A Bayesian approach to model uncertainty in single-cell genomic data

Lina Gerontogianni; Shanshan Ren; Swati Chandna; Thomas E. Bartlett

arxiv: 2508.02061 · v2 · submitted 2025-08-04 · 🧬 q-bio.GN · q-bio.QM

A Bayesian approach to model uncertainty in single-cell genomic data

Shanshan Ren , Thomas E. Bartlett , Lina Gerontogianni , Swati Chandna This is my paper

Pith reviewed 2026-05-19 01:24 UTC · model grok-4.3

classification 🧬 q-bio.GN q-bio.QM

keywords single-cell RNA-seqBayesian clusteringvariational inferencecellular transitionsneurogenesisbreast cancermodel uncertaintyGaussian mixture model

0 comments

The pith

A variational Bayesian framework assigns probabilistic cluster memberships to single-cell genomic data rather than fixed identities.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper seeks to demonstrate that standard clustering methods, which force each cell into one category, hide transitional cell states during development or disease progression. It develops a variational Bayesian Gaussian mixture model that computes the probability a given cell belongs to each possible cluster. This probabilistic view produces more coherent pictures of processes such as neurogenesis and breast cancer advancement. The resulting probabilities then support downstream tasks including differential expression testing and pseudotime ordering. The work also introduces misclustering rate and area-under-curve scores as quantitative checks on clustering quality for single-cell RNA-seq data.

Core claim

This study introduces a variational Bayesian framework for clustering and analysing single-cell genomic data, employing a Bayesian Gaussian mixture model to estimate the probabilistic association of cells with distinct clusters. This approach captures cellular transitions, yielding biologically coherent insights into neurogenesis and breast cancer progression. The inferred clustering probabilities enable further analyses, including Differential Expression Analysis and pseudotime analysis. Furthermore, we propose utilising the misclustering rate and Area Under the Curve in clustering scRNA-seq data as an innovative metric to quantitatively evaluate overall clustering performance.

What carries the argument

Bayesian Gaussian mixture model with variational inference to estimate probabilistic cell-to-cluster associations

Load-bearing premise

Single-cell genomic count data is well-represented by a Bayesian Gaussian mixture model whose variational approximation yields reliable probabilistic cluster assignments without substantial bias from the inference method or data preprocessing choices.

What would settle it

Independent single-cell datasets of neurogenesis or breast cancer progression in which the probabilistic assignments fail to align with established marker genes for transitional states or fail to improve pseudotime ordering relative to hard clustering.

Figures

Figures reproduced from arXiv: 2508.02061 by Lina Gerontogianni, Shanshan Ren, Swati Chandna, Thomas E. Bartlett.

**Figure 1.** Figure 1: Adjacency matrix A ∈ {0, 1} p×n of an asymmetric bipartite network, reproduced from Bartlett et al[1]. A shows an asymmetric bipartite multi-edge network with adjacency matrix A ∈ Z p×n ≥0 ; B models the data matrix A, comprising non-negative integer counts; C is the corresponding network of B. grouping cells into discrete clusters in genomic data analysis, especially in single-cell RNA sequencing, as it o… view at source ↗

**Figure 2.** Figure 2: GMM clusters for breast cancer data in UMAP-LE projection. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: VB-GMM clusters for breast cancer data with in UMAP-LE projection. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: Marker gene plot for breast cancer data in UMAP-LE projection. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: The clusters of the Louvain Method for embryo cortical development data in UMAP-LE [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: Marker gene plot for embryo cortical development data in UMAP-LE projection from [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

**Figure 7.** Figure 7: GMM clusters for embryo cortical development data with [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

**Figure 8.** Figure 8: VB-GMM clusters for embryo cortical development data with [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗

**Figure 9.** Figure 9: Marker gene plot for cortical data in UMAP-LE projection from gestational week 17 to [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗

**Figure 10.** Figure 10: The visualisation plots of different K for GMM in UMAP-LE projection. The visualisation plot of K = 7 (upper left), representing the optimal quantitative K with respect to misclustering rate; The visualisation plot of K = 11 (lower left), representing the optimal quantitative K with respect to NMI; The visualisation plot of K = 4 (upper right), representing the optimal quantitative K with respect to ARI; … view at source ↗

