The post Send Lines of Code from Vim to R/Julia/Python REPL appeared first on Lindons Log.

]]>Editing code in anything other than Vim seems to drain my coding stamina rapidly, but when working with a language such as R that has an interactive REPL (read evaluate print loop) it is often useful to send lines of code thereto for purposes of debugging. I have used the Vim-R-Plugin for some time but recently my differences with R have led me to Julia, and so I have been looking for a way to send code from Vim to Julia. My search has resulted in vim-slime.

vim-slime is absolutely great, it’s a plugin for vim that lets you send text to screen or tmux. This is perfect for programming any language that has an REPL, simply load a session up in a tmux, open up vim, tell it what tmux window and session you want to send text to and away you go. This will work for any language!

A great post can be found here.

I’ve long since desired to be able to do this. It would be great if you could edit the file locally, but have the Julia session execute on a remote machine, say my office computer, so that all the heavy lifting is done by the remote machine allowing me to get on with my own personal things on my local machine. Vim-slime provides an elegant solution. Simply start a tmux session, ssh to your remote machine, start up an interactive REPL of the language of your choice, open another terminal locally and start vim on your code and now you can send lines to your remote R/Julia/Python session! This has dramatically improved my workflow as my office computer is much more powerful than my laptop.

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]]>The post C++ Merge Sort Algorithm appeared first on Lindons Log.

]]>#include <random> #include <iostream> #include <vector> #include <algorithm> using namespace std; vector<int> merge_sort(const vector<int>& input) { if(input.size()<=1) return input; vector<int> output(input.size()); //Split Vector// int midpoint=0.5*input.size(); vector<int> input_left(input.begin(),input.begin()+midpoint); vector<int> input_right(input.begin()+midpoint,input.end()); input_left=merge_sort(input_left); input_right=merge_sort(input_right); merge(input_left.begin(),input_left.end(),input_right.begin(),input_right.end(),output.begin()); return output; } int main(){ //Create unsorted vector of ints vector<int> unsorted(40); iota(unsorted.begin(),unsorted.end(),-20); shuffle(unsorted.begin(),unsorted.end(),default_random_engine()); //Perform merge_sort// vector<int> sorted=merge_sort(unsorted); //Display results// cout << "Unsorted: " << endl; for(auto value:unsorted) cout << value << " "; cout << endl; cout << "Sorted: " << endl; for(auto value:sorted) cout << value << " "; cout << endl; }

Unsorted:

-2 14 7 -15 -9 -17 -8 -1 13 1 -10 -7 16 -19 6 2 -12 -11 8 -18 -14 10 5 4 17 12 15 -16 -5 18 -4 -3 -6 -20 0 3 9 -13 11 19

Sorted:

-20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

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]]>The post Generate Random Inverse Gaussian in R appeared first on Lindons Log.

]]>#include <RcppArmadillo.h> // [[Rcpp::depends(RcppArmadillo)]] using namespace Rcpp; using namespace arma; // [[Rcpp::export]] Col<double> rrinvgauss(int n, double mu, double lambda){ Col<double> random_vector(n); double z,y,x,u; for(int i=0; i<n; ++i){ z=R::rnorm(0,1); y=z*z; x=mu+0.5*mu*mu*y/lambda - 0.5*(mu/lambda)*sqrt(4*mu*lambda*y+mu*mu*y*y); u=R::runif(0,1); if(u <= mu/(mu+x)){ random_vector(i)=x; }else{ random_vector(i)=mu*mu/x; }; } return(random_vector); }

It seems to be faster than existing implementations such as rig from mgcv and rinvgauss from statmod packages.

library(Rcpp) library(RcppArmadillo) library(rbenchmark) library(statmod) library(mgcv) sourceCpp("rrinvgauss.cpp") n=10000 benchmark(rig(n,1,1),rinvgauss(n,1,1),rrinvgauss(n,1,1),replications=100)

rename rrinvgauss as desired.

The post Generate Random Inverse Gaussian in R appeared first on Lindons Log.

]]>The post Generalized Double Pareto Priors for Regression appeared first on Lindons Log.

]]>Consider the regression model \(Y=X\beta+\varepsilon\) where we put a generalized double pareto distribution as the prior on the regression coefficients \(\beta\). The GDP distribution has density

$$\begin{equation}

f(\beta|\xi,\alpha)=\frac{1}{2\xi}\left( 1+\frac{|\beta|}{\alpha\xi} \right)^{-(\alpha+1)}.

\label{}

\end{equation}$$

The GDP distribution can be conveniently represented as a scale mixture of normals. Let

$$\begin{align*}

\beta_{i}|\phi,\tau_{i} &\sim N(0,\phi^{-1}\tau_{i})\\

\tau_{i}|\lambda_{i}&\sim Exp(\frac{\lambda_{i}^{2}}{2})\\

\lambda_{i}&\sim Ga(\alpha,\eta)\\

\end{align*}$$

then \(\beta|\phi \sim GDP(\xi=\frac{\eta}{\sqrt{\phi}\alpha},\alpha)\).

