Overview
This vignette curates methodological reading for Step 1.b of DeCovarT: inferring gene regulatory network (GRN) structure that can inform cell-type precision matrices () (and hence (=^{-1})). The living checklist lives in GitHub issue #1; here we summarise the same layout with bibliographic cross-references.
Pandoc @citekey citations below resolve only against the Zotero/BBT export in inst/REFERENCES.bib. Papers not yet in that library are linked by DOI (Markdown) until they are added to Zotero and re-exported.
DeCovarT’s synthetic pipeline already builds a graph-constrained () from a random adjacency (synthetic scenarios vignette). Empirical inference must decide whether that graph is undirected (GGM / glasso), directed (Gaussian BN), prior-informed, dynamic, or multi-omic.
i) General reviews — correlation, causation, direct vs indirect effects
Marginal co-expression is a poor proxy for regulation. Shared upstream TFs, cell-type programmes, and common stimuli induce correlation without a direct edge; time lags and nonlinear kinetics can hide true edges. Conditional graphical models target direct dependence via the precision (), while causal / interventional designs are needed for direction and mechanism (Gene regulatory network inference in the era of single-cell multi-omics; Svinin and Glaab (2025); Network inference in systems biology).
Badia-i-Mompel et al. (2023) review single-cell multi-omics GRN tools and prior databases. Svinin and Glaab (2025) place association, prior-informed, and causal network analyses in one taxonomy (see figures in issue #1). Together they motivate separating undirected skeletons (ii) from DAG / perturbation work (iii).
ii) Undirected probabilistic strategies
ii.a) General inference
Gaussian graphical models (GGMs) encode conditional independence in (). The MLE exists only under sample-size / chordality conditions formalised by Uhler et al. (2012): if (n) is too small relative to clique size, the covariance MLE fails. Sparse (_1) estimation (graphical lasso; Friedman et al. (2008)) is the workhorse when (pn) (Chiquet 2015; Menéndez et al. 2010).
Two practical workflows appear repeatedly:
- Seeds then expand — select a small discriminating gene set (e.g. low condition number / high centroid separation, in the spirit of Aliee and Theis (2021)), then grow a prior-informed network with tools such as NeKo.
- Structure then parameters — estimate a sparse skeleton (glasso, NeighbourNet, CeSpGRN), then fit continuous weights, possibly under hard zeros on (_{ij}) for ((i,j)E).
Count-aware extensions replace the Gaussian likelihood by Poisson–log-normal (PLN) or copula models (chiquetPoissonLognormalModelVersatile2021?; Chiquet et al. 2018), and CeSpGRN, which matters when DeCovarT moves from Gaussian convolutions to sequencing counts.
ii.b) Regularisation and topological constraints
Three standard devices:
\max_{\Theta \succ 0} \log\det\Theta - \operatorname{tr}(S\Theta) \quad\text{s.t.}\quad \Theta_{ij}=0\;\forall\,(i,j)\notin E
(hard covariance selection);
\max_{\Theta \succ 0} \log\det\Theta - \operatorname{tr}(S\Theta) - \sum_{i\neq j}\lambda_{ij}\,|\Theta_{ij}|
(edge-specific / hybrid penalties); and structured group or fused graphical lasso for hubs, modules, or shared biological constraints (Chiquet 2015). Trees and chordal graphs remain PD-friendly (Uhler et al. (2012)). DeCovarT’s affine spectral shift of a binary adjacency is a constructive PD completion compatible with hard zeros off (E).
ii.c) Algorithm unrolling and deep / structured variational inference
Algorithm unrolling turns iterative sparse-precision solvers into trainable layers (Monga et al. (2021); GLAD). For hierarchical count ecosystems, Chaussard et al. (2024) introduce PLN-Tree: a tree-structured extension of PLN with backward Markov variational inference and identifiability results—an attractive route when taxonomy or gene modules should inform ().
ii.d) Prior knowledge and PPI as precision weights
Priors from TF–target, chromatin, and PPI databases can enter as (_{ij}) masks or as Bayesian edge probabilities (Badia-i-Mompel et al. (2023); Svinin and Glaab (2025); NeKo). This is the undirected analogue of pathway contextualisation in iii: association graphs become biologically constrained without claiming full causality.
iii) Directed approaches — DAGs, causality, perturbations
Gaussian Bayesian networks orient edges under Markov and faithfulness assumptions; interventions (Perturb-seq and related CRISPR screens) supply the environments needed for stronger causal claims (Peters et al. (2016); Svinin and Glaab (2025)). Pathway tools (NPA, SignalingProfiler, LIONESS/PANDA, NLBayes) contextualise expression on signed graphs but often stop short of a full weighted GRN MLE—treat them as priors or downstream scorers rather than drop-in replacements for ().
From factorised Gaussian BNs to a joint multivariate normal
For each node, (X_i(i)(i+*_{ij}X_j,,_i^2)). Stacking these linear structural equations yields a unique joint (_G(,)). In R,
bnlearn::gbn2mvnorm()(Scutari 2026) returns that mean and covariance from a fitted Gaussian BN—the directed counterpart of (=^{-1}) in an undirected GGM.
iv) Benchmarks
Comparative reverse-engineering studies and DREAM-style challenges remain the default yardsticks for undirected and directed GRN callers (Penfold & Wild (2011); Network inference in systems biology; Menéndez et al. (2010)). Emulation tests for TF–target function complement topological accuracy when gold-standard DAGs are incomplete. Any DeCovarT-related network estimator should report against these protocols before biological claims.
v) Perspectives
v.i) Multi-omics
Collaborative / multi-layer glasso and tensor-fusion GNNs couple transcriptomic layers with metabolomic, methylation, or miRNA views. They generalise the single-precision story to coupled () blocks—relevant when references cease to be expression-only.
v.ii) Boolean meets Bayesian
Noisy logic models (e.g. NLBayes) combine Boolean pathway structure with Bayesian TF-activity inference. They are powerful for regulon-level questions but are not substitutes for a dense weighted precision used inside DeCovarT’s Gaussian convolution.
v.iii) Dynamics and real-time capture
Static snapshots average over regulatory delays. Time-series reviews (Natalia et al. (2023)), Dictys, and related dynamic biomarker methods target (_t) along trajectories. Real-time or densely time-stamped assays would further reduce confounding between co-expression and causation.
v.iv) Graph neural networks
OOD-robust GNNs and invariant causal prediction (Peters et al. (2016)) address batch / cell-type shift when message-passing replaces classical glasso. Unrolling (ii.c) remains preferable when interpretability of each layer as an optimisation step matters for statistical GRN recovery.
Link to DeCovarT simulation
simulate_hierarchical_grn_moments() currently draws a random-graph adjacency and completes it to a PD precision. The literature above suggests three near-term upgrades, in increasing ambition:
- replace the random skeleton by a prior-informed or glasso skeleton on real purified profiles;
- use edge-specific (_{ij}) (PPI / TF) instead of a uniform
precision_scale; - move from a shared Gaussian () to PLN / PLN-Tree or cell-type-specific copula models when count structure dominates.