bayesian methods for variable selection with applications to high-dimensional data...
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BAYESIAN METHODS FOR VARIABLE SELECTION WITHAPPLICATIONS TO HIGH-DIMENSIONAL DATA
Part 1: Mixture Priors for Linear Settings
Marina Vannucci
Rice University, USA
PASI-CIMAT04/28-30/2010
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 1 / 36
Mixture Priors for Linear Settings
Part 1: Mixture Priors for Linear Settings
Linear regression models (univariate and multivariate responses)
Matlab code on simulated data
Extensions to categorical responses and survival outcomes
Applications to high-throughput data from bioinformatics
Models that incorporate biological information
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 2 / 36
Mixture Priors for Linear Settings Linear Regression Models
Regression Model
Yn×1 = 1α+ Xn×pβββp×1 + ε, ε ∼ N(0, σ2I)
Introduce latent variable γγγ = (γ1, . . . , γp)′ to select variables{γj = 1 if variable j included in modelγj = 0 otherwise
Specify priors for model parameters:
βj |σ2 ∼ (1 − γj)δ0(βj) + γjN(0, σ2hj)
α|σ2 ∼ N(α0,h0σ2)
σ2 ∼ IG(ν/2, λ/2)
p(γγγ) =
p∏
j=1
wγj (1 − w)1−γj .
where δ0(·) is the Dirac function.Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 3 / 36
Mixture Priors for Linear Settings Linear Regression Models
Posterior Distribution
Combine data and prior information into a posterior distribution ⇒interest in posterior distribution
p(γγγ|Y,X) ∝ p(γγγ)∫
f (Y|X, α, βββ, σσσ)p(α|σ)p(βββ|σ, γγγ)p(σ)dαdβββdσ
p(γγγ|Y,X) ∝ g(γγγ)
|X′(γγγ)X(γγγ)|
−1/2(νλ+ S2γγγ)−(n+ν)/2p(γγγ)
X(γγγ) =
(X(γγγ)H
12(γγγ)
Ipγγγ
), Y =
(Y0
)
S2γγγ = Y′Y − Y′X(γγγ)(X
′(γγγ)X(γγγ))
−1X′(γγγ)Y
the residual sum of squares from the least squares regression of Y onX(γγγ). Fast updating schemes use Cholesky or QR decompositions withefficient algorithms to remove or add columns.
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 4 / 36
Mixture Priors for Linear Settings Linear Regression Models
Model Fitting via MCMC
With p variables there are 2p different γγγ values. We useMetropolis as stochastic search.At each MCMC iteration we generate a candidate γγγnew byrandomly choosing one of these moves:
(i) Add or Delete : randomly choose one of the indices in γγγold andchange its value.
(ii) Swap : choose independently and at random a 0 and a 1 in γγγold andswitch their values.
The proposed γγγnew is accepted with probability
min{
p(γγγnew |X,Y)
p(γγγold |X,Y),1}.
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 5 / 36
Mixture Priors for Linear Settings Linear Regression Models
Posterior inference
The stochastic search results in a list of visited models (γγγ(0), γγγ(1), . . .)and their corresponding relative posterior probabilities
p(γγγ(0)|X,Y),p(γγγ(1)|X,Y) . . .
Select variables:in the “best” models, i.e. the γγγ’s with highest p(γγγ|X,Y) orwith largest marginal posterior probabilities
p(γj = 1|X,Y) =
∫p(γj = 1, γγγ(−j)|X,Y)dγγγ(−j)
≈∑
γγγ:γj=1
p(
Y|X, γγγ(t))
p(γ(t))
or more simply by empirical frequencies in the MCMC output
p(γj = 1|X,Y) = E(γj = 1|X,Y) ≈ #{γγγ(t) = 1}
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 6 / 36
Mixture Priors for Linear Settings Linear Regression Models
Multivariate Response
Yn×q = 1α′ + Xn×pBp×q + E, Ei ∼ N(0,ΣΣΣ)
Variable selection via γγγ as
BBBj |ΣΣΣ ∼ (1 − γj)I0 + γjN(0,hjΣΣΣ),
with BBBj the j-th row of BBB and I0 a vector of point masses at 0.
