pith. sign in

arxiv: 2606.18535 · v1 · pith:XH2F4CSEnew · submitted 2026-06-16 · 📊 stat.ME · cs.LG· math.ST· stat.TH

Shrinkage priors for Bayesian Substitute Confounders

Yordan P. Raykov , Hengrui Luo , Justin D. Strait , Wasiur R. KhudaBukhsh This is my paper

Pith reviewed 2026-06-26 22:59 UTC · model grok-4.3

classification 📊 stat.ME cs.LGmath.STstat.TH
keywords substitute confoundersBayesian factor modelsshrinkage priorscausal inferenceunmeasured confoundingmulti-cause studiesoverlap preservationposterior concentration
0
0 comments X

The pith

Shrinkage priors on Bayesian factor models produce substitute confounders that preserve multi-cause overlap for consistent causal estimation.

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

The paper shows how to learn lower-dimensional substitute scores from the joint distribution of multiple observed causes that still capture the shared assignment variation needed for causal adjustment. It introduces a Bayesian factor framework equipped with shrinkage priors that favor factors supported by several causes and discourage single-cause or overlap-collapsing scores. Under conditions of posterior concentration and overlap-preserving geometry, the regression-adjusted estimators recover mean potential outcomes whenever the latent-variable identification assumptions are satisfied. This approach addresses cases where flexible assignment models over-encode the treatment vector or fit noise rather than shared confounding structure.

Core claim

A Bayesian factor assignment framework with shrinkage priors learns sparse substitute confounders that retain coarse multi-cause dependence. The theory is stated at the level of posterior concentration, factor score contraction, and overlap-preserving assignment geometry and does not rely on a particular shrinkage prior. Under these conditions, the proposed regression-adjusted estimators are consistent for mean potential outcomes when the corresponding latent variable identification assumptions hold.

What carries the argument

Bayesian factor assignment framework with shrinkage priors that favor low-dimensional factors supported by multiple causes, discourage single-cause factors, and induce an ordering of latent factors through progressive shrinkage.

If this is right

  • The regression-adjusted estimators remain consistent for mean potential outcomes under the stated posterior concentration and latent identification conditions.
  • Shrinkage priors systematically discourage factors that encode only single-cause variation or collapse treatment overlap.
  • The framework induces an ordering among latent factors through progressive shrinkage without requiring a specific prior form.
  • In applications, sparse substitute scores can recover much of the adjustment obtained by conditioning directly on observed biomarkers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same shrinkage mechanism could be tested on other multi-cause observational datasets where direct biomarker measurement is unavailable.
  • Diagnostics for factor collapse could be extended to monitor overlap geometry in real time during posterior sampling.
  • The ordering induced by progressive shrinkage may offer a built-in way to select the number of substitute factors without separate cross-validation.

Load-bearing premise

The latent variable identification assumptions hold and the posterior concentrates on factors that preserve overlap geometry.

What would settle it

A simulation or dataset in which the latent confounder satisfies the identification assumptions yet the fitted factors produce scores whose posterior fails to contract while maintaining overlap, resulting in inconsistent estimators for the mean potential outcomes.

Figures

Figures reproduced from arXiv: 2606.18535 by Hengrui Luo, Justin D. Strait, Wasiur R. KhudaBukhsh, Yordan P. Raykov.

Figure 1
Figure 1. Figure 1: Synthetic grid summary. Top row: increasing nominal strength of the localised [PITH_FULL_IMAGE:figures/full_fig_p025_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Adaptive-shrinkage grid. Left: causal MSE as the target moves toward the con￾founded direction. Middle: selected directional penalty. Right: ratio between predictive￾CV causal error and the matched oracle causal error, reported for the two predictive-CV methods. 6.3 Alzheimer’s Disease Neuroimaging Initiative Next, we consider baseline data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI): a mul… view at source ↗
Figure 3
Figure 3. Figure 3: Causal-MSE landscapes for two reference points in the adaptive-shrinkage grid. The heat maps evaluate causal MSE over global shrinkage λ0 and directional shrinkage λ1, using the true confounding projector to isolate the regularization geometry. Stars mark the directional oracle, triangles the isotropic ABW oracle on the boundary λ1 = 0, and crosses the predictive-CV selection [PITH_FULL_IMAGE:figures/full… view at source ↗
Figure 4
Figure 4. Figure 4: Baseline hippocampal volume across diagnostic groups in ADNI, illustrating the [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Posterior diagnostic contrasts for baseline hippocampal volume in the CSF [PITH_FULL_IMAGE:figures/full_fig_p031_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Full synthetic-grid diagnostics. Rows report score recovery, residual focal depen [PITH_FULL_IMAGE:figures/full_fig_p044_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Original Scenario A geometry. The confounding signal is localised on a small [PITH_FULL_IMAGE:figures/full_fig_p045_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Original Scenario B geometry. The confounder block is smaller and weaker than [PITH_FULL_IMAGE:figures/full_fig_p045_8.png] view at source ↗
read the original abstract

