Bayesian variable selection for globally sparse probabilistic PCA

Bouveyron, Charles; Latouche, Pierre; Mattei, Pierre-Alexandre

doi:10.1214/18-ejs1450

Cited by 21 publications

(40 citation statements)

References 67 publications

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“…In order to recover the sparsity pattern for each of the K classes, we leverage the strategy proposed by Bouveyron et al for performing Bayesian variable selection in probabilistic PCA. To this end, we complete the previous modeling by considering priors for both the sparsity patterns p ( v 1 ),…, p ( v K ), and the projection parameters p ( W 1 ),…, p ( W K ).…”

Section: Model Inferencementioning

confidence: 99%

“…Regarding the projection parameters following Bouveyron et al, we derive a closed‐form expression of the marginal likelihood p ( X k | v k ). To this end, we consider a Gaussian prior for the matrix W .…”

Section: Model Inferencementioning

confidence: 99%

“…To do so, for each class k = 1,…, K , the noise ε is assumed to decompose as follows:

ε_{| Z = k} = v_{k} ε_{1} + {\bar{v}}_{k} ε_{2},

where

ε_{1} false| Z = k \sim scriptN ((), 0, σ_{k 1}^{2} I_{p})

is the noise of the inactive variables and

ε_{2} false| Z = k \sim scriptN ((), 0, σ_{k 2}^{2} I_{p})

is the noise of the active variables, both for the k th class. Making use of theorem 2 of Bouveyron et al and assuming that

σ_{k 1}^{2} \to 0

, we end up with a closed‐form formulation for the evidence:

\begin{matrix} right \log p (X_{k} | v_{k}, α_{k}) & left = - \frac{{false\sum}_{i = 1}^{n_{k}} | | x_{k i}^{{\bar{v}}_{k}} - μ_{k}^{{\bar{v}}_{k}} | {false|}^{2}}{2 σ_{k 2}^{2}} - n_{k} (p - q_{k}) \log σ_{k 2} + \frac{n_{k} q_{k}}{2} \log α_{k} & right \\ right & left + \sum_{i = 1}^{n_{k}} (\log K_{(q_{k} - d_{k}) / 2} (α_{k} | | x_{k i}^{v_{k}} - μ_{k}^{v<...>})) \end{matrix}

…”

Section: Model Inferencementioning

confidence: 99%

“…Making use of theorem 2 of Bouveyron et al and assuming that

σ_{k 1}^{2} \to 0

, we end up with a closed‐form formulation for the evidence:

\begin{matrix} right \log p (X_{k} | v_{k}, α_{k}) & left = - \frac{{false\sum}_{i = 1}^{n_{k}} | | x_{k i}^{{\bar{v}}_{k}} - μ_{k}^{{\bar{v}}_{k}} | {false|}^{2}}{2 σ_{k 2}^{2}} - n_{k} (p - q_{k}) \log σ_{k 2} + \frac{n_{k} q_{k}}{2} \log α_{k} & right \\ right & left + \sum_{i = 1}^{n_{k}} (\log K_{(q_{k} - d_{k}) / 2} (α_{k} | | x_{k i}^{v_{k}} - μ_{k}^{v_{k}} | |) - q_{k} \log | | x_{k i}^{v_{k}} - μ_{k}^{v_{k}} | |), \end{matrix}

where K ν denotes the modified Bessel function of the second kind . The hyperparameter α k can be found by solving a univariate convex optimization problem, as in Bouveyron et al…”

Section: Model Inferencementioning

confidence: 99%

“…Regarding the prior on the sparsity pattern, while Bouveyron et al only used the uniform noninformative prior p ( v k ) ∝ 1, we extend their approach by also considering sparsity inducing priors. Specifically, we consider a product of Bernoulli distributions with the same parameter.…”

Section: Model Inferencementioning

confidence: 99%

See 4 more Smart Citations

Class‐specific variable selection in high‐dimensional discriminant analysis through Bayesian Sparsity

Orlhac

Mattei

Bouveyron

et al. 2018

Journal of Chemometrics

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Although the ongoing digital revolution in fields such as chemometrics, genomics, or personalized medicine gives hope for considerable progress in these areas, it also provides more and more high‐dimensional data to analyze and interpret. A common usual task in those fields is discriminant analysis, which however may suffer from the high dimensionality of the data. The recent advances, through subspace classification or variable selection methods, allowed to reach either excellent classification performances or useful visualizations and interpretations. Obviously, it is of great interest to have both excellent classification accuracies and a meaningful variable selection for interpretation. This work addresses this issue by introducing a subspace discriminant analysis method which performs a class‐specific variable selection through Bayesian sparsity. The resulting classification methodology is called sparse high‐dimensional discriminant analysis (sHDDA). Contrary to most sparse methods which are based on the Lasso, sHDDA relies on a Bayesian modeling of the sparsity pattern and avoids the painstaking and sensitive cross‐validation of the sparsity level. The main features of sHDDA are illustrated on simulated and real‐world data. In particular, an exemplar application to cancer characterization based on medical imaging using radiomic feature extraction is proposed.

