Count-Data Graphical Models with Latent Variables

Why Count Features Need Their Own Network Models

Counts are common in multi-omics data: they may represent sequencing reads, detected molecules, or events within a sample. A Gaussian graphical model is not automatically appropriate for such measurements. Counts are nonnegative integers, often right-skewed, and their variability can substantially exceed what a simple Poisson model allows. A count-data graphical model addresses the scale of the measurements while asking a network question: which features remain conditionally dependent after accounting for the others?

In a graph for observed features, an edge represents conditional dependence, not merely correlation. In a joint distribution, two features have no edge when they are conditionally independent given all the other modeled features. This is the same graph interpretation used in graphical models generally; what changes here is the probability model for counts and, when needed, the treatment of unobserved shared variation.

Poisson Conditional Models and Their Assumptions

A Poisson distribution has probability mass function \(P(X=x)=\exp(-\mu)\mu^x/x!\), for \(x=0,1,\ldots\) and mean \(\mu>0\). Its variance is also \(\mu\). A log-linear conditional model uses the log of a feature’s mean to express its dependence on the other features. For a feature \(X_i\), a common working specification is

Here \(X_{-i}\) denotes all features other than \(X_i\). Holding the other features fixed, a one-unit increase in \(X_j\) multiplies the conditional mean of \(X_i\) by \(\exp(\theta_{ij})\). A zero coefficient means that \(X_j\) has no contribution to this particular conditional mean. In a compatible undirected joint model, the two nodewise coefficients for a pair must agree, and a nonzero pair coefficient corresponds to an edge.

The conditional equation alone does not guarantee that all the nodewise conditionals come from one valid joint distribution. This compatibility issue matters especially when estimating regressions separately. One joint Poisson Markov random field with these conditional distributions has the form

where \(Z\) is the sum of the numerator over all count vectors and must be finite. Its conditional distribution for \(X_i\) is Poisson with mean \(\exp(\theta_i+\sum_{j\ne i}\theta_{ij}x_j)\): fixing all other counts leaves a kernel proportional to \(\exp\{x_i(\theta_i+\sum_{j\ne i}\theta_{ij}x_j)-\log(x_i!)\}\). One sufficient condition for a finite normalizing constant is that all pairwise coefficients satisfy \(\theta_{ij}\leq 0\), with finite node parameters. This restriction allows only nonpositive direct conditional effects in this joint model. A nodewise Poisson regression may produce positive coefficients, but those estimates do not thereby establish a compatible joint Poisson model.

Estimating Sparse Conditional Dependence

When \(p\) is large relative to the number of samples, estimating every possible edge is unstable. A standard strategy is to fit a Poisson regression for each node and penalize its coefficients to encourage a sparse neighborhood. For observations \(x^{(1)},\ldots,x^{(n)}\), the node-\(i\) conditional log-likelihood is

An \(L_1\)-penalized fit maximizes \(\ell_i(\beta_i)-\lambda\sum_{j\ne i}|\beta_{ij}|\), where \(\lambda\geq 0\) controls sparsity. The intercept is not penalized. A feature whose estimated coefficient is zero is excluded from that node’s fitted neighborhood. Since regressions are fitted separately, the two estimates for a pair can disagree. An analyst must specify how to form an undirected edge—for example, include it if either directional fit selects it, or require both to select it. These choices trade off sensitivity and specificity; neither turns incompatible nodewise fits into a guaranteed joint distribution.

Latent Variables and Overdispersed Counts

A Poisson conditional model assumes its conditional variance equals its conditional mean. Counts can nevertheless be overdispersed after averaging over unobserved sample-to-sample variation. For example, samples may differ in an unmeasured biological state that affects several features at once. A latent-factor model represents such shared variation explicitly. In the basic Poisson-lognormal model, each sample has a latent vector \(Z\sim N(0,I_q)\), and

The conditional independence assumption applies given \(Z\); it does not imply that the observed counts are independent after \(Z\) is integrated out. The factors can induce marginal covariance among features. Let \(\mu_i=E(X_i)\). The Gaussian moment-generating function gives \(\mu_i=\exp(\alpha_i+\|\lambda_i\|^2/2)\). For \(i\ne j\), conditional independence and averaging over \(Z\) yield

The variance includes the Poisson sampling term \(\mu_i\) and an additional nonnegative term from latent variation. The covariance can be positive, zero, or negative depending on the factor loadings’ inner product. These marginal associations are induced by shared latent variation; they do not, by themselves, identify direct conditional edges among observed features.

Reading the Network and Checking the Model

A useful distinction is whether the scientific target is shared variation or residual conditional dependence. A factor model with conditionally independent counts describes dependence induced by latent factors, but it has no direct observed-feature edges conditional on those factors. A count network can instead target dependencies between observed features conditional on the other observed features. Combining latent factors and residual edges is possible, but requires a specified model that separates the two sources of dependence and is identifiable from the data. A correlation network, a latent-factor model, and a conditional-independence graph answer different questions.

Before interpreting a fitted network, examine whether its distributional assumptions are plausible. Compare observed count means and variances with fitted values, inspect residual patterns across fitted means and samples, and check whether a small number of extreme counts dominate the fit. Underdispersion, excess zeros, or strong sample-level heterogeneity may indicate that the basic Poisson model is inadequate. Latent factors can capture one form of overdispersion, but they do not automatically address every source of misspecification.

• Compatibility failure: separately fitted nodewise models can assign different coefficients to the same pair or produce coefficients that cannot belong to a normalizable joint Poisson model. Report the edge-selection and symmetrization rule, and avoid claiming a joint model when only working conditional regressions were fitted.
• Latent-factor ambiguity: different combinations of loadings, factor scales, and residual dependencies can explain similar marginal associations. Factor constraints, adequate sample size, and sensitivity checks are important; factor labels should not be treated as uniquely identified biological mechanisms without further evidence.
• Overinterpretation: a selected edge is a model-based conditional association, not a causal effect. A missing edge may reflect limited power, regularization, or misspecification rather than true conditional independence.

Compared with Gaussian graphical models, Poisson conditional models respect the count support and use log-linear conditional means, but a simple Poisson model does not allow overdispersion conditional on its predictors. The Poisson-lognormal alternative accommodates extra variation and shared latent dependence, but its observed marginal distribution is not itself a product of independent Poisson variables, and factor-based interpretation requires care. Neither approach is universally preferable: the choice depends on whether the target is a sparse conditional network, latent shared variation, or both.