What is “oracle” in statistics?

Question

When I read several statistical papers, they mention "oracle" property or "oracle" estimator. What do they mean by "oracle"? I understand this oracle is not a company name, but have no idea what this means.

For example, Candes and Tao 2007 is about "oracle" inequality in a context of large dimensional estimation of linear model (p>>n). https://projecteuclid.org/euclid.aos/1201012958)

Zou 2006 discusses "oracle" property of Lasso. (http://www.tandfonline.com/doi/abs/10.1198/016214506000000735)

I am not familiar with this field but I have to read one paper which discusses high dimensional test in statistics.

Answer 1

"Oracle" refers to something that has access to the ground truth. It has the perfect information of which in practice, we rarely have some luxury. In such paper, you can typically see that it reflects a comparison with a model with perfect information.

From Candes and Terry Tao's paper:

• "Even though $n$ may be much smaller than $p$, our estimator achieves a loss within a logarithmic factor of the ideal mean squared error one would achieve with an oracle which would supply perfect information about which coordinates are nonzero, and which were above the noise level.
• To see why this is true, suppose one had available an oracle letting us know in advance the location of the S nonzero entries of the parameter vector

From Hai Zhou's paper:

• We show that the adaptive lasso enjoys the oracle properties; namely, it performs as well as if the true underlying model were given in advance

Notice that oracle procedure/properties is also defined in Hai Zhou's paper reflecting the ability to rediscover the ground truth.