Multi-cause observational studies contain information about unmeasured confounding through the dependence structure among causes. However, literal imputation of the unobserved confounder is often more complex than learning a lower-dimensional substitute score that preserves the shared assignment variation needed for stable causal adjustment. The deconfounder (Wang and Blei, 2019) and related substitute confounder methods exploit this idea, but flexible assignment models can fit the joint distribution of the causes while producing scores that over-encode the treatment vector, collapse overlap, or capture single-cause variation. We develop a Bayesian factor assignment framework for learning sparse substitute confounders that retain coarse multi-cause dependence with shrinkage priors. The theory is stated at the level of posterior concentration, factor score contraction, and overlap-preserving assignment geometry and therefore does not rely on a particular shrinkage prior. Under these conditions, the proposed regression-adjusted estimators are consistent for mean potential outcomes when the corresponding latent variable identification assumptions hold. Shrinkage priors provide a natural tool for latent structural learning: they favour low-dimensional factors supported by multiple causes, discourage effectively single-cause factors, and induce an ordering of the latent factors through progressive shrinkage. Synthetic experiments illustrate the roles of signal strength, outcome validity, and geometry-aware regularization. In an Alzheimer's Disease Neuroimaging Initiative (ADNI) baseline analysis, sparse substitute scores recover much of the adjustment obtained by directly conditioning on invasive cerebrospinal-fluid biomarkers, while collapse diagnostics identify when fitted factors reduce to individual observed measurements.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a Bayesian factor assignment model equipped with shrinkage priors to learn sparse substitute confounders that capture shared multi-cause variation while avoiding over-encoding of the treatment vector or collapse of overlap. The central theoretical result states consistency of the resulting regression-adjusted estimators for mean potential outcomes, conditional on posterior concentration, factor-score contraction, and overlap-preserving assignment geometry together with the usual latent-variable identification assumptions; the argument is deliberately formulated at this general level and does not depend on any particular shrinkage prior. Synthetic experiments examine the roles of signal strength, outcome validity, and geometry-aware regularization, while an ADNI baseline analysis compares the sparse substitute scores against direct conditioning on cerebrospinal-fluid biomarkers.

Significance. If the stated posterior-concentration and geometry-preservation conditions can be verified in practice, the framework supplies a principled, prior-agnostic route to substitute-confounder estimation that directly targets the multi-cause dependence structure required for stable adjustment. The emphasis on shrinkage-induced ordering of latent factors and the explicit diagnostics for factor collapse constitute concrete methodological contributions. The ADNI illustration provides an existence proof that the approach can recover much of the adjustment obtained from invasive biomarkers, which is a non-trivial empirical demonstration in a high-stakes domain.