To see this first note that \(\beta_{i}|\phi,\lambda_{i}\) has a Laplace or Double Exponential distribution with rate parameter \(\sqrt{\phi}\lambda_{i}\).

$$\begin{align*}

p(\beta_{i}|\phi,\lambda_{i})&=\int p(\beta_{i}|\phi,\tau_{i})p(\tau_{i}|\lambda_{i})d\tau_{i}\\

\psi(t)&=\int e^{it\beta_{i}} \int p(\beta_{i}|\phi,\tau_{i})p(\tau_{i}|\lambda_{i})d\tau_{i} d\beta_{i}\\

&=\int \int e^{it\beta_{i}}p(\beta_{i}|\phi,\tau_{i})d\beta_{i}p(\tau_{i}|\lambda_{i})d\tau_{i}\\

&=\int e^{-\frac{1}{2}\frac{\tau_{i}}{\phi}t^{2}}p(\tau_{i}|\lambda_{i})d\tau_{i}\\

&=\frac{\lambda_{i}^{2}}{2} \int e^{-\frac{1}{2}(\frac{t^{2}}{\phi}+\frac{\lambda_{i}^{2}}{2})\tau_{i}}d\tau_{i}\\

&=\frac{\phi\lambda_{i}^{2}}{t^{2}+\phi\lambda_{i}^{2}},

\end{align*}$$

which is the characteristic function of a Double Exponential distribution with rate parameter \(\sqrt{\phi}\lambda_{i}\).

Lastly

$$\begin{align*}

p(\beta_{i}|\phi)&=\int p(\beta_{i}|\phi,\lambda_{i})p(\lambda_{i})d\lambda_{i}\\

&=\frac{1}{2}\sqrt{\phi}\frac{\eta^{\alpha}}{\Gamma(\alpha)}\frac{\Gamma(\alpha+1)}{(\eta+\sqrt{\phi}|\beta_{i}|)^{\alpha+1}}\\

&=\frac{1}{2}\frac{\sqrt{\phi}\alpha}{\eta}\left( 1+\frac{\sqrt{\phi}\alpha}{\eta}\frac{|\beta_{i}|}{\alpha} \right)^{-(\alpha+1)},

\end{align*}$$

which is the density of a \(GDP(\xi=\frac{\eta}{\sqrt{\phi}\alpha},\alpha)\).

\(\tau_{i}\) and \(\lambda_{i}\) are treated as missing data for each \(i\).

\begin{align*}

Q(\beta,\phi||\beta^{(t)},\phi^{(t)})&=c+\mathbb{E}_{\tau,\lambda}\left[ \log p(\beta,\phi|Y,\tau,\lambda)|\beta^{(t)},\phi^{(t)} \right]\\

&=\frac{n+p-3}{2}\log\phi – \frac{\phi}{2}||Y-X\beta||^{2}-\frac{\phi}{2}\sum_{i=1}^{p}\beta_{i}^{2}\mathbb{E}\left[ \frac{1}{\tau_{i}} \right]\\

\end{align*}

For the iterated expectation one needs the distribution \(\tau_{i}|\lambda_{i},\beta_{i},\phi\) and \(\lambda_{i}|\beta_{i},\phi\).

\begin{align*}

p(\tau_{i}|\beta_{i},\lambda_{i},\phi)&\propto p(\beta_{i}|\phi,\tau_{i})p(\tau_{i}|\lambda_{i})\\

&\propto \left( \frac{1}{\tau_{i}} \right)^{\frac{1}{2}}e^{-\frac{1}{2}(\frac{\phi \beta_{i}^{2}}{\tau_{i}}+\lambda_{i}^{2}\tau_{i})}

\end{align*}

This is the kernel of a Generalized Inverse Gaussian distribution, specifically \(p(\tau_{i}|\beta_{i},\lambda_{i},\phi)=GIG(\tau_{i}:\lambda_{i}^{2},\phi \beta_{i}^{2},\frac{1}{2})\).

By a standard change of variables it follows that \(p(\frac{1}{\tau_{i}}|\beta_{i},\lambda_{i},\phi)=IG(\frac{1}{\tau_{i}}:\sqrt{\frac{\lambda_{i}^{2}}{\phi \beta_{i}^{2}}},\lambda_{i}^{2})\) and so \(\mathbb{E}\left[ \frac{1}{\tau_{i}}|\lambda_{i},\beta^{(t)},\phi^{(t)} \right]=\frac{\lambda_{i}}{\sqrt{\phi^{(t)}}|\beta_{i}^{(t)}|}\).

Recall that \(p(\beta_{i}|\phi,\lambda_{i})\) has a double exponential distribution with rate \(\sqrt{\phi}\lambda_{i}\).

