Likelihood-Based Inference and Model Selection for Stochastic Gene Expression in Probability-Generating-Function Space

Selecting stochastic gene-expression models from single-cell counts requires accurate parameter inference and efficient model selection. Likelihood methods in count space can be costly when full stationary count distributions are unavailable, whereas approximate methods may lose accuracy. Probability generating functions (PGFs) offer a compact analytical alternative, but existing PGF workflows are generally not likelihood based and therefore rely on computationally intensive cross-validation. We develop a likelihood-based PGF framework for both tasks. Correlated empirical PGF values are used to construct a Gaussian quasi-likelihood for parameter inference and PGF-based Bayesian information criterion (BIC) for model selection. We show that the empirical PGF is exactly unbiased and that the parameter estimator is consistent, converges at the inverse-square-root sample-size rate, and is first-order asymptotically unbiased. For large samples and a uniquely preferred model, PGF-BIC selects the same model as leave-one-out cross-validation in PGF space.