minor comments (3)
  1. [Abstract / §1] The abstract and introduction both refer to “overlap-preserving assignment geometry” without a concise definition or pointer to the precise mathematical statement; a short displayed equation or boxed definition early in the paper would improve readability.
  2. [§5] Table or figure captions for the synthetic experiments should explicitly state the sample size, number of Monte Carlo replications, and the precise metric used to assess “collapse,” so that readers can judge the reported improvements without returning to the main text.
  3. [§6] The ADNI analysis reports that sparse scores “recover much of the adjustment,” but does not provide a quantitative comparison (e.g., change in estimated treatment effect or width of confidence intervals) relative to the biomarker-adjusted benchmark; adding this would strengthen the practical claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript. The referee summary and significance statement accurately reflect the paper's contributions and the level at which the consistency result is stated. No major comments were raised in the report, and the recommendation is for minor revision. We will incorporate any editorial or minor clarifications in the revised version.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central consistency claim for regression-adjusted estimators is explicitly conditioned on external latent variable identification assumptions plus general posterior concentration, factor score contraction, and overlap-preserving geometry properties. It states that the theory does not rely on any particular shrinkage prior and is framed at the level of these abstract properties rather than specific fitted quantities or self-referential definitions. No equations reduce the claimed result to inputs by construction, no fitted parameters are relabeled as predictions, and no load-bearing self-citation chain is invoked. The derivation chain remains self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central consistency claim rests on standard Bayesian posterior concentration results plus domain-specific latent-variable identification conditions that are not derived inside the paper.

free parameters (1)
  • shrinkage hyperparameters
    Control the strength of progressive shrinkage that induces sparsity and factor ordering; chosen or tuned in the model.
axioms (2)
  • domain assumption latent variable identification assumptions hold
    Invoked to guarantee consistency of the regression-adjusted estimators.
  • domain assumption posterior concentration on overlap-preserving factors
    Stated as the level at which the theory operates; required for the geometry claims.
invented entities (1)
  • sparse substitute confounder factors no independent evidence
    purpose: Lower-dimensional proxies for unmeasured confounding that retain shared multi-cause variation
    Core modeling device introduced by the framework; no independent falsifiable handle supplied in the abstract.

pith-pipeline@v0.9.1-grok · 5807 in / 1355 out tokens · 27339 ms · 2026-06-26T22:59:37.532757+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

241 extracted references · 9 linked inside Pith

  1. [1]

    Foundations and Trends in Econometrics , year =

    Jushan Bai , title =. Foundations and Trends in Econometrics , year =

  2. [2]

    Journal of the American Statistical Association , volume=

    The blessings of multiple causes , author=. Journal of the American Statistical Association , volume=. 2019 , publisher=

    2019

  3. [3]

    Neurobiology of Aging , volume=

    The dynamics of Alzheimer's disease biomarkers in the Alzheimer's Disease Neuroimaging Initiative cohort , author=. Neurobiology of Aging , volume=. 2010 , publisher=

    2010

  4. [4]

    Generalizability of findings from a clinical sample to a community-based sample: A comparison of

    Gianattasio, Kan Z and Bennett, Erin E and Wei, Jingkai and Mehrotra, Megha L and Mosley, Thomas and Gottesman, Rebecca F and Wong, Dean F and Stuart, Elizabeth A and Griswold, Michael E and Couper, David and Glymour, M Maria and Power, Melinda C and Alzheimer's Disease Neuroimaging Initiative , journal=. Generalizability of findings from a clinical sampl...

  5. [5]

    Screening and enrollment of underrepresented ethnocultural and educational populations in the Alzheimer's Disease Neuroimaging Initiative (

    Ashford, Miriam T and Raman, Rema and Miller, Garrett and Donohue, Michael C and Okonkwo, Ozioma C and Mindt, Monica Rivera and Nosheny, Rachel L and Coker, Godfrey A and Petersen, Ronald C and Aisen, Paul S and Weiner, Michael W and Alzheimer's Disease Neuroimaging Initiative , journal=. Screening and enrollment of underrepresented ethnocultural and educ...

  6. [6]

    2017 , publisher=

    Fundamentals of nonparametric Bayesian inference , author=. 2017 , publisher=

    2017

  7. [7]

    Linear Algebra and its Applications , volume=

    Generic global indentification in factor analysis , author=. Linear Algebra and its Applications , volume=. 1997 , publisher=

    1997

  8. [8]

    New approaches to Bayesian consistency , author=

  9. [9]

    Linear algebra and its applications , volume=

    Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics , author=. Linear algebra and its applications , volume=. 1977 , publisher=

    1977

  10. [11]

    Biometrika , volume=

    Identifiability of causal effects with multiple causes and a binary outcome , author=. Biometrika , volume=. 2022 , publisher=