Hence from \(p(\lambda_{i}|\beta_{i},\phi)\propto p(\beta_{i}|\lambda_{i},\phi)p(\lambda_{i})\) it follows that \(\lambda_{i}|\beta_{i},\phi \sim Ga(\alpha+1,\eta+\sqrt{\phi}|\beta_{i}|)\), then performing the expectation with respect to \(\lambda_{i}\) yields

\begin{align*}

\mathbb{E}\left[ \frac{1}{\tau_{i}}|\beta^{(t)},\phi^{(t)} \right]=\left( \frac{\alpha+1}{\eta+\sqrt{\phi^{t}}|\beta_{i}^{(t)}|} \right)\left( \frac{1}{\sqrt{\phi^{(t)}}|\beta_{i}^{(t)}|} \right)

\end{align*}

Writing \(D^{(t)}=diag(\mathbb{E}[\frac{1}{\tau_{1}}],\dots,\mathbb{E}[\frac{1}{\tau_{p}}])\) the function to maximize is

\begin{align*}

Q(\beta,\phi||\beta^{(t)},\phi^{(t)})&=c+\mathbb{E}_{\tau,\lambda}\left[ \log p(\beta,\phi|Y,\tau,\lambda)|\beta^{(t)},\phi^{(t)} \right]\\

&=\frac{n+p-3}{2}\log\phi – \frac{\phi}{2}||Y-X\beta||^{2}-\frac{\phi}{2}\beta^{‘}D^{(t)}\beta,\\

\end{align*}

which is maximized by letting

\begin{align*}

\beta^{(t+1)}&=(X^{‘}X+D^{(t)})^{-1}X^{‘}Y\\

\phi^{(t+1)}&=\frac{n+p-3}{Y^{‘}(I-X(X^{‘}X+D^{(t)})^{-1}X^{‘})Y}\\

&=\frac{n+p-3}{||Y-X\beta^{(t+1)}||^{2}+\beta^{(t+1)’}D^(t)\beta^{(t+1)}}\\

\end{align*}

#include <RcppArmadillo.h> // [[Rcpp::depends(RcppArmadillo)]] using namespace Rcpp; using namespace arma; double gdp_log_posterior_density(int no, int p, double alpha, double eta, const Col<double>& yo, const Mat<double>& xo, const Col<double>& B,double phi); // [[Rcpp::export]] List gdp_em(NumericVector ryo, NumericMatrix rxo, SEXP ralpha, SEXP reta){ //Define Variables// int p=rxo.ncol(); int no=rxo.nrow(); double eta=Rcpp::as<double >(reta); double alpha=Rcpp::as<double >(ralpha); //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)=gdp_log_posterior_density(no,p,alpha,eta,yo,xo,B,phi); do{ t=t+1; Babs=abs(B); D=diagmat(sqrt(((eta+sqrt(phi)*Babs)%(sqrt(phi)*Babs))/(alpha+1))); B=D*solve(D*xoxo*D+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)=gdp_log_posterior_density(no,p,alpha,eta,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 gdp_log_posterior_density(int no, int p, double alpha, double eta, const Col<double>& yo, const Mat<double>& xo, const Col<double>& B,double phi){ double lpd; double xi=eta/(sqrt(phi)*alpha); lpd=(double)0.5*((double)no-1)*log(phi/(2*M_PI))-p*log(2*xi)-(alpha+1)*sum(log(1+abs(B)/(alpha*xi)))-0.5*phi*dot(yo-xo*B,yo-xo*B)-log(phi); return(lpd); }

rm(list=ls()) library(Rcpp) library(RcppArmadillo) sourceCpp("src/gdp_em.cpp") #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 GDP gdp=gdp_em(yo,xo,100,100) #Posterior Density Increasing at Every Iteration? gdp$lpd_trace[2:dim(gdp$lpd_trace)[1],1]-gdp$lpd_trace[1:(dim(gdp$lpd_trace)[1]-1),1]>=0 mean(gdp$lpd_trace[2:dim(gdp$lpd_trace)[1],1]-gdp$lpd_trace[1:(dim(gdp$lpd_trace)[1]-1),1]>=0) #Plot Results plot(gdp$B,ylab=expression(beta[GDP]),main="GDP MAP Estimate of Regression Coefficients")

WEST M. (1987). On scale mixtures of normal distributions, Biometrika, 74 (3) 646-648. DOI: http://dx.doi.org/10.1093/biomet/74.3.646

Artin Armagan, David Dunson, & Jaeyong Lee (2011). Generalized double Pareto shrinkage Statistica Sinica 23 (2013), 119-143 arXiv: 1104.0861v4

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

Also see this similar post on the Bayesian lasso.

The post Generalized Double Pareto Priors for Regression appeared first on Lindons Log.

]]>The post EM Algorithm for Bayesian Lasso R Cpp Code appeared first on Lindons Log.

]]>$$\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*}$$.

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*}$$

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 \(\Lambda^{t}=diag(\sqrt{\frac{\lambda}{\phi^{t}}}\frac{1}{|\beta_{1}^{t}|}, \dots, \sqrt{\frac{\lambda}{\phi^{t}}}\frac{1}{|\beta_{p}^{t}|})\).

$$\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*}$$

#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); }

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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]]>The post Compile R and OpenBLAS from Source Guide appeared first on Lindons Log.