Need to work with matrix-variate distributions (Dawid, 1981):
Y − 1ααα′ − XB ∼ N (In,ΣΣΣ)
ααα− ααα0 ∼ N (h0,ΣΣΣ)Bγγγ − B0γγγ ∼ N (Hγγγ,ΣΣΣ)
ΣΣΣ ∼ IW(δ,Q).
with IW an inverse-Wishart with parameters δ and Q to be specified.Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 7 / 36
Mixture Priors for Linear Settings Linear Regression Models
Posterior Distribution
Combine data and prior information into a posterior distribution ⇒interest in posterior distribution
p(γγγ|Y,X) ∝ p(γγγ)∫
f (Y|X, ααα,B,ΣΣΣ)p(ααα|ΣΣΣ)p(B|ΣΣΣ, γγγ)p(ΣΣΣ)dαααdBdΣΣΣ
p(γγγ|Y,X) ∝ g(γγγ) =
|X′(γγγ)X(γγγ)|
−q/2|Qγγγ |−(n+δ+q−1)/2p(γγγ)
X(γγγ) =
(X(γγγ)H
12(γγγ)
Ipγγγ
), Y =
(Y0
)
Qγγγ = Q + Y′Y − Y′X(γγγ)(X′(γγγ)X(γγγ))
−1X′(γγγ)Y
It can be calculated via QR-decomposition (Seber, ch.10, 1984). Useqrdelete and qrinsert algorithms to remove or add a column.
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 8 / 36
Mixture Priors for Linear Settings Linear Regression Models
Prediction
Prediction of future Y f given the corresponding Xf can be done:
as posterior weighted average of model predictions (BMA)
p(Y f |X,Y) =∑
γγγp(Yf |X,Y, γγγ)p(γγγ|X,Y)
with p(Yf |X,Y, γγγ) a matrix-variate T distribution with mean Xf Bγγγ
Yf =∑
γγγ
(XfγγγBγγγ
)p(γγγ|X,Y)
Bγγγ = (X′γγγXγγγ + H−1
γγγ )−1X′γγγY
as LS or Bayes predictions on single best models
as LS or Bayes predictions with “threshold” models (eg, “median”model) obtained from estimated marginal probabilities of inclusion.
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 9 / 36
Mixture Priors for Linear Settings Linear Regression Models
Prior Specification
Priors on ααα and ΣΣΣ vague and largely uninformative
ααα′ − ααα′0 ∼ N (h,ΣΣΣ), ααα0 ≡ 0,h → ∞,
ΣΣΣ ∼ IW(δ,Q), δ = 3,Q = k I
Choices for Hγγγ :
Hγγγ = c ∗ (X′γγγXγγγ)−1 (Zellner g-prior)
Hγγγ = c ∗ diag(X′γγγXγγγ)−1
Hγγγ = c ∗ Iγγγ
Choice of wj = p(γj = 1) : wj = w , w ∼ Beta(a,b) (sparsity). Also,choices that reflect prior information (e.g., gene networks).
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 10 / 36
Mixture Priors for Linear Settings
Advantages of Bayesian Approach
Past and collateral information through priors
n << p
Rich modeling via Markov chain Monte Carlo (MCMC) (for p large)
Optimal model averaging prediction
Extends to multivariate response
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 11 / 36
Mixture Priors for Linear Settings
Main References
GEORGE, E.I. and MCCULLOCH, R.E. (1993). Variable Selection viaGibbs Sampling. Journal of the American Statistical Association, 88,881–889.
GEORGE, E.I. and MCCULLOCH, R.E. (1997). Approaches for BayesianVariable Selection. Statistica Sinica, 7, 339–373.
MADIGAN, D. and YORK, J. (1995). Bayesian Graphical Models forDiscrete Data. International Statistical Review, 63, 215–232
BROWN, P.J., VANNUCCI, M. and FEARN, T. (1998). MultivariateBayesian Variable Selection and Prediction. Journal of the RoyalStatistical Society, Series B, 60, 627–641.
BROWN, P.J., VANNUCCI, M. and FEARN, T. (2002). Bayes modelaveraging with selection of regressors. Journal of the Royal StatisticalSociety, Series B, 64(3), 519–536.
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 12 / 36
Mixture Priors for Linear Settings
Additional References
Use of g-priors:LIANG, F., PAULO, R., MOLINA, G., CLYDE, M. and BERGER, J. (2008).Mixture of g priors for Bayes variable section. Journal of the AmericanStatistical Association, 103, 410-423.