    2022

  11. [12]

    Nature genetics , volume=

    A unified mixed-model method for association mapping that accounts for multiple levels of relatedness , author=. Nature genetics , volume=. 2006 , publisher=

    2006

  12. [13]

    Biometrika , volume=

    Kernel methods for causal functions: dose, heterogeneous and incremental response curves , author=. Biometrika , volume=. 2024 , publisher=

    2024

  13. [14]

    Journal of Machine Learning Research , volume=

    Directed Cyclic Graphs for Simultaneous Discovery of Time-Lagged and Instantaneous Causality from Longitudinal Data Using Instrumental Variables , author=. Journal of Machine Learning Research , volume=

  14. [15]

    Mueller, Susanne G and Weiner, Michael W and Thal, Leon J and Petersen, Ronald C and Jack, Clifford and Jagust, William and Trojanowski, John Q and Toga, Arthur W and Beckett, Laurel , journal=. The. 2005 , publisher=

    2005

  15. [16]

    Journal of medical Internet research , volume=

    Real-life gait performance as a digital biomarker for motor fluctuations: the Parkinson@ Home validation study , author=. Journal of medical Internet research , volume=. 2020 , publisher=

    2020

  16. [17]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , pages=

    Augmented balancing weights as linear regression , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , pages=. 2025 , publisher=

    2025

  17. [18]

    Artificial intelligence and statistics , pages=

    Handling sparsity via the horseshoe , author=. Artificial intelligence and statistics , pages=. 2009 , organization=

    2009

  18. [19]

    Bayesian Analysis , volume=

    Inference with Normal-Gamma prior distributions in regression problems , author=. Bayesian Analysis , volume=

  19. [20]

    Advances in Neural Information Processing Systems , volume=

    Using embeddings to correct for unobserved confounding in networks , author=. Advances in Neural Information Processing Systems , volume=

  20. [22]

    Annals of Statistics , volume =

    Tensor Decompositions and Sparse Log--linear Models , author =. Annals of Statistics , volume =

  21. [23]

    and van der Vaart, Aad W

    Ghosal, Subhashis and Ghosh, Jayanta K. and van der Vaart, Aad W. , title =. Annals of Statistics , year =

  22. [24]

    arXiv preprint arXiv:2406.19604 , year=

    Geodesic Causal Inference , author=. arXiv preprint arXiv:2406.19604 , year=

    arXiv

  23. [25]

    Journal of the National Cancer institute , volume=

    Smoking and lung cancer: recent evidence and a discussion of some questions , author=. Journal of the National Cancer institute , volume=. 1959 , publisher=

    1959

  24. [26]

    The American Economic Review , volume=

    Nonparametric bounds on treatment effects , author=. The American Economic Review , volume=. 1990 , publisher=

    1990

  25. [27]

    Annals of internal medicine , volume=

    Sensitivity analysis in observational research: introducing the E-value , author=. Annals of internal medicine , volume=. 2017 , publisher=

    2017

  26. [28]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    Making sense of sensitivity: Extending omitted variable bias , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2020 , publisher=

    2020

  27. [30]

    Biometrika , volume=

    Identifying causal effects with proxy variables of an unmeasured confounder , author=. Biometrika , volume=. 2018 , publisher=

    2018

  28. [31]

    Biometrika , pages=

    Measurement bias and effect restoration in causal inference , author=. Biometrika , pages=. 2014 , publisher=

    2014

  29. [32]

    Journal of the American statistical Association , volume=

    Identification of causal effects using instrumental variables , author=. Journal of the American statistical Association , volume=. 1996 , publisher=

    1996

  30. [33]

    Review of Economic Studies , volume=

    Inference on treatment effects after selection among high-dimensional controls , author=. Review of Economic Studies , volume=. 2014 , publisher=

    2014

  31. [34]

    2018 , publisher=

    Double/debiased machine learning for treatment and structural parameters , author=. 2018 , publisher=

    2018

  32. [35]

    Journal of Econometrics , volume=

    Robust inference on average treatment effects with possibly more covariates than observations , author=. Journal of Econometrics , volume=. 2015 , publisher=

    2015

  33. [36]