**Figure 11.** Figure 11: The visualisation plots of different K for VB-GMM in UMAP-LE projection. The visualisation plot of K = 5 (left), representing the optimal quantitative K with respect to misclustering rate and ARI; The visualisation plot of K = 11 (middle), representing the optimal quantitative K with respect to NMI; The visualisation plot of K = 14 (right), representing the optimal visually detected K [PITH_FULL_IMAGE:f… view at source ↗

**Figure 12.** Figure 12: AUC heatmap of GMM (left) and VB-GMM (right) on neuron data. [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗

**Figure 13.** Figure 13: GMM clusters for breast cancer data with [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗

**Figure 14.** Figure 14: GMM clusters for breast cancer data with [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗

**Figure 15.** Figure 15: VB-GMM clusters for breast cancer data with [PITH_FULL_IMAGE:figures/full_fig_p017_15.png] view at source ↗

**Figure 16.** Figure 16: VB-GMM for breast cancer data with k = 8 in UMAP-LE projection. Gene names log2FC AveExpr t P.Value adj.P.Val B EMP1 -1.373 7.114 -43.119 <0.001 <0.001 796.007 KLF6 -1.510 7.693 -41.820 <0.001 <0.001 753.928 EDN1 -1.228 6.713 -39.897 <0.001 <0.001 693.481 TM4SF1 -1.495 8.430 -39.374 <0.001 <0.001 676.741 RTN4 -1.145 7.923 -38.868 <0.001 <0.001 661.305 YWHAH -1.225 7.800 -37.093 <0.001 <0.001 607.637 CYR61… view at source ↗

read the original abstract

Network models provide a powerful framework for analysing single-cell count data, facilitating the characterisation of cellular identities, disease mechanisms, and developmental trajectories. However, uncertainty modeling in unsupervised learning with genomic data remains insufficiently explored. Conventional clustering methods assign a singular identity to each cell, potentially obscuring transitional states during differentiation or mutation. This study introduces a variational Bayesian framework for clustering and analysing single-cell genomic data, employing a Bayesian Gaussian mixture model to estimate the probabilistic association of cells with distinct clusters. This approach captures cellular transitions, yielding biologically coherent insights into neurogenesis and breast cancer progression. The inferred clustering probabilities enable further analyses, including Differential Expression Analysis and pseudotime analysis. Furthermore, we propose utilising the misclustering rate and Area Under the Curve in clustering scRNA-seq data as an innovative metric to quantitatively evaluate overall clustering performance. This methodological advancement enhances the resolution of single-cell data analysis, enabling a more nuanced characterisation of dynamic cellular identities in development and disease.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper applies variational Bayesian GMM to get probabilistic cell assignments in scRNA-seq and proposes misclustering rate plus AUC as metrics, but the Gaussian likelihood on count data risks biasing the transition probabilities.

read the letter

The one thing to know is that this paper applies a variational Bayesian Gaussian mixture model to single-cell RNA-seq data so that cells receive probabilities of belonging to different clusters instead of a single hard label. They run it on neurogenesis and breast cancer datasets and report that the probabilities produce biologically coherent pictures of transitions, which then support differential expression and pseudotime work. They also suggest misclustering rate and AUC as evaluation metrics for this kind of clustering output. That combination is the core of what they present. The framing around uncertainty in cell identity during developmental or disease shifts is reasonable and the downstream use of the probabilities is straightforward. The metric proposal adds a modest practical element that some groups might adopt when they need to score soft assignments. The soft spot sits in the likelihood choice. Single-cell counts are discrete, zero-inflated, and overdispersed. A Gaussian mixture, even after common normalizations, assumes symmetric continuous errors and constant variance. That mismatch can shift posterior probabilities for cells sitting between clusters. The abstract gives no sign they moved to a negative binomial or zero-inflated model, and there is no reported check that the variational approximation recovers known transition markers without systematic distortion. If the full methods section contains those diagnostics or comparisons to count-appropriate alternatives, the results gain weight. Without them the central claims rest on an assumption that may not hold evenly across the data. This work is aimed at computational biologists who already run clustering on scRNA-seq and want probabilistic outputs for trajectory studies. A reader comfortable with variational methods will see it as an application rather than a deep methodological advance, but the concrete examples and metric suggestion could still be worth a look. It deserves a serious referee. The topic is relevant, the method is implementable, and referees can focus directly on the likelihood justification and validation strength. I would send it for review.

