# Multilevel maximum likelihood estimation with application to covariance matrices

## TL;DR

This paper demonstrates that restricting maximum likelihood estimation to a subspace can reduce asymptotic variance, improving covariance matrix fitting in high-dimensional data assimilation tasks.

## Contribution

It introduces a multilevel maximum likelihood framework and applies it to spectral diagonal and sparse inverse covariance models, showing computational benefits.

## Key findings

- Restricting to a subspace decreases asymptotic variance.
- Smaller parameter sets reduce sampling noise.
- Hierarchical approach improves covariance estimation in high dimensions.

## Abstract

The asymptotic variance of the maximum likelihood estimate is proved to decrease when the maximization is restricted to a subspace that contains the true parameter value. Maximum likelihood estimation allows a systematic fitting of covariance models to the sample, which is important in data assimilation. The hierarchical maximum likelihood approach is applied to the spectral diagonal covariance model with different parameterizations of eigenvalue decay, and to the sparse inverse covariance model with specified parameter values on different sets of nonzero entries. It is shown computationally that using smaller sets of parameters can decrease the sampling noise in high dimension substantially.