# Maximum likelihood estimation in Gaussian models under total positivity

arXiv: https://arxiv.org/abs/1702.04031 · HTML: https://mlsanity.com/paper/1702.04031/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Feb 2017.

## Built on by (most cited first, 14 of 14)

- [Fast Projected Newton-like Method for Precision Matrix Estimation under Total Positivity](https://mlsanity.com/paper/2112.01939/) (2021)
- [Learning Graphical Factor Models with Riemannian Optimization](https://mlsanity.com/paper/2210.11950/) (2022)
- [Sparse Graph Learning Under Laplacian-Related Constraints](https://mlsanity.com/paper/2111.08161/) (2021)
- [Learning Large-Scale MTP2 Gaussian Graphical Models via Bridge-Block Decomposition](https://mlsanity.com/paper/2309.13405/) (2023)
- [Laplacian Constrained Precision Matrix Estimation: Existence and High Dimensional Consistency](https://mlsanity.com/paper/2111.00590/) (2021)
- [Network Topology Inference with Sparsity and Laplacian Constraints](https://mlsanity.com/paper/2309.00960/) (2023)
- [Learning with latent group sparsity via heat flow dynamics on networks](https://mlsanity.com/paper/2201.08326/) (2022)
- [Adaptive Estimation of Graphical Models under Total Positivity](https://mlsanity.com/paper/2210.15471/) (2022)
- [Leveraging Low-rank Factorizations of Conditional Correlation Matrices in Graph Learning](https://mlsanity.com/paper/2506.10628/) (2025)
- [Learning Gaussian Graphical Models under Total Positivity via Spectral Graph Sparsification](https://mlsanity.com/paper/2605.17154/) (2026)
- [A note on incorrect inferences in non-binary qualitative probabilistic networks](https://mlsanity.com/paper/2208.09344/) (2022)
- [Learning Kronecker-Structured Graphs from Smooth Signals](https://mlsanity.com/paper/2505.09822/) (2025)
- [Learning under Latent Group Sparsity via Diffusion on Networks](https://mlsanity.com/paper/2507.15097/) (2025)
- [Spectral Sparsification of Laplacian-Constrained Gaussian and H"usler-Reiss Graphical Models](https://mlsanity.com/paper/2606.16681/) (2026)

## Abstract (arXiv)

We analyze the problem of maximum likelihood estimation for Gaussian distributions that are multivariate totally positive of order two (MTP2). By exploiting connections to phylogenetics and single-linkage clustering, we give a simple proof that the maximum likelihood estimator (MLE) for such distributions exists based on at least 2 observations, irrespective of the underlying dimension. Slawski and Hein, who first proved this result, also provided empirical evidence showing that the MTP2 constraint serves as an implicit regularizer and leads to sparsity in the estimated inverse covariance matrix, determining what we name the ML graph. We show that we can find an upper bound for the ML graph by adding edges corresponding to correlations in excess of those explained by the maximum weight spanning forest of the correlation matrix. Moreover, we provide globally convergent coordinate descent algorithms for calculating the MLE under the MTP2 constraint which are structurally similar to iterative proportional scaling. We conclude the paper with a discussion of signed MTP2 distributions.