Transcript Bayesian Factor Regression Models in the “Large p, Small n

Bayesian Factor Regression Models
in the “Large p, Small n” Paradigm
Mike West, Duke University
Presented by: John Paisley
Duke University
Outline
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Empirical Factor Regression (SVD)
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Latent Factor Regression
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Sparse Factor Regression
Linear Regression &
Empirical Factor Regression
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Linear Regression
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SVD Regression
D is a diagonal matrix of singular values
Empirical Factor Regression
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By definition,
Regression is now done in factor space using
generalized shrinkage (ridge regression)
priors on , e.g. RVM
Problem of inversion:
many-to-one mapping
has
is canonical “least-norm” inverse
Example: Biscuit Dough Data
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NIR spectroscopy reflectance values are predictors
Response is fat content of dough samples
39 training, 39 testing: data are pooled and testing data
responses treated as missing values to be imputed
Top 16 factors used, based on size of singular values
Example: Biscuit Dough Data (2)
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Left: Fitted and predicted vs true values
Right: Least-norm inverse of beta
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~ 1700 nm range is absorbance region for fat
As can be seen, solution is not sparse
Latent Factor Regression
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Loosen
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to
Under proper constraints on B, this finds common structure in X
and isolates idiosyncrasies to noise
Now, variation in X has less effect on y
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The implied prior is 
When variance, Phi  0,
this reverts to empirical
linear regression
Sparse Latent Factor Regression
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WRT gene expression profiling, “multiple biological
factors underlie patterns of gene expression variation, so
latent factor approaches are natural – we imagine that
latent factors reflect individual biological functions… This
is a motivating context for sparse models.”
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Columns of B represents the genes involved in a particular
biological factor.
Rows of B represent a particular gene’s involvement across
biological factors.
Example: Gene Expression Data
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p = 6128 genes measured using Affymetrix DNA microarrays
n = 49 breast cancer tumor samples
k = 25 factors
Factor 3 separates by
red: estrogen receptor
positive tumors
blue: ER negative
Example: Gene Expression Data
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Comparison with results obtained using empirical SVD factors
Conclusion
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Sparse factor regression modeling is a
promising framework for dimensionality
reduction of predictors.
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Only those factors that are relevant (e.g.
factor 3) are of interest. Therefore, only those
genes with non-zero values in that column of
B are meaningful.