Bayesian Lasso
$$\begin{align*}
p(Y_{o}|\beta,\phi)&=N(Y_{o}|1\alpha+X_{o}\beta,\phi^{-1} I_{n{o}})\\
\pi(\beta_{i}|\phi,\tau_{i}^{2})&=N(\beta_{i}|0, \phi^{-1}\tau_{i}^{2})\\
\pi(\tau_{i}^{2})&=Exp \left( \frac{\lambda}{2} \right)\\
\pi(\phi)&\propto \phi^{-1}\\
\pi(\alpha)&\propto 1\\
\end{align*}$$
Marginalizing over \(\alpha\) equates to centering the observations and losing a degree of freedom and working with the centered \( Y_{o} \).
Mixing over \(\tau_{i}^{2}\) leads to a Laplace or Double Exponential prior on \(\beta_{i}\) with rate parameter \(\sqrt{\phi\lambda}\) as seen by considering the characteristic function
$$\begin{align*}
\varphi_{\beta_{i}|\phi}(t)&=\int e^{jt\beta_{i}}\pi(\beta_{i}|\phi)d\beta_{i}\\
&=\int \int e^{jt\beta_{i}}\pi(\beta_{i}|\phi,\tau_{i}^{2})\pi(\tau_{i}^{2})d\tau_{i} d\beta_{i}\\
&=\frac{\lambda}{2} \int e^{-\frac{1}{2}\frac{t^{2}}{\phi}\tau_{i}^{2}}e^{-\frac{\lambda}{2}\tau_{i}^{2}}d\tau_{i}\\
&=\frac{\lambda}{\frac{t^{2}}{\phi}+\lambda}=\frac{\lambda\phi}{t^{2}+\lambda\phi}
\end{align*}$$.
EM Algorithm
The objective is to find the mode of the joint posterior \(\pi(\beta,\phi|Y_{o})\). It is easier, however, to find the joint mode of \(\pi(\beta,\phi|Y_{o},\tau^{2})\) and use EM to exploit the scale mixture representation.
$$\begin{align*}
\log \pi(\beta,\phi|Y_{o},\tau^{2})=c+ \frac{n_o+p-3}{2}\log \phi -\frac{\phi}{2}||Y_{o}-X_{o}\beta||^{2}-\sum_{i=1}^{p}\frac{\phi}{2}\frac{1}{\tau_{i}^{2}}\beta^{2}_{i}
\end{align*}$$
Expectation
The expecation w.r.t. \(\tau_{i}^{2}\) is handled as by
$$
\begin{align*}
&\frac{\lambda}{2}\int \frac{1}{\tau_{i}^{2}}\left( \frac{\phi}{2\pi\tau_{i}^{2}} \right)^{\frac{1}{2}}e^{-\frac{1}{2}\phi\beta_{i}^{2}\frac{1}{\tau_{i}^{2}}}e^{-\frac{\lambda}{2}\tau_{i}^{2}}d\tau_{i}^{2}\\
&\frac{\lambda}{2}\int \left( \frac{\phi}{2\pi[\tau_{i}^{2}]^{3}} \right)^{\frac{1}{2}}e^{-\frac{1}{2}\phi\beta_{i}^{2}\frac{1}{\tau_{i}^{2}}}e^{-\frac{\lambda}{2}\tau_{i}^{2}}d\tau_{i}^{2}\\
\end{align*}$$
This has the kernel of an Inverse Gaussian distribution with shape parameter \(\phi \beta_{i}^{2}\) and mean \(\sqrt{\frac{\phi}{\lambda}}|\beta_{i}|\)
$$\begin{align*}
&\frac{{\lambda}}{2|\beta_{i}|}\int \left( \frac{\beta_{i}^{2}\phi}{2\pi[\tau_{i}^{2}]^{3}} \right)^{\frac{1}{2}}e^{-\frac{1}{2}\phi\beta_{i}^{2}\frac{1}{\tau_{i}^{2}}}e^{-\frac{\lambda}{2}\tau_{i}^{2}}d\tau_{i}^{2}\\
&\frac{\lambda}{2|\beta_{i}|}e^{-\sqrt{\lambda\phi\beta_{i}^{2}}}\int \left( \frac{\beta_{i}^{2}\phi}{2\pi[\tau_{i}^{2}]^{3}} \right)^{\frac{1}{2}}e^{-\frac{1}{2}\phi\beta_{i}^{2}\frac{1}{\tau_{i}^{2}}}e^{-\frac{\lambda}{2}\tau_{i}^{2}}e^{\sqrt{\lambda\phi\beta_{i}^{2}}}d\tau_{i}^{2}\\
&\frac{\lambda}{2|\beta_{i}|}e^{-\sqrt{\lambda\phi\beta_{i}^{2}}}\\