    Journal of the American Statistical Association , volume=

    Matrix completion methods for causal panel data models , author=. Journal of the American Statistical Association , volume=. 2021 , publisher=

    2021

  34. [37]

    Political Analysis , volume=

    Generalized synthetic control method: Causal inference with interactive fixed effects models , author=. Political Analysis , volume=. 2017 , publisher=

    2017

  35. [38]

    Journal of the American Statistical Association , volume=

    Estimation and inference of heterogeneous treatment effects using random forests , author=. Journal of the American Statistical Association , volume=. 2018 , publisher=

    2018

  36. [41]

    Journal of the Royal Statistical Society: Series B (Methodological) , volume=

    Assessing sensitivity to an unobserved binary covariate in an observational study with binary outcome , author=. Journal of the Royal Statistical Society: Series B (Methodological) , volume=. 1983 , publisher=

    1983

  37. [42]

    Journal of Econometrics , volume=

    Overlap in observational studies with high-dimensional covariates , author=. Journal of Econometrics , volume=. 2021 , publisher=

    2021

  38. [43]

    1997 , publisher=

    Foundations of modern probability , author=. 1997 , publisher=

    1997

  39. [44]

    Journal of Multivariate Analysis , volume =

    Olav Kallenberg , title =. Journal of Multivariate Analysis , volume =

  40. [45]

    arXiv preprint arXiv:2503.05024 , year=

    Kernel-based estimators for functional causal effects , author=. arXiv preprint arXiv:2503.05024 , year=

    arXiv

  41. [46]

    arXiv preprint arXiv:2003.04948 , year=

    Towards clarifying the theory of the deconfounder , author=. arXiv preprint arXiv:2003.04948 , year=

    arXiv 2003

  42. [47]

    Journal of the American Statistical Association , volume=

    The blessings of multiple causes: Rejoinder , author=. Journal of the American Statistical Association , volume=. 2019 , publisher=

    2019

  43. [49]

    Biometrika , volume=

    Causal diagrams for empirical research , author=. Biometrika , volume=. 1995 , publisher=

    1995

  44. [50]

    , author=

    Estimating causal effects of treatments in randomized and nonrandomized studies. , author=. Journal of educational Psychology , volume=. 1974 , publisher=

    1974

  45. [51]

    2019 , publisher=

    Ogburn, Elizabeth L and Shpitser, Ilya and Tchetgen, Eric J Tchetgen , journal=. 2019 , publisher=

    2019

  46. [52]

    2013 , month = jan, note =

    Tim Austin , title =. 2013 , month = jan, note =

    2013

  47. [53]

    Journal of Machine Learning Research , year =

    Samuel Heaps , title =. Journal of Machine Learning Research , year =

  48. [54]

    Journal of the American Statistical Association , volume=

    Dirichlet--Laplace priors for optimal shrinkage , author=. Journal of the American Statistical Association , volume=. 2015 , publisher=

    2015

  49. [55]

    Bernardo and Adrian F

    Jose M. Bernardo and Adrian F. M. Smith , title =. 2000 , edition =

    2000

  50. [56]

    Dunson , title =

    Anirban Bhattacharya and David B. Dunson , title =. Biometrika , year =

  51. [57]

    arXiv preprint arXiv:1312.6114 , year=

    Auto-encoding variational bayes , author=. arXiv preprint arXiv:1312.6114 , year=

    Pith/arXiv arXiv

  52. [58]

    1932 , publisher=

    Selected studies of the principle of relative frequency in language , author=. 1932 , publisher=

    1932

  53. [59]

    Statistics & Probability Letters , volume=

    A note on the multiplicative gamma process , author=. Statistics & Probability Letters , volume=. 2017 , publisher=

    2017

  54. [60]

    Microarrays, empirical

    Efron, Bradley , journal=. Microarrays, empirical. 2008 , publisher=

    2008

  55. [61]

    IEEE Transactions on signal processing , volume=

    K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation , author=. IEEE Transactions on signal processing , volume=. 2006 , publisher=

    2006

  56. [62]

    PLoS Genetics , volume=

    Analysis of population structure: a unifying framework and novel methods based on sparse factor analysis , author=. PLoS Genetics , volume=. 2010 , publisher=