]]>2.1 Get R

2.2 Specific Instructions for DSS Users

3. Validation

4. Benchmark

This guide is intended to aid any R and Linux user who desires a threaded version of BLAS. In particular I hope this will allow other grad students, who like me do not have many user privileges on their office computer, to follow suit and exploit multiple cores to speed up their linear algebra computations within R. The following will be performed on **Scientific Linux 6.4** but has should be completely general. If you are a **Ubuntu** user, then there is an elegant and streamlined process for changing BLAS libraries and a recommended post about it here. I use **Fedora** on my laptop, and the following has also been tested thereupon.

My office computer has a quadcore processor with two threads per core but I also have access to a departmental computer with 4 sockets and 12 cores per socket (1 thread per core), so it really makes sense to use a threaded version of BLAS. If you are curious about the hardware on your own computer you can run the command “cat /proc/cpuinfo” or “lscpu”.

Unfortunately my office computer is part of a network upon which I do not have permissions to change ‘/usr/lib64/R/lib/libRblas.so’. Moreover R appears to be running serially: if you start up R and get the PID (process ID) from ‘top’ or ‘ps aux | grep R’ or something and then execute ‘cat /proc/PID/status | grep Threads’ you can see there is only one thread available.

[msl33@cabbage ~]$ cat /proc/13605/status | grep Threads Threads: 1

(where 13605 was the process ID of my R process. That is using the default R on the network. One could appeal to the network administrator to change things for you but they probably won’t because a parallel BLAS implementation may cause problems for other users who require a serial BLAS, such as those that use the multicore environment to perform inherently parallel algorithms such as parallel tempering instead of using idle cores to speed up the linear algebra. There are also some known conflicts with the multicore package in R. There is, however, nothing stopping the user from compiling one’s own custom R build in one’s home directory and just changing the executable path thereto. In addition, you then have the power and freedom customize R to your needs – at the moment I have some very large matrices which would benefit from a threaded BLAS but at some point I may want to revert to a tuned serial BLAS such at ATLAS for certain parallel algorithms.

Firstly, go ahead and create a directory in which to keep all your custom software.

[msl33@cabbage ~]$ pwd /home/grad/msl33 [msl33@cabbage ~]$ mkdir software

Make a directory “openblas” in the “software directory.

[msl33@cabbage ~]$ cd software/ [msl33@cabbage software]$ mkdir openblas

Next, grab the tarball from the OpenBLAS homepage. Change directory into where you downloaded the tarball and extract the files from it.

[msl33@cabbage ~]$ cd Downloads/ [msl33@cabbage Downloads]$ tar -xvf xianyi-OpenBLAS-v0.2.9-0-gf773f49.tar.gz

While this is running, fill a kettle with some water and turn it on, this stage is very important.

Change directory into where you extracted the files and verify that NO_AFFINITY=1 is uncommented in the Makefile.rule. If so proceed and run make.

[msl33@cabbage ~/Downloads]$ cd xianyi-OpenBLAS-347dded/ [msl33@cabbage xianyi-OpenBLAS-347dded]$ cat Makefile.rule | grep NO_AFFINITY NO_AFFINITY = 1 [msl33@cabbage xianyi-OpenBLAS-347dded]$ make

Now is a good time to “make” some tea with the water prepared earlier. When done successfully one will see

Now, as instructed above, install to the “software” directory made earlier.

[msl33@cabbage xianyi-OpenBLAS-347dded]$ make PREFIX=/home/grad/msl33/software/openblas install ... Install OK!

In openblas/lib there will be a file “libopenblas.so”, needed for later. That’s it for openblas, next we will do R.

Let’s create an R directory in software. Go onto the R homepage, then download, then choose a mirror and grab the tarball of the latest version. Download it to your “software” directory and extract it as before with “tar -xvf R-3.1.1.tar.gz”. Once extracted, remove the tarball and change directory into R-3.1.1. Before running the configure script one might bring some customizations into consideration in the name of efficiency. One might consider upping the optimization level from 2 to 3 and adding march or mtune by editing “config.site” and changing “## CFLAGS=” on line 53 to “CFLAGS=’-O3 -march=native'” and making similar changes for FFLAGS and CXXFLAGS. It is noted in the R Installation and Administration documentation that these can produce worthwhile speedups but come with a warning that the build will be less reliable, with segfaults and numerical errors creeping in. It is safest to leave things regular (reccommended link) but I’ll take the risk. Now, if you are not using a computer on the duke statistical science network, run the configure script, otherwise see the additional instructions before running configure.

[msl33@cabbage R-3.1.1]$ ./configure --prefix=/home/grad/msl33/software/R --enable-R-shlib --enable-BLAS-shlib --enable-memory-profiling --with-tcltk=no

[On the DSS computers some further instructions are required to locate headers and libraries. The first time I tried to make on my office computer I encountered this error. “jni.h” could not be found. The error was resolved by locating it and then export the environment variable JAVA_HOME.