\end{align*}$$
Normalization as follows
$$\begin{align*}
&\frac{\lambda}{2}\int \left( \frac{\phi}{2\pi\tau_{i}^{2}} \right)^{\frac{1}{2}}e^{-\frac{1}{2}\phi\beta_{i}^{2}\frac{1}{\tau_{i}^{2}}}e^{-\frac{\lambda}{2}\tau_{i}^{2}}d\tau_{i}^{2}\\
&\frac{\lambda}{2}\int \tau_{i}^{2}\left( \frac{\phi}{2\pi[\tau_{i}^{2}]^{3}} \right)^{\frac{1}{2}}e^{-\frac{1}{2}\phi\beta_{i}^{2}\frac{1}{\tau_{i}^{2}}}e^{-\frac{\lambda}{2}\tau_{i}^{2}}d\tau_{i}^{2}\\
\end{align*}$$
$$\begin{align*}
&\frac{\lambda}{2|\beta_{i}|}e^{-\sqrt{\lambda\phi\beta_{i}^{2}}}\sqrt{\frac{\phi}{\lambda}}|\beta_{i}|\\
\end{align*}$$
\( \Rightarrow \mathbb{E}\left[ \frac{1}{\tau_{i}^{2}} \right]=\sqrt{\frac{\lambda}{\phi^{t}}}\frac{1}{|\beta_{i}^{t}|}\).
Let \(\Lambda^{t}=diag(\sqrt{\frac{\lambda}{\phi^{t}}}\frac{1}{|\beta_{1}^{t}|}, \dots, \sqrt{\frac{\lambda}{\phi^{t}}}\frac{1}{|\beta_{p}^{t}|})\).
Maximization
$$\begin{align*}
&Q(\beta,\phi||\beta^{t},\phi^{t})=c+ \frac{n_o+p-3}{2}\log \phi -\frac{\phi}{2}||Y_{o}-X_{o}\beta||^{2} – \frac{\phi}{2}\beta^{T}\Lambda^{t}\beta\\
&=c+ \frac{n_o+p-3}{2}\log \phi -\frac{\phi}{2}||\beta-(X_{o}^{T}X_{o}+\Lambda^{t})^{-1}X_{o}^{T}Y_{o}||^{2}_{(X_{o}^{T}X_{o}+\Lambda^{t})}-\frac{\phi}{2}Y_{o}^{T}(I_{n_{o}}-X_{o}^{T}(X_{o}^{T}X_{o}+\Lambda^{t})^{-1}X_{o})Y_{o}\\
\end{align*}$$
$$\begin{align*}
\beta^{t+1}&=(X_{o}^{T}X_{o}+\Lambda^{t})^{-1}X_{o}^{T}Y_{o}\\
\end{align*}$$
$$\begin{align*}
\phi^{t+1}=\frac{n_{o}+p-3}{Y_{o}^{T}(I_{n_{o}}-X_{o}^{T}(X_{o}^{T}X_{o}+\Lambda^{t})^{-1}X_{o})Y_{o}}
\end{align*}$$
RCpp C++ Code
#include <RcppArmadillo.h> // [[Rcpp::depends(RcppArmadillo)]] using namespace Rcpp; using namespace arma; double or_log_posterior_density(int no, int p, double lasso, const Col<double>& yo, const Mat<double>& xo, const Col<double>& B,double phi); // [[Rcpp::export]] List or_lasso_em(NumericVector ryo, NumericMatrix rxo, SEXP rlasso){ //Define Variables// int p=rxo.ncol(); int no=rxo.nrow(); double lasso=Rcpp::as<double >(rlasso); //Create Data// arma::mat xo(rxo.begin(), no, p, false); arma::colvec yo(ryo.begin(), ryo.size(), false); yo-=mean(yo); //Pre-Processing// Col<double> xoyo=xo.t()*yo; Col<double> B=xoyo/no; Col<double> Babs=abs(B); Mat<double> xoxo=xo.t()*xo; Mat<double> D=eye(p,p); Mat<double> Ip=eye(p,p); double yoyo=dot(yo,yo); double deltaB; double deltaphi; double phi=no/dot(yo-xo*B,yo-xo*B); double lp; //Create Trace Matrices Mat<double> B_trace(p,20000); Col<double> phi_trace(20000); Col<double> lpd_trace(20000); //Run EM Algorithm// cout << "Beginning EM Algorithm" << endl; int t=0; B_trace.col(t)=B; phi_trace(t)=phi; lpd_trace(t)=or_log_posterior_density(no,p,lasso,yo,xo,B,phi); do{ t=t+1; lp=sqrt(lasso/phi); Babs=abs(B); D=diagmat(sqrt(Babs)); B=D*solve(D*xoxo*D+lp*Ip,D*xoyo); phi=(no+p-3)/(yoyo-dot(xoyo,B)); //Store Values// B_trace.col(t)=B; phi_trace(t)=phi; lpd_trace(t)=or_log_posterior_density(no,p,lasso,yo,xo,B,phi); deltaB=dot(B_trace.col(t)-B_trace.col(t-1),B_trace.col(t)-B_trace.col(t-1)); deltaphi=phi_trace(t)-phi_trace(t-1); } while((deltaB>0.00001 || deltaphi>0.00001) && t<19999); cout << "EM Algorithm Converged in " << t << " Iterations" << endl; //Resize Trace Matrices// B_trace.resize(p,t); phi_trace.resize(t); lpd_trace.resize(t); return Rcpp::List::create( Rcpp::Named("B") = B, Rcpp::Named("B_trace") = B_trace, Rcpp::Named("phi") = phi, Rcpp::Named("phi_trace") = phi_trace, Rcpp::Named("lpd_trace") = lpd_trace ) ; } double or_log_posterior_density(int no, int p, double lasso, const Col<double>& yo, const Mat<double>& xo, const Col<double>& B,double phi){ double lpd; lpd=(double)0.5*((double)no-1)*log(phi/(2*M_PI))-0.5*phi*dot(yo-xo*B,yo-xo*B)+0.5*(double)p*log(phi*lasso)-sqrt(phi*lasso)*sum(abs(B))-log(phi); return(lpd); }
An Example in R
rm(list=ls()) #Generate Design Matrix set.seed(3) no=100 foo=rnorm(no,0,1) sd=4 xo=cbind(foo+rnorm(no,0,sd),foo+rnorm(no,0,sd),foo+rnorm(no,0,sd),foo+rnorm(no,0,sd),foo+rnorm(no,0,sd),foo+rnorm(no,0,sd),foo+rnorm(no,0,sd),foo+rnorm(no,0,sd)) for(i in 1:40) xo=cbind(xo,foo+rnorm(no,0,sd),foo+rnorm(no,0,sd),foo+rnorm(no,0,sd),foo+rnorm(no,0,sd),foo+rnorm(no,0,sd),foo+rnorm(no,0,sd),foo+rnorm(no,0,sd)) #Scale and Center Design Matrix xo=scale(xo,center=T,scale=F) var=apply(xo^2,2,sum) xo=scale(xo,center=F,scale=sqrt(var/no)) #Generate Data under True Model p=dim(xo)[2] b=rep(0,p) b[1]=1 b[2]=2 b[3]=3 b[4]=4 b[5]=5 xo%*%b yo=xo%*%b+rnorm(no,0,1) yo=yo-mean(yo) #Run Lasso or_lasso=or_lasso_em(yo,xo,100) #Posterior Density Increasing at Every Iteration? or_lasso$lpd_trace[2:dim(or_lasso$lpd_trace)[1],1]-or_lasso$lpd_trace[1:(dim(or_lasso$lpd_trace)[1]-1),1]>=0 mean(or_lasso$lpd_trace[2:dim(or_lasso$lpd_trace)[1],1]-or_lasso$lpd_trace[1:(dim(or_lasso$lpd_trace)[1]-1),1]>=0) #Plot Results plot(or_lasso$B,ylab=expression(beta[lasso]),main="Lasso MAP Estimate of Regression Coefficients")
Park, T., & Casella, G. (2008). The Bayesian Lasso Journal of the American Statistical Association, 103 (482), 681-686 DOI: 10.1198/016214508000000337
Figueiredo M.A.T. (2003). Adaptive sparseness for supervised learning, IEEE Transactions on Pattern Analysis and Machine Intelligence, 25 (9) 1150-1159. DOI: http://dx.doi.org/10.1109/tpami.2003.1227989
Better Shrinkage Priors:
Armagan A., Dunson D.B. & Lee J. GENERALIZED DOUBLE PARETO SHRINKAGE., Statistica Sinica, PMID: 24478567
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