Improving MCMC mixing:BOTTOLO, L. and RICHARDSON, S. (2009). Evolutionary stochasticsearch. Journal of Computational and Graphical Statistics, underrevision. The authors propose an evolutionary Monte Carlo schemecombined with a parallel tempering approach that prevents the chainfrom getting stuck in local modes.
Multiplicity:SCOTT, J. and BERGER, J. (2008). Bayes and empirical-Bayesmultiplicity adjustment in the variable-selection problem. The Annals ofStatistics, to appear. The marginal prior on γγγ contains a non-linearpenalty which is a function of p and therefore, as p grows, with thenumber of true variables remaining fixed, the posterior distribution of wconcentrates near 0.
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 13 / 36
Mixture Priors for Linear Settings Matlab Code
Code from my Website
bvsme fast: Bayesian Variable Selection with fast form of QRupdating
Metropolis search
gPrior or diagonal and non-diagonal selection prior
Bernoulli priors or Beta-Binomial prior
Predictions by LS, BMA and BMA with selection
http://stat.rice.edu/∼marina
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 14 / 36
Mixture Priors for Linear Settings Categorical Outcomes
Probit Models with Binary Response
Response with G = 2 classes: zi ∈ {0,1} associated with a set ofp predictors Xi , i = 1, . . . ,n.
Data augmentation: Latent (unobserved) yi linearly associatedwith the Xi ’s
yi = α+ X′iβββ + ǫi , ǫi ∼ N(0, σ2 = 1), i = 1, . . . ,n.
with intercept α and coefficient vector βββp×1.
Association
zi =
{0 if yi < 01 if otherwise
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 15 / 36
Mixture Priors for Linear Settings Categorical Outcomes
Probit Models with Multinomial Response
Response with G classes: zi ∈ {0,1, ...,G − 1} associated with aset of p predictors Xi , i = 1, · · · ,n (gene expressions).
Data augmentation: Latent (unobserved) vector Yi linearlyassociated with the Xi ’s
Yi = ααα′ + X′iB + Ei , Ei ∼ N(0,ΣΣΣ), i = 1, . . . ,n.
with intercepts ααα(G−1)×1 and coefficient matrix Bp×(G−1).
Association
zi =
{0 if yig < 0 for each gg if yig = max
1≤g≤G−1{yig}
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 16 / 36
Mixture Priors for Linear Settings Categorical Outcomes
Variable Selection
We introduce a binary latent vector γγγ for variable selection{γj = 1 if variable j discriminate the samplesγj = 0 otherwise
A mixture prior is placed on the j th row of B, given γγγ
B j ∼ (1 − γj)I0 + γjN(0, cΣΣΣ)
Assume γj ’s are independent Bernoulli variables
Combine data and priors into posterior p(γγγ|X,Y). Inference iscomplicated because response variable is latent.
ΣΣΣ ∼ IW (δ; Q), ααα ∼ N(0,h0ΣΣΣ), large h0.
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 17 / 36
Mixture Priors for Linear Settings Categorical Outcomes
MCMC Algorithm
We sample (γγγ,Y) by Metropolis within Gibbs
Metropolis step to update γγγ from [γγγ|X,Z,Y]. We update γγγ(old) toγγγ(new) by:
(a) Add/delete : randomly choose a γj and change its value.(b) Swap : randomly choose a 0 and a 1 in γγγold and switch values.
The new candidate γγγ(new) is accepted with probability
min{p(γγγ(new)|X,Z,Y)
p(γγγ(old)|X,Z,Y),1}
We sample (Y|γγγ,X,Z) from a truncated normal or t distributionwith truncation based on Z.
