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Pathway-based Bayesian factor models for gene expression data

Lorenzo Mauri, Federica Stolf, Amy H. Herring, Cameron Miller, David B. Dunson

TL;DR

BASIL is developed, a scalable Bayesian factor modeling framework that incorporates gene pathway annotations into latent variable analysis for RNA-sequencing data and consistently outperforms state-of-the-art approaches, accurately reconstructing gene-gene covariance, selecting the correct latent dimension, and identifying biologically coherent modules.

Abstract

Interpreting gene expression data requires methods that can uncover coordinated patterns corresponding to biological pathways. Traditional approaches such as principal component analysis and factor models reduce dimensionality, but latent components may have unclear biological meaning. Current approaches to incorporate pathway annotations impose restrictive assumptions, require extensive hyperparameter tuning, and do not provide principled uncertainty quantification, hindering the robustness and reproducibility of results. Here, we develop Bayesian Analysis with gene-Sets Informed Latent space (BASIL), a scalable Bayesian factor modeling framework that incorporates gene pathway annotations into latent variable analysis for RNA-sequencing data. BASIL places structured priors on factor loadings, shrinking them toward combinations of annotated gene sets, enhancing biological interpretability and stability, while simultaneously learning new unstructured components. BASIL provides accurate covariance estimates and uncertainty quantification, without resorting to computationally expensive Markov chain Monte Carlo sampling. An automatic empirical Bayes procedure eliminates the need for manual hyperparameter tuning, promoting reproducibility and usability in practice. In simulations and large-scale human transcriptomic datasets, BASIL consistently outperforms state-of-the-art approaches, accurately reconstructing gene-gene covariance, selecting the correct latent dimension, and identifying biologically coherent modules.

Pathway-based Bayesian factor models for gene expression data

TL;DR

BASIL is developed, a scalable Bayesian factor modeling framework that incorporates gene pathway annotations into latent variable analysis for RNA-sequencing data and consistently outperforms state-of-the-art approaches, accurately reconstructing gene-gene covariance, selecting the correct latent dimension, and identifying biologically coherent modules.

Abstract

Interpreting gene expression data requires methods that can uncover coordinated patterns corresponding to biological pathways. Traditional approaches such as principal component analysis and factor models reduce dimensionality, but latent components may have unclear biological meaning. Current approaches to incorporate pathway annotations impose restrictive assumptions, require extensive hyperparameter tuning, and do not provide principled uncertainty quantification, hindering the robustness and reproducibility of results. Here, we develop Bayesian Analysis with gene-Sets Informed Latent space (BASIL), a scalable Bayesian factor modeling framework that incorporates gene pathway annotations into latent variable analysis for RNA-sequencing data. BASIL places structured priors on factor loadings, shrinking them toward combinations of annotated gene sets, enhancing biological interpretability and stability, while simultaneously learning new unstructured components. BASIL provides accurate covariance estimates and uncertainty quantification, without resorting to computationally expensive Markov chain Monte Carlo sampling. An automatic empirical Bayes procedure eliminates the need for manual hyperparameter tuning, promoting reproducibility and usability in practice. In simulations and large-scale human transcriptomic datasets, BASIL consistently outperforms state-of-the-art approaches, accurately reconstructing gene-gene covariance, selecting the correct latent dimension, and identifying biologically coherent modules.
Paper Structure (25 sections, 6 theorems, 29 equations, 7 figures, 1 table)

This paper contains 25 sections, 6 theorems, 29 equations, 7 figures, 1 table.

Table of Contents

  1. Main Text
  2. Results
  3. Discussion
  4. Acknowledgments
  5. Author contributions
  6. Data availability
  7. Code availability
  8. Methods
  9. Supplementary Material for "Pathway-based Bayesian factor models for gene expression data"

Key Result

Proposition 1

Under Assumptions assumption:data--assumption:hyperparameters, as $n \to \infty$, with probability at least $1-o(1)$, we have where $|r_n| \lesssim \frac{1}{\sqrt{n}} + \frac{1}{\sqrt{p_n}}$.

Figures (7)

  • Figure 1: BASIL overview. a, BASIL is a two-layer matrix factorization approach: in the top layer the gene expression matrix ($Y$) is factorized as the product of a small number of latent variables ($M$) and their corresponding gene weights ($\Lambda$); in the hierarchical layer BASIL exploits prior gene ontology -- the binary gene set membership matrix $C$ -- in the definition of $\Lambda$. b, Empirical gene-gene correlation matrix computed from a subset of 100 genes for a gene expression dataset (global fever data) and correlation matrices estimated with BASIL and PLIER, reconstructed through latent variable factorization. The scatter plots show observed versus predicted correlation values.
  • Figure 2: Simulated data results.a, Performance of BASIL, ROTATE and PLIER in estimation accuracy for the gene co-expression matrix in low and high ontology signal scenarios over the $25$ replications. b, Mean runtime over $25$ replications of BASIL, ROTATE and PLIER with datasets simulated under different values for the number of genes in the 'high ontology signal' scenario. The results are in seconds on a logarithmic scale. c, BASIL and PLIER estimations of the number of latent factors across the two scenarios. We report median and interquartile range over $25$ replications. d, Frequentist coverage of 95% credible intervals for the gene covariance matrix by BASIL over the $25$ replications across the two scenarios. e, Ratio between the estimated variances for $\Gamma$ and $\Psi$ as the ontology signal increases. The plot shows the results over $25$ replications for each setting.
  • Figure 3: Whole-blood RNA-seq data correlation analysis.a, Empirical gene-gene correlation matrix for a subset of 100 genes and correlation matrices estimated with BASIL and PLIER. The two middle panels show the BASIL posterior mean of the correlation matrix without and with zeroing out the entries for which the corresponding $95\%$ credible interval included 0. b, Gene network plot for BASIL including only statistically significant correlations and PLIER. Positive (negative) correlations are shown in red (blue).
  • Figure 4: Global fever data known pathway and unstructured components. a, Top 5 pathways for the first ten factors of $\Gamma$ after zeroing out the entries for which the corresponding 95% credible intervals included 0. The plot shows the posterior mean and 95% posterior credible interval for each element. Positive (negative) values are shown in red (blue). b, Proportions of variance explained by known ($C\Gamma$) versus unknown ($\Psi$) pathways for a subset of 50 genes.
  • Figure 5: Enrichment analysis for global fever data. a, Network maps for the first two factors, following filtering and community detection, overlaid with the enrichment scores. b, Factor annotations for the first ten factors.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Proposition 1
  • Proposition 2
  • proof : Proof of Proposition \ref{['prop:sigma_hat_consistency']}
  • proof : Proof of Proposition \ref{['prop:prior_variances']}
  • Proposition 3: Proposition 3.5 of fable
  • Proposition 4
  • proof : Proof of Proposition \ref{['prop:norm_Y']}
  • Proposition 5
  • proof : Proof of Proposition \ref{['prop:norm_Y']}
  • Proposition 6
  • ...and 1 more