Referee Report

2 major / 1 minor

Summary. The manuscript presents a variational Bayesian framework based on a Gaussian mixture model for unsupervised clustering of single-cell genomic count data. The approach aims to model uncertainty in cell identities via probabilistic cluster assignments rather than hard clustering, thereby capturing transitional states during processes such as neurogenesis and breast cancer progression. The inferred probabilities are used to support downstream tasks including differential expression analysis and pseudotime inference, and the authors propose misclustering rate together with AUC as quantitative metrics for evaluating clustering performance on scRNA-seq data.

Significance. If the variational approximation produces reliable soft assignments that are not materially distorted by the Gaussian likelihood on normalized count data, the method could offer a principled way to quantify uncertainty in single-cell clustering and improve resolution of dynamic cellular trajectories. The explicit proposal of misclustering rate and AUC as evaluation metrics is a concrete, falsifiable contribution that could be adopted more broadly if shown to correlate better with biological ground truth than conventional indices.

major comments (2)

[Abstract / Methods] Abstract and Methods: The central claim that the variational Bayesian GMM yields reliable probabilistic assignments capturing cellular transitions rests on the assumption that a Gaussian likelihood (after normalization) adequately represents single-cell count data. Single-cell counts are discrete, zero-inflated, and overdispersed; the Gaussian model implicitly assumes symmetric continuous errors and homoscedasticity, which can bias posterior probabilities near cluster boundaries. The manuscript should either replace the likelihood with a count-appropriate model (e.g., negative binomial) or provide quantitative checks (e.g., posterior predictive diagnostics or comparison of transition probabilities against known marker gradients) demonstrating that the approximation does not introduce systematic bias in the neurogenesis and breast-cancer results.
[Results] Results: The abstract asserts that the framework yields 'biologically coherent insights' into neurogenesis and breast cancer progression and that the probabilities enable further analyses, yet no quantitative validation details, baseline comparisons (e.g., against standard GMM, Leiden, or scVI), or error analysis are supplied in the visible text. Without these, the support for the utility claim remains qualitative and the load-bearing assertion that the method improves resolution of transitional states cannot be evaluated.

minor comments (1)

[Methods / Evaluation] The manuscript should define the misclustering rate and AUC explicitly (including how ground-truth labels are obtained for the AUC calculation) and compare them against established metrics such as adjusted Rand index or normalized mutual information on the same datasets.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments. We address each major point below and indicate the revisions that will be incorporated into the next version of the manuscript.

read point-by-point responses

Referee: [Abstract / Methods] Abstract and Methods: The central claim that the variational Bayesian GMM yields reliable probabilistic assignments capturing cellular transitions rests on the assumption that a Gaussian likelihood (after normalization) adequately represents single-cell count data. Single-cell counts are discrete, zero-inflated, and overdispersed; the Gaussian model implicitly assumes symmetric continuous errors and homoscedasticity, which can bias posterior probabilities near cluster boundaries. The manuscript should either replace the likelihood with a count-appropriate model (e.g., negative binomial) or provide quantitative checks (e.g., posterior predictive diagnostics or comparison of transition probabilities against known marker gradients) demonstrating that the approximation does not introduce systematic bias in the neurogenesis and breast-cancer results.

Authors: We agree that the Gaussian likelihood after normalization represents an approximation to the discrete, zero-inflated nature of raw scRNA-seq counts. Our implementation follows the standard preprocessing pipeline used across the field (log-transformation, scaling, and selection of highly variable genes) to render the data suitable for continuous mixture modeling. To directly evaluate whether this approximation introduces systematic bias in the soft assignments, the revised manuscript will add posterior predictive checks: we will draw replicated datasets from the fitted variational posterior and compare their marginal distributions against the observed normalized data. In addition, we will quantify the alignment between inferred cluster-transition probabilities and known marker-gene gradients along the neurogenesis trajectory. These diagnostics will be reported in a new subsection of the Results. revision: yes
Referee: [Results] Results: The abstract asserts that the framework yields 'biologically coherent insights' into neurogenesis and breast cancer progression and that the probabilities enable further analyses, yet no quantitative validation details, baseline comparisons (e.g., against standard GMM, Leiden, or scVI), or error analysis are supplied in the visible text. Without these, the support for the utility claim remains qualitative and the load-bearing assertion that the method improves resolution of transitional states cannot be evaluated.

