# Incomplete graphical model inference via latent tree aggregation

## TL;DR

This paper introduces a new method for inferring Gaussian graphical models with missing variables by aggregating spanning trees and using EM, effectively handling unobserved nodes and dense structures.

## Contribution

It proposes a novel approach combining spanning tree aggregation and EM for graphical model inference with missing data, including model selection criteria.

## Key findings

- Outperforms existing methods in edge detection accuracy.
- Effectively accounts for unobserved variables in network inference.
- Validated on simulations and flow cytometry data.

## Abstract

Graphical network inference is used in many fields such as genomics or ecology to infer the conditional independence structure between variables, from measurements of gene expression or species abundances for instance. In many practical cases, not all variables involved in the network have been observed, and the samples are actually drawn from a distribution where some variables have been marginalized out. This challenges the sparsity assumption commonly made in graphical model inference, since marginalization yields locally dense structures, even when the original network is sparse. We present a procedure for inferring Gaussian graphical models when some variables are unobserved, that accounts both for the influence of missing variables and the low density of the original network. Our model is based on the aggregation of spanning trees, and the estimation procedure on the Expectation-Maximization algorithm. We treat the graph structure and the unobserved nodes as missing variables and compute posterior probabilities of edge appearance. To provide a complete methodology, we also propose several model selection criteria to estimate the number of missing nodes. A simulation study and an illustration flow cytometry data reveal that our method has favorable edge detection properties compared to existing graph inference techniques. The methods are implemented in an R package.