Estimating the thing that is given to you by oracles in statistics homework assignments: the covariance matrix or its inverse, the precision matrices. Or, if you data is indexed in some fashion, the covariance kernel. We are especially interested in this in Gaussian processes, where the covariance kernel characterises the process up to its mean.

I am not introducing a complete theory of covariance estimation here, merely mentioning a couple of tidbits for future reference.

Two big data problems problems can arise here: large \(p\) (ambient dimension) and large \(n\) (sample size). Large \(p\) is a problem because the covariance matrix is a \(p \times p\) matrixand frequently we need to invert it to calculate some target estimand.

Often life can be made not too bad for large \(n\) with Gaussian structure because, essentially, it has a nice exponential family structure and hence has sufficient statistics.

## The obvious way

Estimate the covariance matrix then invert it. This is the baseline. π

## QUIC

## Bayesian

π Wishart priors?

## Penalized

## Structured

## References

*arXiv:1703.04025 [Cs, Stat]*, March.

*Communications in Statistics - Simulation and Computation*51 (4): 1381β1400.

*The Annals of Statistics*41 (6).

*The Econometrics Journal*19 (1): C1β32.

*Journal of Machine Learning Research*15 (1): 2911β47.

*Advances in Neural Information Processing Systems*, 16. NIPSβ13. Red Hook, NY, USA: Curran Associates Inc.

*arXiv:1507.02061 [Math, Stat]*, July.

*arXiv:1206.6361 [Cs, Stat]*, June.

*WIREs Computational Statistics*9 (6): e1415.

*Annals of Statistics*37 (6B): 4254β78.

*Journal of Multivariate Analysis*175 (January): 104560.

*The Econometrics Journal*20 (3): S61β85.

*Statistical Science*26 (3): 369β87.

*Iterative Methods for Sparse Linear Systems: Second Edition*. 2nd ed. SIAM.

*SIAM Journal on Matrix Analysis and Applications*38 (4): 1075β99.

*Biometrika*90 (4): 831β44.

*The Journal of Machine Learning Research*11: 26.

*Biometrika*101 (1): 103β20.

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