Authors: The manuscript already reports quantitative performance via the proposed misclustering rate and AUC on both simulated and real data, together with downstream differential-expression and pseudotime results that demonstrate utility. Nevertheless, to make the comparative evaluation fully explicit, the revised Results section will include side-by-side benchmarks against a standard (non-Bayesian) GMM, Leiden clustering, and scVI, using the same misclustering-rate and AUC metrics. These additions will directly quantify the improvement in resolution of transitional states. revision: yes

Circularity Check

0 steps flagged

Standard variational Bayesian GMM application shows no circular derivation

full rationale

The paper applies a variational Bayesian Gaussian mixture model to single-cell count data for probabilistic clustering, as described in the abstract. No equations, derivations, or self-citations are presented that reduce the claimed outputs (probabilistic assignments, DE analysis, pseudotime) to inputs by construction. The framework is positioned as a standard unsupervised learning method whose results enable downstream analyses, with proposed metrics (misclustering rate, AUC) serving as independent evaluation tools rather than tautological re-expressions of fitted parameters. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard domain assumptions for modeling count data with mixtures and variational approximations; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (2)

domain assumption Single-cell genomic count data can be modeled as arising from a Gaussian mixture distribution in a suitable latent space.
The paper directly employs a Bayesian Gaussian mixture model for clustering.
domain assumption Variational inference provides a sufficiently accurate approximation to the posterior for probabilistic cluster assignments.
The framework is described as variational Bayesian without further qualification.

pith-pipeline@v0.9.0 · 5706 in / 1340 out tokens · 31819 ms · 2026-05-19T01:24:37.011953+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

employing a Bayesian Gaussian mixture model to estimate the probabilistic association of cells with distinct clusters... likelihood is a mixture of Gaussians... posterior estimate of the cluster assignment probability
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

regularised graph Laplacian... singular value decomposition... asymptotically multivariate Gaussian distributed in the spectral embeddings
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat_induction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Dirichlet distribution is used to model the mixture weights... Variational Bayesian estimation of a Gaussian Mixture Model (VB-GMM)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Breast cancer at-risk: n = 25636 cells, p = 16822 RNA transcripts

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Embryo cortical development: n = 41734 cells, p = 16862 RNA transcripts. This test dataset incorporates all the embryo cortical data in V1 region from gestational week 17 to gestational week 22, representing the broadest range of gestational weeks allowed for research with human embryos due to ethical constraints. 13 4.2 Specification of our statistical m...

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A stochastic network approach to clustering and visualising single-cell genomic count data

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Feature selection and dimension reduction for single-cell RNA-Seq based on a multinomial model

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Endothelin-1 Enriched Tumor Phenotype Predicts Breast Cancer Recurrence

Tamkus D, Sikorskii A, Gallo KA, Wiese DA, Leece C, Madhukar BV, et al. Endothelin-1 Enriched Tumor Phenotype Predicts Breast Cancer Recurrence. International Scholarly Research Notices. 2013;2013(1):385398

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Andrews MG, Subramanian L, Salma J, Kriegstein AR. How mechanisms of stem cell polarity shape the human cerebral cortex. Nature Reviews Neuroscience. 2022;23(12):711-24

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A radial glia gene marker, fatty acid binding protein 7 (FABP7), is involved in proliferation and invasion of glioblastoma cells

De Rosa A, Pellegatta S, Rossi M, Tunici P , Magnoni L, Speranza MC, et al. A radial glia gene marker, fatty acid binding protein 7 (FABP7), is involved in proliferation and invasion of glioblastoma cells. PloS one. 2012;7(12):e52113

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Dok5 is involved in the signaling pathway of neurotrophin-3 against TrkC-induced apoptosis

Pan Y , Zhang J, Liu W, Shu P , Yin B, Yuan J, et al. Dok5 is involved in the signaling pathway of neurotrophin-3 against TrkC-induced apoptosis. Neuroscience letters. 2013;553:46-51

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Variational inference for Dirichlet process mixtures

Blei DM, Jordan MI. Variational inference for Dirichlet process mixtures. Journal of Bayesian Analysis. 2006;1(1):121-44

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Newson R. Parameters behind “nonparametric” statistics: Kendall’s tau, Somers’ D and median differences. The Stata Journal. 2002;2(1):45-64. B Additional information Funding This work was supported in part by the Research Innovation Fund awarded by Birkbeck, University of London, UK. License 20

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Breast cancer at-risk: n = 25636 cells, p = 16822 RNA transcripts

work page

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Embryo cortical development: n = 41734 cells, p = 16862 RNA transcripts. This test dataset incorporates all the embryo cortical data in V1 region from gestational week 17 to gestational week 22, representing the broadest range of gestational weeks allowed for research with human embryos due to ethical constraints. 13 4.2 Specification of our statistical m...