[msl33@cabbage software]$ locate jni.h /usr/lib/jvm/java-1.7.0-sun-1.7.0.11/include/jni.h [msl33@cabbage software]$ export JAVA_HOME=/usr/lib/jvm/java-1.7.0-sun-1.7.0.11/

In addition, when running the configure script the readline headers/libs could not be found. We’ll just borrow them from some other software. Add to CFLAGS, FFLAGS, CXXFLAGS “-I/opt/EPD_Free/include -L/opt/EPD_Free/lib” in addition to any other flags that you have set. Also make a lib directory and copy them in.

[msl33@cabbage R-3.1.1]$ mkdir lib [msl33@cabbage R-3.1.1]$ cp /opt/EPD_Free/lib/libreadline.* lib/ [msl33@cabbage R-3.1.1]$ cp /opt/EPD_Free/lib/libncurses* lib/

Now run the configure line above.]

Once the configure has completed, you’ll see a summary below like

Now issue the command “make”, it will take some time. Once make has finished, you can execute “make install” to populate the software/R directory created earlier but you don’t need to. Change directories to lib and make a backup of libRblas.so and create a symbolic link to the openblas library that was made earlier.

[msl33@cabbage ~]$ cd software/R-3.1.1/lib [msl33@cabbage lib]$ pwd /home/grad/msl33/software/R-3.1.1/lib [msl33@cabbage lib]$ mv libRblas.so libRblas.so.keep [msl33@cabbage lib]$ ln -s /home/grad/msl33/software/openblas/lib/libopenblas.so libRblas.so

That was the last step.

The R executable in the bin directory should now use openblas. Note this is the R executable you now need to run in order to use the custom built R with openblas. Just typing R in terminal will load the old /usr/lib64… which we students didn’t have the permissions to alter. You can, however, create an alias in your .bashrc file by inserting the line ‘alias R=”/home/grad/msl33/software/R-3.1.1/bin/./R”‘. Now when you type R in a terminal it will load the new R and not the old one. One can check that R executable depends on the correct linked shared blas library with the “ldd” command.

[msl33@cabbage bin]$ pwd /home/grad/msl33/software/R-3.1.1/bin [msl33@cabbage bin]$ ./R CMD ldd exec/./R | grep blas libRblas.so => /home/grad/msl33/software/R-3.1.1/lib/libRblas.so (0x00007f62e3fb7000) [msl33@cabbage bin]$ ls -lt ../lib | grep openblas lrwxrwxrwx 1 msl33 grad 53 Jul 16 15:35 libRblas.so -> /home/grad/msl33/software/openblas/lib/libopenblas.so

In addition, execute “./R” from the “bin” directory (or just R if you set up the alias) and grab the process id.

[msl33@cabbage bin]$ ps aux | grep R | grep software | awk '{print $2}' 2412 [msl33@cabbage bin]$ cat /proc/`ps aux | grep R | grep software | awk '{print $2}'`/status | grep Threads Threads: 8 [msl33@cabbage bin]$

Evidently the R session now has 8 threads available. Finally, lets perform an eigen-decomposition and look at the cpu usage using top. You’ll see it light up all of your cores.

Using this benchmark the reference BLAS took 32.1 seconds whilst openBLAS took 7.1 seconds.

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]]>The post Monotonicity of EM Algorithm Proof appeared first on Lindons Log.

]]>Here the monotonicity of the EM algorithm is established.

$$ f_{o}(Y_{o}|\theta)=f_{o,m}(Y_{o},Y_{m}|\theta)/f_{m|o}(Y_{m}|Y_{o},\theta)$$

$$ \log L_{o}(\theta)=\log L_{o,m}(\theta)-\log f_{m|o}(Y_{m}|Y_{o},\theta) \label{eq:loglikelihood} $$

where \( L_{o}(\theta)\) is the likelihood under the observed data and \(L_{o,m}(\theta)\) is the likelihood under the complete data. Taking the expectation of the second line with respect to the conditional distribution of \(Y_{m}\) given \(Y_{o}\) and the current parameters \(\theta^{k}\) yields

$$\log L_{o}(\theta)= \mathbb{E}_{Y_{m}}\left[\log L_{o,m}(\theta)|Y_{o},\theta^{k}\right]-\mathbb{E}_{Y_{m}}\left[\log f_{m|o}(Y_{m}|Y_{o},\theta)|Y_{o},\theta^{k} \right]$$

which is used to construct the difference between the log-likelihood of a new value of \(\theta\) and the current value \(\theta^{k}\) as

\begin{equation}

\begin{split}

\log L_{o}(\theta)-&\log L_{o}(\theta^{k})=\mathbb{E}_{Y_{m}}\left[ \log L_{o,m}(\theta)|Y_{o},\theta^{k}\right]-\mathbb{E}_{Y_{m}}\left[ \log L_{o,m}(\theta^{k})|Y_{o},\theta^{k}\right] \\