    2010

  57. [63]

    Biometrika , volume=

    Bayesian cumulative shrinkage for infinite factorizations , author=. Biometrika , volume=. 2020 , publisher=

    2020

  58. [64]

    arXiv preprint arXiv:1310.4792 , year=

    A latent factor model with a mixture of sparse and dense factors to model gene expression data with confounding effects , author=. arXiv preprint arXiv:1310.4792 , year=

    Pith/arXiv arXiv

  59. [65]

    Journal of the American Statistical Association , volume=

    Partially collapsed Gibbs samplers: Theory and methods , author=. Journal of the American Statistical Association , volume=. 2008 , publisher=

    2008

  60. [66]

    Biometrika , pages=

    Sparse Bayesian infinite factor models , author=. Biometrika , pages=

  61. [67]

    arXiv preprint arXiv:1802.03426 , year=

    Umap: uniform manifold approximation and projection for dimension reduction , author=. arXiv preprint arXiv:1802.03426 , year=

    Pith/arXiv arXiv

  62. [68]

    Journal of Machine Learning Research , volume=

    Visualizing data using t-SNE , author=. Journal of Machine Learning Research , volume=

  63. [69]

    Wang, Lianming and Dunson, David B , journal=. Fast. 2011 , publisher=

    2011

  64. [70]

    URL http://yann

    The MNIST database of handwritten digits, 1998 , author=. URL http://yann. lecun. com/exdb/mnist , volume=

    1998

  65. [71]

    Notes on optimization on

    Tagare, Hemant D , booktitle=. Notes on optimization on. 2011 , publisher=

    2011

  66. [72]

    Raftery, Adrian E and Lewis, Steven M , journal=. [. 1992 , publisher=

    1992

  67. [73]

    M. J. Bayarri and J. O. Berger and F. Liu , title =. Bayesian Analysis , number =

  68. [74]

    The Journal of Machine Learning Research , volume=

    Pymanopt: A python toolbox for optimization on manifolds using automatic differentiation , author=. The Journal of Machine Learning Research , volume=. 2016 , publisher=

    2016

  69. [75]

    Khatri, CG and Mardia, Kanti V , journal=. The von. 1977 , publisher=

    1977

  70. [76]

    Annals of Mathematics , pages=

    Bessel functions of matrix argument , author=. Annals of Mathematics , pages=. 1955 , publisher=

    1955

  71. [77]

    Ghahramani, Zoubin and Hinton, Geoffrey E and others , year=. The

  72. [78]

    Journal of proteome research , Year =

    Interactive visual exploration of 3D mass spectrometry imaging data using hierarchical stochastic neighbor embedding reveals spatiomolecular structures at full data resolution , Author =. Journal of proteome research , Year =

  73. [79]

    The Journal of Immunology , Year =

    Using visualization of t-distributed stochastic neighbor embedding to identify immune cell subsets in mouse tumors , Author =. The Journal of Immunology , Year =

  74. [80]

    1985 , Pages =

    Exchangeability and related topics , Author =. 1985 , Pages =

    1985

  75. [81]

    Journal of the Royal Statistical Society: Series B (Methodological) , Year =

    On the statistical analysis of dirty pictures , Author =. Journal of the Royal Statistical Society: Series B (Methodological) , Year =

  76. [82]

    Neural computation , Year =

    GTM: The generative topographic mapping , Author =. Neural computation , Year =

  77. [83]

    IEEE transactions on pattern analysis and machine intelligence , Year =

    Combinatorial clustering and the beta negative binomial process , Author =. IEEE transactions on pattern analysis and machine intelligence , Year =

  78. [84]

    Bernoulli , Year =

    Posteriors, conjugacy, and exponential families for completely random measures , Author =. Bernoulli , Year =

  79. [85]

    IEEE Transactions on Signal Processing , Year =

    Compressive sensing on manifolds using a nonparametric mixture of factor analyzers: Algorithm and performance bounds , Author =. IEEE Transactions on Signal Processing , Year =

  80. [86]

    Signal processing , Year =

    Independent component analysis, a new concept? , Author =. Signal processing , Year =

Showing first 80 references.