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A stochastic network approach to clustering and visualising single-cell genomic count data

Bartlett TE, Chandna S, Roy S. A stochastic network approach to clustering and visualising single-cell genomic count data. arXiv preprint arXiv:230302498. 2023

work page 2023

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Feature selection and dimension reduction for single-cell RNA-Seq based on a multinomial model

Townes FW, Hicks SC, Aryee MJ, Irizarry RA. Feature selection and dimension reduction for single-cell RNA-Seq based on a multinomial model. Genome biology. 2019;20:1-16. 19

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Regularized spectral clustering under the degree-corrected stochastic blockmodel

Qin T, Rohe K. Regularized spectral clustering under the degree-corrected stochastic blockmodel. Advances in neural information processing systems. 2013;26

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Modularity and community structure in networks

Newman ME. Modularity and community structure in networks. Proceedings of the national academy of sciences. 2006;103(23):8577-82

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A statistical interpretation of spectral embedding: the generalised random dot product graph

Rubin-Delanchy P , Cape J, Tang M, Priebe CE. A statistical interpretation of spectral embedding: the generalised random dot product graph. Journal of the Royal Statistical Society Series B: Statistical Methodology. 2022;84(4):1446-73

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Mammary stem cells and progenitors: targeting the roots of breast cancer for prevention

Tharmapalan P , Mahendralingam M, Berman HK, Khokha R. Mammary stem cells and progenitors: targeting the roots of breast cancer for prevention. The EMBO journal. 2019;38(14):e100852

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The pivotal roles of the epithelial membrane protein family in cancer invasiveness and metastasis

Ahmat Amin MKB, Shimizu A, Ogita H. The pivotal roles of the epithelial membrane protein family in cancer invasiveness and metastasis. Cancers. 2019;11(11):1620

work page 2019

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KLF6-SV1 drives breast cancer metastasis and is associated with poor survival

Hatami R, Sieuwerts AM, Izadmehr S, Y ao Z, Qiao RF , Papa L, et al. KLF6-SV1 drives breast cancer metastasis and is associated with poor survival. Science translational medicine. 2013;5(169):169ra12-2

work page 2013

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Endothelin-1 Enriched Tumor Phenotype Predicts Breast Cancer Recurrence

Tamkus D, Sikorskii A, Gallo KA, Wiese DA, Leece C, Madhukar BV, et al. Endothelin-1 Enriched Tumor Phenotype Predicts Breast Cancer Recurrence. International Scholarly Research Notices. 2013;2013(1):385398

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How mechanisms of stem cell polarity shape the human cerebral cortex

Andrews MG, Subramanian L, Salma J, Kriegstein AR. How mechanisms of stem cell polarity shape the human cerebral cortex. Nature Reviews Neuroscience. 2022;23(12):711-24

work page 2022

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A radial glia gene marker, fatty acid binding protein 7 (FABP7), is involved in proliferation and invasion of glioblastoma cells

De Rosa A, Pellegatta S, Rossi M, Tunici P , Magnoni L, Speranza MC, et al. A radial glia gene marker, fatty acid binding protein 7 (FABP7), is involved in proliferation and invasion of glioblastoma cells. PloS one. 2012;7(12):e52113

work page 2012

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Dok5 is involved in the signaling pathway of neurotrophin-3 against TrkC-induced apoptosis

Pan Y , Zhang J, Liu W, Shu P , Yin B, Yuan J, et al. Dok5 is involved in the signaling pathway of neurotrophin-3 against TrkC-induced apoptosis. Neuroscience letters. 2013;553:46-51

work page 2013

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Variational inference for Dirichlet process mixtures

Blei DM, Jordan MI. Variational inference for Dirichlet process mixtures. Journal of Bayesian Analysis. 2006;1(1):121-44

work page 2006

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nonparametric

Newson R. Parameters behind “nonparametric” statistics: Kendall’s tau, Somers’ D and median differences. The Stata Journal. 2002;2(1):45-64. B Additional information Funding This work was supported in part by the Research Innovation Fund awarded by Birkbeck, University of London, UK. License 20

work page 2002