+&\mathbb{E}_{Y_{m}}\left[ \log f_{m|o}(Y_{m}|Y_{o},\theta^{k})|Y_{o},\theta^{k} \right]-\mathbb{E}_{Y_{m}}\left[ \log f_{m|o}(Y_{m}|Y_{o},\theta)|Y_{o},\theta^{k} \right],\\

\end{split}

\end{equation}

or by adopting common notation as

\begin{equation}

\log L_{o}(\theta)-\log L_{o}(\theta^{k})=Q(\theta;\theta^{k})-Q(\theta^{k};\theta^{k})+H(\theta^{k};\theta^{k})-H(\theta;\theta^{k}).\\

\end{equation}

Consider the last two “\( H\)” terms, then by Jensen’s inequality

\begin{align*}

&-\mathbb{E}_{Y_{m}}\left[ \log f_{m|o}(Y_{m}|Y_{o},\theta)- \log f_{m|o}(Y_{m}|Y_{o},\theta^{k})|Y_{o},\theta^{k} \right]\\

&=-\mathbb{E}_{Y_{m}}\left[\log \frac{ f_{m|o}(Y_{m}|Y_{o},\theta)}{ f_{m|o}(Y_{m}|Y_{o},\theta^{k})}|Y_{o},\theta^{k} \right]\\

&\geq-\log \mathbb{E}_{Y_{m}}\left[ \frac{ f_{m|o}(Y_{m}|Y_{o},\theta)}{ f_{m|o}(Y_{m}|Y_{o},\theta^{k})}|Y_{o},\theta^{k} \right]\\

&=-\log \int f_{m|o}(Y_{m}|Y_{o},\theta)dY_{m}\\

&=0 \; \; \; \; \; \; \;\forall \theta\in \Theta.

\end{align*}

It follows that \(\log L_{o}(\theta)-\log L_{o}(\theta^{k})\geq 0\) by choosing \(\theta\) such that \(Q(\theta;\theta^{k})-Q(\theta^{k};\theta^{k})\geq 0\).

Ruslan R Salakhutdinov, Sam T Roweis, & Zoubin Ghahramani (2012). On the Convergence of Bound Optimization Algorithms arXiv arXiv: 1212.2490v1

Wu C.F.J. (1983). On the Convergence Properties of the EM Algorithm, The Annals of Statistics, 11 (1) 95-103. DOI: 10.1214/aos/1176346060

McLachlan G. & Peel D. DOI: 10.1002/0471721182

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]]>The post C++11 versus R Standalone Random Number Generation Performance Comparison appeared first on Lindons Log.

]]>#define MATHLIB_STANDALONE #include <iostream> #include <vector> #include <random> #include <chrono> #include "Rmath.h" int main(int argc, char *argv[]) { int ndraws=100000000; std::vector<double> Z(ndraws); std::mt19937 engine; std::normal_distribution<double> N(0,1); auto start = std::chrono::steady_clock::now(); for(auto & z : Z ) { z=N(engine); } auto end = std::chrono::steady_clock::now(); std::chrono::duration<double> elapsed=end-start; std::cout << elapsed.count() << " seconds - C++11" << std::endl; start = std::chrono::steady_clock::now(); for(auto & z : Z ) { z=rnorm(0,1); } end = std::chrono::steady_clock::now(); elapsed=end-start; std::cout << elapsed.count() << " seconds - R Standalone" << std::endl; return 0; }

Compiling and run with:

[michael@michael coda]$ g++ normal.cpp -o normal -std=c++11 -O3 -lRmath [michael@michael coda]$ ./normal

5.2252 seconds - C++11 6.0679 seconds - R Standalone

11.2132 seconds - C++11 12.4486 seconds - R Standalone

6.31157 seconds - C++11 6.35053 seconds - R Standalone

As expected the C++11 implementation is faster but not by a huge amount. As the computational cost of my code is dominated by other linear algebra procedures of O(n^3) I’d actually be willing to use the R standalone library because the syntax is more user friendly.

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]]>The post Stochastic Optimization in R by Parallel Tempering appeared first on Lindons Log.

]]>I’ve written a few posts now about using parallel tempering to sample from complicated multi-modal target distributions but there are also other benefits and uses to this algorithm. There is a nice post on Darren Wilkinson’s blog about using tempered posteriors for marginal likelihood calculations. There is also another area where parallel tempering finds application, namely in stochastic optimization. I first encountered parallel tempering whilst doing my MPhys degree at the University of Warwick but at that time it was employed as a stochastic optimization algorithm to find the minimum energy configuration of a Lennard-Jones cluster as opposed to a sampling algorithm. All that is required is one observation to turn this sampling algorithm into a stochastic optimization algorithm. Lets break this observation down into a few steps.

Consider sampling from a simple exponential distribution $$f(E)\propto e^{-\beta E}1_{(0,\infty )}(E),$$

with rate parameter beta. For now lets fix beta=5. One could sample from this distribution using the same Rmpi parallel tempering code given in my previous post by simply changing the target distribution to the exponential above. The histograms of mcmc draws from four tempered distribution would then look something like this:

Note the scale on the x-axis. The two important observations mentioned earlier are

The second point is important because although the sampling algorithm is creating draws that are not the minimum value of E, by increasing the rate parameter one can force these draws to be arbitrarily close to E-min.