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Section: Model Inferencementioning

confidence: 99%

Section: Model Inferencementioning

confidence: 99%

“…To do so, for each class k = 1,…, K , the noise ε is assumed to decompose as follows:

ε_{| Z = k} = v_{k} ε_{1} + {\bar{v}}_{k} ε_{2},

where

ε_{1} false| Z = k \sim scriptN ((), 0, σ_{k 1}^{2} I_{p})

is the noise of the inactive variables and

ε_{2} false| Z = k \sim scriptN ((), 0, σ_{k 2}^{2} I_{p})

is the noise of the active variables, both for the k th class. Making use of theorem 2 of Bouveyron et al and assuming that

σ_{k 1}^{2} \to 0

, we end up with a closed‐form formulation for the evidence:

\begin{matrix} right \log p (X_{k} | v_{k}, α_{k}) & left = - \frac{{false\sum}_{i = 1}^{n_{k}} | | x_{k i}^{{\bar{v}}_{k}} - μ_{k}^{{\bar{v}}_{k}} | {false|}^{2}}{2 σ_{k 2}^{2}} - n_{k} (p - q_{k}) \log σ_{k 2} + \frac{n_{k} q_{k}}{2} \log α_{k} & right \\ right & left + \sum_{i = 1}^{n_{k}} (\log K_{(q_{k} - d_{k}) / 2} (α_{k} | | x_{k i}^{v_{k}} - μ_{k}^{v<...>})) \end{matrix}

…”

Section: Model Inferencementioning

confidence: 99%

“…Making use of theorem 2 of Bouveyron et al and assuming that

σ_{k 1}^{2} \to 0

, we end up with a closed‐form formulation for the evidence:

\begin{matrix} right \log p (X_{k} | v_{k}, α_{k}) & left = - \frac{{false\sum}_{i = 1}^{n_{k}} | | x_{k i}^{{\bar{v}}_{k}} - μ_{k}^{{\bar{v}}_{k}} | {false|}^{2}}{2 σ_{k 2}^{2}} - n_{k} (p - q_{k}) \log σ_{k 2} + \frac{n_{k} q_{k}}{2} \log α_{k} & right \\ right & left + \sum_{i = 1}^{n_{k}} (\log K_{(q_{k} - d_{k}) / 2} (α_{k} | | x_{k i}^{v_{k}} - μ_{k}^{v_{k}} | |) - q_{k} \log | | x_{k i}^{v_{k}} - μ_{k}^{v_{k}} | |), \end{matrix}

where K ν denotes the modified Bessel function of the second kind . The hyperparameter α k can be found by solving a univariate convex optimization problem, as in Bouveyron et al…”

Section: Model Inferencementioning

confidence: 99%

Section: Model Inferencementioning

confidence: 99%

See 3 more Smart Citations

Class‐specific variable selection in high‐dimensional discriminant analysis through Bayesian Sparsity

Orlhac

Mattei

Bouveyron

et al. 2018

Journal of Chemometrics

Self Cite

View full text Add to dashboard Cite

show abstract

Rank‐based Bayesian variable selection for genome‐wide transcriptomic analyses

Emilie

Fleischer

Vitelli

2022

Statistics in Medicine

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Variable selection is crucial in high‐dimensional omics‐based analyses, since it is biologically reasonable to assume only a subset of non‐noisy features contributes to the data structures. However, the task is particularly hard in an unsupervised setting, and a priori ad hoc variable selection is still a very frequent approach, despite the evident drawbacks and lack of reproducibility. We propose a Bayesian variable selection approach for rank‐based unsupervised transcriptomic analysis. Making use of data rankings instead of the actual continuous measurements increases the robustness of conclusions when compared to classical statistical methods, and embedding variable selection into the inferential tasks allows complete reproducibility. Specifically, we develop a novel extension of the Bayesian Mallows model for variable selection that allows for a full probabilistic analysis, leading to coherent quantification of uncertainties. Simulation studies demonstrate the versatility and robustness of the proposed method in a variety of scenarios, as well as its superiority with respect to several competitors when varying the data dimension or data generating process. We use the novel approach to analyze genome‐wide RNAseq gene expression data from ovarian cancer patients: several genes that affect cancer development are correctly detected in a completely unsupervised fashion, showing the usefulness of the method in the context of signature discovery for cancer genomics. Moreover, the possibility to also perform uncertainty quantification plays a key role in the subsequent biological investigation.

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Spike and slab Bayesian sparse principal component analysis

Ning,

Ning

2024

Stat Comput

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Bayesian variable selection for globally sparse probabilistic PCA

Cited by 21 publications

References 67 publications

Class‐specific variable selection in high‐dimensional discriminant analysis through Bayesian Sparsity

Class‐specific variable selection in high‐dimensional discriminant analysis through Bayesian Sparsity

Rank‐based Bayesian variable selection for genome‐wide transcriptomic analyses

Spike and slab Bayesian sparse principal component analysis

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