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 18 / 36
Mixture Priors for Linear Settings Categorical Outcomes
Posterior Inference
Select variables that are in the “best” models
γγγ∗ = argmax1≤t≤M
{p(γγγ(t)|X,Z, Y)
}, with Y =
1M
M∑
t=1
Y(t)
Select variables with largest marginal probabilities
p(γj = 1|X,Z, Y)
Predict future Yf by a posterior predictive mean
Yf =∑
γγγYf (γγγ)π(γγγ|Y,X,Z)
with Yf (γγγ) = 1α′ + Xf (γγγ)Bγγγ and ααα, Bγγγ based on Y
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 19 / 36
Mixture Priors for Linear Settings Matlab Code
Code from my website
bvsme prob: Bayesian Variable Selection for classification withfast form of QR updating
binary/multinomial/ordinal response
Metropolis search
gPrior or diagonal and non-diagonal selection prior
Bernoulli priors or Beta-Binomial prior
Predictions by LS, BMA and BMA with selection
http://stat.rice.edu/∼marina
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 20 / 36
Mixture Priors for Linear Settings Categorical Outcomes
Logit Models
More naturally interpretable in terms of odds ratios.Marginalization not possible.For binary data, a data augmented model is
zi = x′iβββ + ǫi ,
with ǫi a scale mixture of normals with marginal logistic,
ǫi ∼ N(0, λi)
λi = (2ψi)2
ψi ∼ KS,
with KS the Kolmogorov-Smirnov distribution.Variable selection is achieved by imposing mixture priors on βj ’s.Sampling schemes improve mixing by joint updates of correlatedparameters, i.e, (γγγ, βββ) using a Metropolis-Hastings with proposalthe full conditional of βββ and the add-delete-swap Metropolis for γγγ.Also, (z, λλλ) from truncated logistics and rejection sampling.
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 21 / 36
Mixture Priors for Linear Settings Survival Outcomes
Accelerated Failure Time models
We use accelerated failure time (AFT) models
log(Ti) = α+ X′iβββ + εi , i = 1, . . . ,n.
Observe yi = min(ti , ci ) and δi = I{ti ≤ ci}, where ci censoring time.
We introduce augmented data W = (w1, . . . ,wn)′ to impute the
censored survival times{
wi = log(yi) if δi = 1wi > log(yi) if δi = 0
We consider different distributional assumptions for εi .
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 22 / 36
Mixture Priors for Linear Settings Survival Outcomes
Introduce latent vector γγγ for variable selection.MCMC steps consist of(1) Metropolis search to update γγγ from f (γγγ|X,W).(2) Impute censored failure times, wi with δi = 0, from f (wi |W−i ,X, γγγ).
Inference on variables based on p(γj = 1|X, W) or p(γγγ|X, WWW ).
Prediction of survival time for future patients
Wf =∑
γγγ
(1α′ + Xf (γγγ)βγγγ
)p(γγγ|X, W).
Predictive survivor function
P(Tf > t |Xf ,X, W) ≈∑
γγγP(
W > w |Xf ,X, W, γγγ)
p(γγγ|X, W).
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 23 / 36
Mixture Priors for Linear Settings Matlab Code
Code from my website
bvsme surv: Bayesian Variable Selection for AFT models withright censoring
Metropolis search
diagonal selection prior
Bernoulli priors or Beta-Binomial prior
http://stat.rice.edu/∼marina
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 24 / 36
Mixture Priors for Linear Settings
Main References
ALBERT, J.H. and CHIB, S. (1993), ”Bayesian Analysis of Binary andPolychotomous Response Data”, JASA, 88(422), 669-679.
SHA, N., VANNUCCI, M., TADESSE, M.G., BROWN, P.J., DRAGONI, I.,DAVIES, N., ROBERTS, T. C., CONTESTABILE, A., SALMON, N.,BUCKLEY, C. and FALCIANI, F. (2004). Bayesian variable selection inmultinomial probit models to identify molecular signatures of diseasestage, Biometrics, 60, 812-819.
HOLMES, C.C. and HELD, L. (2006). Bayesian auxiliary variable modelsfor binary and multinomial regression. Bayesian Analysis, 1(1), 145-166.
SHA, N., TADESSE, M.G. and VANNUCCI, M. (2006). Bayesian variableselection for the analysis of microarray data with censored outcome.Bioinformatics, 22(18), 2262-2268.
