GP birthday example

[Aki Vehtari]

I'm moving the discussion of GP birthday example here as it is off-topic for the issue "Add squared exponential covariance function..."
https://github.com/stan-dev/math/issues/191#issuecomment-151571062

On 27.10.2015 19:05, Michael Betancourt wrote:
> Here’s my proposal as for the birthday problem as discussed at the meeting.
> Note that I left out the 5th, 6th, and 7th kernels as they do not seem to induce
> a positive-definite covariance so I must not be understanding correctly.

Your are correct that these do not induce a positive-definite covariance, but they don't need to, as the sum of the kernels in the example induces a positive-definite covariance. 5th, 6th, and 7th kernels induce a "zero-definite" covariance, but since the 8th kernel adds non-zero value to the diagonal the overall result is positive definite.

Aki

[Bob Carpenter]

It's related to the discussion of how we add these
functions.

I'm OK with operations that combine matrices that
aren't positive definite and leaving it to the user.
We can recommend ways to stay within the algebra
of constructing and combining positive-definite matrices.

- Bob

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[Seth Flaxman]

I was just playing with the birthday data in Stan. Can someone bring me up to speed with this discussion?

Seth

[Daniel Lee]

On Thu, Oct 29, 2015 at 4:20 PM, Seth Flaxman <fla...@gmail.com> wrote:

I was just playing with the birthday data in Stan. Can someone bring me up to speed with this discussion?

Not sure what you're looking for. Might want to look at this hijacked issue:
https://github.com/stan-dev/math/issues/191

[Aki Vehtari]

> On 27.10.2015 19:05, Michael Betancourt wrote:
> > Here’s my proposal as for the birthday problem as discussed at the meeting.
> > Note that I left out the 5th, 6th, and 7th kernels as they do not seem to induce
> > a positive-definite covariance so I must not be understanding correctly.

We had email discussion with Mike, and I realized that the 5th and 6th kernel descriptions have typo in BDA3. Instead of I(t) they should have I(t,t'), where the value is 1 if the condition (weekday or weekend) is true for bth t and t', and 0 otherwise. 7th component correct as it is not described as a kernel, but as a linear model where I(t) is ok as defined after the equation for the 7th component.

Aki

[Michael Betancourt]

Yeah, with I(t) replace by I(t, t’) or I(t) I(t’) everything makes
sense, with the symmetric indicators defining block structure
in the covariance function.

[Seth Flaxman]

Oh, I was asking more for the contextand what was discussed at the meeting, i.e. is someone (Daniel?) taking on the much needed task of adding more built-in GP functionality to Stan? And is the birthday example actually close to running?

Seth

[Daniel Lee]

We discussed GPs for a while at the meeting, but the result was two proposals on the table:

- the user specifies a covariance function for two vectors and throws this into a functor that will generate a covariance matrix

- the user calls specific covariance functions and composes the covariance function they want

Michael wrote up an example of the first.

Bob wrote up an example of the second.

Rayleigh and I are working on a single function for the squared exponential covariance function and nothing else at this point. I think this function could be used for both proposals if factored correctly.

I think that's the only external info I have.