Incorrect equations in SGPR_notes

[joacorapela]

Bug / performance issue / build issue

I believe there is an error in the next-to-last equation in the documentation page at https://gpflow.readthedocs.io/en/master/notebooks/theory/SGPR_notes.html, which makes the last prediction equation invalid. I provide details at http://www.gatsby.ucl.ac.uk/~rapela/gpflow/wrongPredictionEq.pdf. Can this be right?

[joacorapela]

The next-to-the-last equation in the documentation page is correct, as shown here.

[st--]

Hi @joacorapela , it's great that you're carefully going through the docs, and thanks for raising what you saw as a mistake!
I've just gone through both your writeup and the docs, I think you had a minor mistake in your note 5. - you write you used Kuu⁻¹ Λ Kuu⁻¹ = L⁻ᵀ B⁻¹ L⁻¹, but in the documentation page it actually reads Kuu⁻¹ Λ⁻¹ Kuu⁻¹ = L⁻ᵀ B⁻¹ L⁻¹, note the Λ vs Λ⁻¹. With this, both sides of the second-to-last equation should match up.

[joacorapela]

Thanks @st-- . The third to the last equation in the GPflow documentation page correctly states Kuu⁻¹ Λ Kuu⁻¹ = L⁻ᵀ and as proved here the GPflow documentation is correct. What do you think?

[st--]

There's clearly a bug somewhere. I'm not quite sure where it is. One way that I find helpful is to think of covariance matrices as having "units". So e.g. Kfu Kuu⁻¹ Kuf has units of (cov) (cov)⁻¹ (cov) which cancels out to (cov) which is appropriate for Qff.

We can see that the equations for m and Λ are inconsistent with the definition q(u) = N(m, Λ) which would imply Λ is a covariance matrix, but based on its definition below, Λ clearly has "units" (cov)⁻¹, which matches up with the mean, m = Λ⁻¹ Kuu⁻¹ Kuf y σ⁻², the units are (mean) = [Λ⁻¹] (cov)⁻¹ (cov) (mean) (cov)⁻¹, so the unit of Λ⁻¹ has to be (cov) for it to be consistent.

[st--]

In other words, Kuu⁻¹ Λ Kuu⁻¹ isn't consistent, it should be Kuu⁻¹ Λ⁻¹ Kuu⁻¹ based on the "units".

[joacorapela]

Thanks for your comment @st--. I believe there is a typo in the third to the last equation of this GPflow note. The right hand side of this equation should contain instead of , as proven here. With this correction, the proof of the predictive mean (second to the last equation of this GPflow note) becomes simple, as shown here. I created a pull request about this. Do you agree @st-- ?

[st--]

I think we'd resolved this now ? If anyone disagrees please shout and we can reopen it:)