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 25 / 36
Mixture Priors for Linear Settings Applications to High-Throughput Data
DNA microarrays
DNA fragments arrayed on glass slide or chip
Parallel quantification of thousands of genes in a singleexperiment
Identify biomarkers for treatment strategies and diagnostic tools
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 26 / 36
Mixture Priors for Linear Settings Applications to High-Throughput Data
Statistical Analyses
Identification of differentially expressed genes (gene selection forsample classification)
Discovery of subtypes of tissue/disease that respond differently totreatment (gene selection and sample clustering)
Prediction of continuous responses (clinical outcome, survivaltime)
The major challenge is the high-dimensionality of the data.p ≫ n
Widely used approaches: t-test, ANOVA, Cox model on singlegenes (ignores joint effect of genes; multiple testing issue) ordimension reduction techniques, PCA, PLS (leads to linearcombinations; cannot assess original variables). Lately emphasison subset selection methods (LASSO, Bayesian models).
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 27 / 36
Mixture Priors for Linear Settings Applications to High-Throughput Data
Identification of Biomarkers of Disease Stage
Data consist of 11 early stage (duration less than 2 years) and 9late stage (over 15 years) rheumatoid arthritis patients.
mRNA samples extracted from peripheral blood and hybridized tocustom-made cDNA arrays.
755 gene expressions. Logged and std-ed data
Bernoulli prior with expected model size 10
We ran six MCMC chains with very different starting γ vectors.
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 28 / 36
Mixture Priors for Linear Settings Applications to High-Throughput Data
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Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 29 / 36
Mixture Priors for Linear Settings Applications to High-Throughput Data
Selected genes by best 10 models of each chain and of their union
Small sets of functionally related genes involved in cytoskeletonremodeling and motility, and with lymphocytes’ ability to respondto activation.
.05(1/20) misclassification error.
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Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 30 / 36
Mixture Priors for Linear Settings Protein Mass Spectrometry
Peak Selection for Protein Mass spectra
Cancer classification based on mass spectra at 15,000 m/z ratios.x-axis: ratio of weight of a molecule to its electrical charge (m/z),y-axis: intensity ∼ abundance of that molecule in the sample.Goal: identification of peaks (proteins or protein fragments)related to a clinical outcome or disease status
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 31 / 36
Mixture Priors for Linear Settings Protein Mass Spectrometry
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Serum spectra on 50 (10+11+29) subjects (SELDI-TOF). Ordinalresponse - tumor grade (ovarian cancer).
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 32 / 36
Mixture Priors for Linear Settings Protein Mass Spectrometry
Data Processing
Preprocessing:
- Baseline subtraction- Denoising (often by wavelets)- Peak identification- Normalization- Alignment
Analysis:
- Model fitting- Validation
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 33 / 36
Mixture Priors for Linear Settings Protein Mass Spectrometry
Data processing results in 39 identified peaks.Probit model with Bayesian variable selection applied to 39 peaks.“Best” models with around 7 peptides (6 common).Misclassification errors (2/10,8/11,9/29)
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Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 34 / 36
Mixture Priors for Linear Settings Survival Outcomes
Case Study on Breast Cancer (van’t Veer et al. (2002))
Microarray data on 76 patients, 33 who developed distantmetastases within 5 years and 43 who did not (censored).Training and test sets (38+38 patients). About 5,000 genes.MSE=1.9 (with 11 genes).
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Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 35 / 36
Mixture Priors for Linear Settings
Main References
SHA, N., VANNUCCI, M., TADESSE, M.G., BROWN, P.J., DRAGONI, I.,DAVIES, N., ROBERTS, T. C., CONTESTABILE, A., SALMON, N.,BUCKLEY, C. and FALCIANI, F. (2004). Bayesian variable selection inmultinomial probit models to identify molecular signatures of diseasestage, Biometrics, 60, 812-819.
KWON, D.W., TADESSE, M.G., SHA, N., PFEIFFER, R.M. andVANNUCCI, M. (2007). Identifying biomarkers from mass spectrometrydata with ordinal outcome. Cancer Informatics, 3, 19–28.
SHA, N., TADESSE, M.G. and VANNUCCI, M. (2006). Bayesian variableselection for the analysis of microarray data with censored outcome.Bioinformatics, 22(18), 2262-2268.
TADESSE, M.G., SHA, N., KIM, S. and VANNUCCI, M. (2006).Identification of biomarkers in classification and clustering ofhigh-throughput data. In Bayesian Inference for Gene Expression andProteomics, K. Anh-Do, P. Mueller and M. Vannucci (Eds). CambridgeUniversity Press.
Marina Vannucci (Rice University, USA) Bayesian Variable Selection (Part 1) PASI-CIMAT 04/28-30/2010 36 / 36