Latent), kept alongside the existing one rather than replacing it — without an ARD prior (component count stays fixed at 3), and without yet resolving the rotational-identifiability question that a free $\mathbf{W}$ introduces.

Should we keep calling this "Bayesian PCA"?

Question: Given the loadings aren't Bayesian, can I still call what I'm doing "Bayesian PCA"?

Answer: There's no single owned definition of the term to violate. Bishop's paper is the most-cited use of "Bayesian PCA" and is specifically about ARD priors on the loadings for automatic component selection — not what this project does. Current call: keep the name (it's a reasonable shorthand for a talk), but be explicit — in the talk and in baeysian_pca.md — about the actual scope: the classical decomposition (covariance, loadings, eigenvalues) stays fixed, and the Bayesian treatment applies specifically to the score distribution used to derive the stress. That one sentence of scoping prevents anyone familiar with Bishop's paper from assuming this replicates his ARD-based approach.

Update: resolved by renaming rather than scoping in prose — this track is now called Bayesian PCA (Bolt-On), naming the "uncertainty bolted onto a fixed classical point estimate" scope directly in the title instead of relying on a caveat sentence every time it comes up.