Documentation of qEI

[nabenabe0928]

What happened?

The statement in this doc:

is inconsistent with Eq. (3) in the following paper:

• https://hal.science/hal-00732512v2/document

Please provide a minimal, reproducible example of the unexpected behavior.

It is just about documentation.
Please verify if this statement is correct or the paper's proposition is incorrect.

Please paste any relevant traceback/logs produced by the example provided.

BoTorch Version

stable

Python Version

No response

Operating System

No response

Code of Conduct

• I agree to follow BoTorch's Code of Conduct

[Balandat]

Eq. (3) in the paper is actually an MC estimate, not an analytic form.

To maybe add some small nuance, it seems that Eq. (3) is not technically an MC estimate, but that it involves multivariate normal CDFs which need to be evaluated numerically.

[ltiao]

Hi @nabenabe0928

Thanks for bringing this up! (It seems something weird happened with my previous comment that @Balandat quoted and I can't seem to recover it)

To summarize, the doc suggests that an analytic version for the q > 1 case doesn't exist, while the paper you referenced does provide an analytic form in Eq (3). As @Balandat pointed out, you're correct that this is indeed analytic, but it involves multivariate normal CDF terms, which generally lack a closed-form solution.

I hope this clears things up! If you have any more questions, feel free to ask.

[nabenabe0928]

Thank you very much for the response 🙏🙏
Oh yes, you are right that multivariate normal CDF requires numerical Integration, explaining the statement of no existing closed form.
So actually, the MC version doesn't take the mean of EIs in the way that EH I does?
Does the implementation rather use the CDF?
Like are there any Special techniques specifically for CDF?
I am not very familiar to the numerical method for CDF, but I have implemented the EHVI by myself, so I am familiar to the integration of EHVI.

[esantorella]

This paper explains how EI and qEI are computed.