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Theory

SCRIBE is built on a rigorous mathematical foundation rooted in Bayesian statistics and probabilistic modeling. This section provides an accessible overview of the core theoretical results that underpin the package, presenting the key ideas and their implications without exhaustive algebraic derivations.

For practitioners

You do not need to read these pages to use SCRIBE effectively. The Model Selection guide provides all the practical information needed to choose and run models. The theory pages are for users who want to understand why the models work the way they do.

Foundations: the Negative Binomial core

The default SCRIBE models treat genes as independent Negative Binomial draws whose parameters arise from the biophysics of bursty transcription and mRNA capture. This family covers the vast majority of use cases and forms the backbone of the inference, denoising, and differential expression pipelines.

  • Dirichlet-Multinomial Model — Derives how independent negative binomial counts with a shared success probability factorize into a negative binomial for totals and a Dirichlet-Multinomial for compositions, providing a principled normalization scheme for scRNA-seq data.

  • Hierarchical Gene-Specific p — Relaxes the shared-p assumption by placing a hierarchical prior on gene-specific success probabilities, with a generalized composition sampling procedure that strictly extends the Dirichlet model.

  • Beta Negative Binomial — Extends the NB with per-gene power-law tails via a mean-preserving Beta compound, with a biophysical interpretation as extrinsic noise in burst size and a sparsity-inducing hierarchical prior that defaults to NB behaviour.

  • Two-state promoter (Poisson-Beta) — Extends the NB family with a Poisson-Beta compound likelihood for bursty / bimodal genes that the NB cannot fit. Derives the closed-form steady-state distribution of a binary promoter, shows closure under binomial thinning, recovers the NB in the fast-OFF limit, presents the four parameterizations (natural, ratio, mean-Fano, moment-delta) that fix successive mean-field VI pathologies, and develops the autograd-robust Gauss-Legendre quadrature engine.

  • Anchoring Priors — Resolves two layers of practical non-identifiability in the variable capture model: the capture-expression degeneracy (via biology-informed capture prior) and the mean-overdispersion degeneracy (via data-informed mean anchoring), each justified by a law-of-large-numbers concentration argument.

  • Hierarchical Priors — Introduces the three prior families (Gaussian, Horseshoe, NEG) used for adaptive shrinkage across genes, mixture components, and datasets, with applications to gene-specific p, mu, the zero-inflation gate, and multi-dataset hierarchical models.

Gene regulatory network models

When genes interact through regulatory networks, the independent-gene assumption no longer holds. Under the linear noise approximation, the steady-state distribution of log-abundances becomes a multivariate Gaussian whose covariance encodes the regulatory structure. These models learn that covariance explicitly, enabling recovery of gene-gene correlation programs.

  • GRN Biophysics — Derives that the steady-state distribution of gene expression in a gene regulatory network under the linear noise approximation is a multivariate Gaussian on log-abundances, providing the shared biophysical foundation for both the PLN and LNM observation models.

  • Poisson Log-Normal Model — Develops the PLN observation model where each gene's count is a Poisson draw from the exponentiated log-abundance, yielding log-concave posteriors, natural total-count coupling, and efficient Laplace inference via Woodbury Newton.

  • NB Log-Normal Model — Extends PLN's Poisson observation channel to a Negative Binomial with per-gene dispersion \(r_g\), restoring bursty-transcription overdispersion at the per-gene level while retaining PLN's log-concave posterior and cross-gene covariance structure. Introduces the per-cell rigid- translation gauge and the SVI-cascade + freeze workflow that pins it structurally.

  • Logistic-Normal Multinomial Model — Replaces the Dirichlet prior on compositions with a logistic-normal distribution, enabling arbitrary cross-gene correlations in log-ratio space with inference via Laplace approximation or VAE.

  • Loadings-Matrix Shrinkage Priors — A modular framework for shrinking the columns of the low-rank loadings matrix \(\underline{\underline{W}}\). Lets users keep latent_dim generous and have the prior pick the effective rank adaptively. Ships three column-wise strategies (Gaussian, horseshoe, NEG) plus the math fixes (softplus-floor, subspace correction) that keep the MAP well-defined.

Analysis framework

Once any model is fit---whether from the NB family or the log-normal family---SCRIBE provides a unified set of tools for extracting biological conclusions and validating the fit. These methods operate on the posterior regardless of which generative model produced it.

  • Bayesian Denoising — Derives a closed-form posterior for the true transcript counts given observed UMIs, exploiting Poisson-Gamma conjugacy to recover a shifted negative binomial denoised distribution, with extensions for zero-inflated models and cross-gene correlations.

  • Differential Expression — Develops a fully Bayesian DE framework in compositional (CLR) space with three inference methods (parametric, empirical, shrinkage), complemented by biological-level metrics (LFC, log-variance ratio, Jeffreys divergence) that are free of compositional closure.

  • Model Comparison — Develops WAIC and PSIS-LOO criteria for ranking models by out-of-sample predictive accuracy, with pairwise uncertainty quantification, per-gene elpd decomposition, and optimal model stacking.

  • Goodness-of-Fit Diagnostics — Provides expression-scale-invariant per-gene diagnostics via randomized quantile residuals (RQR) and posterior predictive checks (PPC), enabling principled gene filtering before downstream inference.