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Bayesian adaptive lassos with nonconvex penalization
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Griffin, Jim E. and Brown, Philip J. (2007) Bayesian adaptive lassos with nonconvex penalization. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. Working papers, Vol.2007 (No.2).

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Abstract
The lasso (Tibshirani,1996) has sparked interest in the use of penalization of
the loglikelihood for variable selection, as well as shrinkage. Recently, there have
been attempts to propose penalty functions which improve upon the Lassos properties
for variable selection and prediction, such as SCAD (Fan and Li, 2001) and
the Adaptive Lasso (Zou, 2006). We adopt the Bayesian interpretation of the Lasso
as the maximum a posteriori (MAP) estimate of the regression coefficients, which
have been given independent, double exponential prior distributions. Generalizing
this prior provides a family of adaptive lasso penalty functions, which includes the quasicauchy distribution (Johnstone and Silverman, 2005) as a special case.
The properties of this approach are explored. We are particularly interested in the
more variables than observations case of characteristic importance for data arising
in chemometrics, genomics and proteomics  to name but three. Our methodology
can give rise to multiple modes of the posterior distribution and we show how
this may occur even with the convex lasso. These multiple modes do no more
than reflect the indeterminacy of the model. We give fast algorithms and suggest
a strategy of using a set of perfectly fitting random starting values to explore
different regions of the parameter space with substantial posterior support. Simulations
show that our procedure provides significant improvements on a range of
established procedures and we provide an example from chemometrics.
Item Type:  Working or Discussion Paper (Working Paper) 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Statistics 
Library of Congress Subject Headings (LCSH):  Regression analysis 
Series Name:  Working papers 
Publisher:  University of Warwick. Centre for Research in Statistical Methodology 
Place of Publication:  Coventry 
Official Date:  2007 
Volume:  Vol.2007 
Number:  No.2 
Number of Pages:  30 
Status:  Not Peer Reviewed 
Access rights to Published version:  Open Access 
References:  Abramowitz, M. and Stegun, I. A. (Eds.) (1964) “Handbook of Mathematical 
URI:  http://wrap.warwick.ac.uk/id/eprint/35537 
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