How does this relate to optimization? Consider setting $$E(\theta)=(\theta-40)^2$$ Whereas before where using the Metropolis algorithm one would propose a new value of E, say E’, now the proposal is made in θ, and θ’ is accepted based on u < f(E(θ')) / f(E(θ)). By construction the algorithm gives draws close to E-min, which occurs when θ=40. The traceplot of θ is shown below:

Click here for the code.

The above quadratic was an easy uni-modal example. Let’s try a harder function. Consider the minimum of $$ E(\theta)=3sin(\theta)+(0.1\theta-3)^2,$$ which looks like this:

This function has infinitely many local minima but one global minimum around 30. Local minima make optimization challenging and many optimization algorithms get stuck in these regions as locally it appears the minimum has been reached. This is where the parallel tempering really helps. The traceplots of theta are shown for six tempered distributions below:

Click here for the code.

I’m currently working on another example just for fun, namely finding the lowest energy configuration of an n-particle Lennard-Jones cluster. This is a nice example because one can visualize the process using vmd and it also provides some insight into the origins of such terminology as “tempering”, “annealing” and “temperature” which always look somewhat out of place in the statistics literature.

Consider the function

$$ E(\theta)=10\sin(0.3\theta)\sin(1.3\theta^2) + 0.00001\theta^4 + 0.2\theta+80, $$

which is shown below.

The trace-plots for the parallel tempering optimization are shown below

Examining the mcmc draws the minimum is obtained at theta=-15.81515.

Li Y., Protopopescu V.A., Arnold N., Zhang X. & Gorin A. (2009). Hybrid parallel tempering and simulated annealing method, Applied Mathematics and Computation, 212 (1) 216-228. DOI: 10.1016/j.amc.2009.02.023

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]]>The post Parallel Tempering in R with Rmpi appeared first on Lindons Log.

]]>First one needs to write a density one wishes to sample from

logdensity<-function(theta){ #Distribution one wishes to sample from here. #It may be more convinient to pass a theta as a list sigma2=0.001; Sigma=matrix(0,2,2); Sigma[1,1]=sigma2; Sigma[2,2]=sigma2; density=dmvnorm(theta,c(0,0),Sigma)+dmvnorm(theta,c(-2,0.8),Sigma)+dmvnorm(theta,c(-1,1),Sigma)+dmvnorm(theta,c(1,1),Sigma)+dmvnorm(theta,c(0.5,0.5),Sigma); return(log(density)) }

The density I chose was a mixture of 5 well-separated bi-variate Normals. One should note that it is probably cleanest to pass all the arguments to this function as a list theta. It wasn’t really necessary in this case but if you have a posterior distribution with a number of parameters of varying dimension then it would be much nicer as a list. In a future blog post I may change the target density to be the energy distribution of a Lennard-Jones cluster.

This too is written as a function because Rmpi allows you to pass the function to all slaves and execute it. It was basically the easiest way of writing it for Rmpi.

temper<-function(niter,Bmin,swap.interval){ rank=mpi.comm.rank(); size=mpi.comm.size(); swap=0; swaps.attempted=0; swaps.accepted=0; #Higher ranks run the higher "temperatures" (~smaller fractional powers) B=rep(0,size-1); for(r in 1:size-1){ temp=(r-1)/(size-2); B[r]=Bmin^temp; } #Create a list for proposal moves prop=rep(0,2); theta=matrix(0,niter,2) for(t in 2:niter){ for(c in 1:length(prop)) prop1=theta[t-1,c]+rnorm(1,0,0.1); #Calculate Log-Density at proposed and current position logdensity.current=logdensity(theta[t-1,]) logdensity.prop=logdensity(prop); #Calculate log acceptance probability lalpha=B[rank]*(logdensity.prop-logdensity.current) if(log(runif(1))<lalpha){ #Accept proposed move theta[t,]=prop; logdensity.current=logdensity.prop; }else{ #Otherwise do not move theta[t,]=theta[t-1,]; } if(t%%swap.interval ==0){ for(evenodd in 0:1){ swap=0; logdensity.partner=0; if(rank%%2 == evenodd%%2){ rank.partner=rank + 1; #ranks range from 1:size-1. Cannot have a partner rank == size if(0<rank.partner && rank.partner<size){ #On first iteration, evens receive from above odd #On second iteration, odds receive from above evens logdensity.partner<-mpi.recv.Robj(rank.partner,rank.partner); lalpha = (B[rank]-B[rank.partner])*(logdensity.partner-logdensity.current); swaps.attempted=swaps.attempted+1; if(log(runif(1))<lalpha){ swap=1; swaps.accepted=swaps.accepted+1; } mpi.send.Robj(swap,dest=rank.partner,tag=rank) } if(swap==1){ thetaswap=theta[t,]; mpi.send.Robj(thetaswap,dest=rank.partner,tag=rank) theta[t,]=mpi.recv.Robj(rank.partner,rank.partner) } }else{ rank.partner=rank-1; #ranks range from 1:size-1. Cannot have a partner rank ==0 if(0<rank.partner && rank.partner<size){ #On first iteration, odds send to evens below #On second iteration, evens sent to odds below mpi.send.Robj(logdensity.current,dest=rank.partner,tag=rank); swap=mpi.recv.Robj(rank.partner,rank.partner); } if(swap==1){ thetaswap=theta[t,]; theta[t,]=mpi.recv.Robj(rank.partner,rank.partner); mpi.send.Robj(thetaswap,dest=rank.partner,tag=rank); } } } } } return(theta) }

