Bayesian Factor Regression Models in the

1
Bayesian Factor Regression Models in the Large
p, Small n ParadigmMike West, Duke University

• Presented by John Paisley
• Duke University

2
Outline

• Empirical Factor Regression (SVD)
• Latent Factor Regression
• Sparse Factor Regression

3
Linear Regression Empirical Factor Regression

• Linear Regression

• SVD Regression

D is a diagonal matrix of singular values
4
Empirical Factor Regression

• By definition,
• Regression is now done in factor space using
  generalized shrinkage (ridge regression) priors
  on , e.g. RVM
• Problem of inversion has many-to-one mapping
• is canonical least-norm inverse

5
Example Biscuit Dough Data

• 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

6
Example Biscuit Dough Data (2)

• Left Fitted and predicted vs true values
• Right Least-norm inverse of beta
• 1700 nm range is absorbance region for fat
• As can be seen, solution is not sparse

7
Latent Factor Regression

• Loosen 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
• The implied prior is ?
• When variance, Phi ? 0,
• this reverts to empirical
• linear regression

8
Sparse Latent Factor Regression

• 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.
• Columns of B represents the genes involved in a
  particular biological factor.
• Rows of B represent a particular genes
  involvement across biological factors.

9
Example Gene Expression Data

• 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

10
Example Gene Expression Data

• Comparison with results obtained using empirical
  SVD factors

11
Conclusion

• Sparse factor regression modeling is a promising
  framework for dimensionality reduction of
  predictors.
• 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.