The bulk of the above code is the communication of each processor with its next nearest neighbors. Metropolis moves will be attempted every *swap.interval* iterations, an argument one can pass to the function. When this code block is entered, even rank processors will partner with their higher ranked odd neighbours (they have a high rank so higher temperature i.e. smaller fractional power – a more “melted down” target density). The higher odd partners will send their lower even partners the value of their density and then the lower even partners will calculate an acceptance probabilty. If the move succeeds the lower rank even processors send their higher rank odd processors a binary swap=1 telling the higher rank odd processors that a send/receive procedure will occur. The lower even rank sends the higher odd rank its parameters and then subsequently the higher odd rank sends its lower even rank its parameters. In this way a metropolis move between processors is achieved. Next, odd rank processors form partners with their higher even ranked neighbours (because we need to swap with processor rank 1, the target density). The same procedure occurs as before but swapping odd for even. More visually, first swaps are attempted between 2-3, 4-5, 6-7 etc and then swaps are attempted between 1-2, 3-4, 5-6. This is almost like a merge-sort style algorithm. One can see how the parameters could be passed from 3 down to 2 and then from 2 down to 1. The main point is that each processor attempts a swap with its nearest-neighbours, the one above and the one below, every *swap.interval* iterations.

With these functions defined one can now proceed to set up the mpi communicator/world.

First spawn some slaves.

library(Rmpi) mpi.spawn.Rslaves(nslaves=6)

If it worked, you should see something like this:

> mpi.spawn.Rslaves(nslaves=6) 6 slaves are spawned successfully. 0 failed. master (rank 0, comm 1) of size 7 is running on: cabbage slave1 (rank 1, comm 1) of size 7 is running on: cabbage slave2 (rank 2, comm 1) of size 7 is running on: cabbage slave3 (rank 3, comm 1) of size 7 is running on: cabbage slave4 (rank 4, comm 1) of size 7 is running on: cabbage slave5 (rank 5, comm 1) of size 7 is running on: cabbage slave6 (rank 6, comm 1) of size 7 is running on: cabbage

(yes, my office computer was named cabbage, lettuce is the one next to me). One can then send the function definitions to the slave processors.

niter=3000 Bmin=0.005 swap.interval=3 #Send to slaves some required data mpi.bcast.Robj2slave(niter) mpi.bcast.Robj2slave(Bmin) mpi.bcast.Robj2slave(swap.interval) #Send to slaves the logdensity function mpi.bcast.Robj2slave(logdensity) #Send to slaves the temper function mpi.bcast.Robj2slave(temper) #Send to slaves the dmvnorm function mpi.bcast.Robj2slave(dmvnorm)

If you want to make sure that the slaves have the correct function definition, one can execute the command *mpi.remote.exec(temper)* and this will return the function definition. That is all, now it can be run.

mcmc=mpi.remote.exec(temper(niter,Bmin,swap.interval))

This returns a list object containing the mcmc draws for each slave.

The end product is something that looks like this

Which are the draws (in black) from the target distribution. It is also useful to build up intuition for parallel tempering to look at what is happening on the other processors. The draws for all processors are shown below:

N.B. Although my computer only has 8 cores I tried running the code 12 slaves. At first I was concerned that the MPI communications would enter a deadlock and the code would hang but it didn’t, so it seems you can scale up the number of slaves above the number of cores.

Notice that the temperature set used in the code has the property that , for c a constant. There is a paper by Kofke(2002) that justifies this temperature set as it yields a constant acceptance ratio between cores under certain conditions. Indeed, the acceptance ratio (the fraction of metropolis moves that succeeded between cores) are roughly constant, as shown below:

[1] 0.7227723 [1] 0.7926793 [1] 0.710171 [1] 0.8037804 [1] 0.7191719 [1] 0.7974797 [1] 0.729673 [1] 0.8223822 [1] 0.8184818 [1] 0.8445845

Earl D.J. & Deem M.W. (2005). Parallel tempering: Theory, applications, and new perspectives, Physical Chemistry Chemical Physics, 7 (23) 3910. DOI: 10.1039/b509983h

Kofke D.A. (2002). On the acceptance probability of replica-exchange Monte Carlo trials, The Journal of Chemical Physics, 117 (15) 6911. DOI: 10.1063/1.1507776

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