## Faster Gibbs sampling MCMC from within R

### Introduction

This post follows on from the previous post on Gibbs sampling in various languages. In that post a simple Gibbs sampler was implemented in various languages, and speeds were compared. It was seen that R is very slow for iterative simulation algorithms characteristic of MCMC methods such as the Gibbs sampler. Statically typed languages such as C/C++ and Java were seen to be fastest for this type of algorithm. Since many statisticians like to use R for most of their work, there is natural interest in the possibility of extending R by calling simulation algorithms written in other languages. It turns out to be straightforward to call C, C++ and Java from within R, so this post will look at how this can be done, and exactly how fast the different options turn out to be. The post draws heavily on my previous posts on calling C from R and calling Java from R, as well as Dirk Eddelbuettel’s post on calling C++ from R, and it may be helpful to consult these posts for further details.

### Languages

#### R

We will start with the simple pure R version of the Gibbs sampler, and use this as our point of reference for understanding the benefits of re-coding in other languages. The background to the problem was given in the previous post and so won’t be repeated here. The code can be given as follows:

gibbs<-function(N=50000,thin=1000)
{
mat=matrix(0,ncol=2,nrow=N)
x=0
y=0
for (i in 1:N) {
for (j in 1:thin) {
x=rgamma(1,3,y*y+4)
y=rnorm(1,1/(x+1),1/sqrt(2*x+2))
}
mat[i,]=c(x,y)
}
names(mat)=c("x","y")
mat
}


This code works perfectly, but is very slow. It takes 458.9 seconds on my very fast laptop (details given in previous post).

#### C

Let us now see how we can introduce a new function, gibbsC into R, which works in exactly the same way as gibbs, but actually calls on compiled C code to do all of the work. First we need the C code in a file called gibbs.c:

#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <R.h>
#include <Rmath.h>

void gibbs(int *Np,int *thinp,double *xvec,double *yvec)
{
int i,j;
int N=*Np,thin=*thinp;
GetRNGstate();
double x=0;
double y=0;
for (i=0;i<N;i++) {
for (j=0;j<thin;j++) {
x=rgamma(3.0,1.0/(y*y+4));
y=rnorm(1.0/(x+1),1.0/sqrt(2*x+2));
}
xvec[i]=x; yvec[i]=y;
}
PutRNGstate();
}


This can be compiled with R CMD SHLIB gibbs.c. We can load it into R and wrap it up so that it is easy to use with the following code:

dyn.load(file.path(".",paste("gibbs",.Platform$dynlib.ext,sep=""))) gibbsC<-function(n=50000,thin=1000) { tmp=.C("gibbs",as.integer(n),as.integer(thin), x=as.double(1:n),y=as.double(1:n)) mat=cbind(tmp$x,tmp$y) colnames(mat)=c("x","y") mat }  The new function gibbsC works just like gibbs, but takes just 12.1 seconds to run. This is roughly 40 times faster than the pure R version, which is a big deal. Note that using the R inline package, it is possible to directly inline the C code into the R source code. We can do this with the following R code: require(inline) code=' int i,j; int N=*Np,thin=*thinp; GetRNGstate(); double x=0; double y=0; for (i=0;i<N;i++) { for (j=0;j<thin;j++) { x=rgamma(3.0,1.0/(y*y+4)); y=rnorm(1.0/(x+1),1.0/sqrt(2*x+2)); } xvec[i]=x; yvec[i]=y; } PutRNGstate();' gibbsCin<-cfunction(sig=signature(Np="integer",thinp="integer",xvec="numeric",yvec="numeric"),body=code,includes="#include <Rmath.h>",language="C",convention=".C") gibbsCinline<-function(n=50000,thin=1000) { tmp=gibbsCin(n,thin,rep(0,n),rep(0,n)) mat=cbind(tmp$x,tmp$y) colnames(mat)=c("x","y") mat }  This runs at the same speed as the code compiled separately, and is arguably a bit cleaner in this case. Personally I’m not a big fan of inlining code unless it is something really very simple. If there is one thing that we have learned from the murky world of web development, it is that little good comes from mixing up different languages in the same source code file! #### C++ We can also inline C++ code into R using the inline and Rcpp packages. The code below originates from Sanjog Misra, and was discussed in the post by Dirk Eddelbuettel mentioned at the start of this post. require(Rcpp) require(inline) gibbscode = ' int N = as<int>(n); int thn = as<int>(thin); int i,j; RNGScope scope; NumericVector xs(N),ys(N); double x=0; double y=0; for (i=0;i<N;i++) { for (j=0;j<thn;j++) { x = ::Rf_rgamma(3.0,1.0/(y*y+4)); y= ::Rf_rnorm(1.0/(x+1),1.0/sqrt(2*x+2)); } xs(i) = x; ys(i) = y; } return Rcpp::DataFrame::create( Named("x")= xs, Named("y") = ys); ' RcppGibbsFn <- cxxfunction( signature(n="int", thin = "int"), gibbscode, plugin="Rcpp") RcppGibbs <- function(N=50000,thin=1000) { RcppGibbsFn(N,thin) }  This version of the sampler runs in 12.4 seconds, just a little bit slower than the C version. #### Java It is also quite straightforward to call Java code from within R using the rJava package. The following code import java.util.*; import cern.jet.random.tdouble.*; import cern.jet.random.tdouble.engine.*; class GibbsR { public static double[][] gibbs(int N,int thin,int seed) { DoubleRandomEngine rngEngine=new DoubleMersenneTwister(seed); Normal rngN=new Normal(0.0,1.0,rngEngine); Gamma rngG=new Gamma(1.0,1.0,rngEngine); double x=0,y=0; double[][] mat=new double[2][N]; for (int i=0;i<N;i++) { for (int j=0;j<thin;j++) { x=rngG.nextDouble(3.0,y*y+4); y=rngN.nextDouble(1.0/(x+1),1.0/Math.sqrt(2*x+2)); } mat[0][i]=x; mat[1][i]=y; } return mat; } }  can be compiled with javac GibbsR.java (assuming that Parallel COLT is in the classpath), and wrapped up from within an R session with library(rJava) .jinit() obj=.jnew("GibbsR") gibbsJ<-function(N=50000,thin=1000,seed=trunc(runif(1)*1e6)) { result=.jcall(obj,"[[D","gibbs",as.integer(N),as.integer(thin),as.integer(seed)) mat=sapply(result,.jevalArray) colnames(mat)=c("x","y") mat }  This code runs in 10.7 seconds. Yes, that’s correct. Yes, the Java code is faster than both the C and C++ code! This really goes to show that Java is now an excellent option for numerically intensive work such as this. However, before any C/C++ enthusiasts go apoplectic, I should explain why Java turns out to be faster here, as the comparison is not quite fair… In the C and C++ code, use was made of the internal R random number generation routines, which are relatively slow compared to many modern numerical library implementations. In the Java code, I used Parallel COLT for random number generation, as it isn’t straightforward to call the R generators from Java code. It turns out that the COLT generators are faster than the R generators, and that is why Java turns out to be faster here… #### C+GSL Of course we do not have to use the R random number generators within our C code. For example, we could instead call on the GSL generators, using the following code: #include <stdio.h> #include <math.h> #include <stdlib.h> #include <gsl/gsl_rng.h> #include <gsl/gsl_randist.h> #include <R.h> void gibbsGSL(int *Np,int *thinp,int *seedp,double *xvec,double *yvec) { int i,j; int N=*Np,thin=*thinp,seed=*seedp; gsl_rng *r = gsl_rng_alloc(gsl_rng_mt19937); gsl_rng_set(r,seed); double x=0; double y=0; for (i=0;i<N;i++) { for (j=0;j<thin;j++) { x=gsl_ran_gamma(r,3.0,1.0/(y*y+4)); y=1.0/(x+1)+gsl_ran_gaussian(r,1.0/sqrt(2*x+2)); } xvec[i]=x; yvec[i]=y; } }  It can be compiled with R CMD SHLIB -lgsl -lgslcblas gibbsGSL.c, and then called as for the regular C version. This runs in 8.0 seconds, which is noticeably faster than the Java code, but probably not “enough” faster to make it an important factor to consider in language choice. ### Summary In this post I’ve shown that it is relatively straightforward to call code written in C, C++ or Java from within R, and that this can give very significant performance gains relative to pure R code. All of the options give fairly similar performance gains. I showed that in the case of this particular example, the “obvious” Java code is actually slightly faster than the “obvious” C or C++ code, and explained why, and how to make the C version slightly faster by using the GSL. The post by Dirk shows how to call the GSL generators from the C++ version, which I haven’t replicated here. ## Gibbs sampler in various languages (revisited) ### Introduction Regular readers of this blog will know that in April 2010 I published a short post showing how a trivial bivariate Gibbs sampler could be implemented in the four languages that I use most often these days (R, python, C, Java), and I discussed relative timings, and how one might start to think about trading off development time against execution time for more complex MCMC algorithms. I actually wrote the post very quickly one night while I was stuck in a hotel room in Seattle – I didn’t give much thought to it, and the main purpose was to provide simple illustrative examples of simple Monte Carlo codes using non-uniform random number generators in the different languages, as a starting point for someone thinking of switching languages (say, from R to Java or C, for efficiency reasons). It wasn’t meant to be very deep or provocative, or to start any language wars. Suffice to say that this post has had many more hits than all of my other posts combined, is still my most popular post, and still attracts comments and spawns other posts to this day. Several people have requested that I re-do the post more carefully, to include actual timings, and to include a few additional optimisations. Hence this post. For reference, the original post is here. A post about it from the python community is here, and a recent post about using Rcpp and inlined C++ code to speed up the R version is here. ### The sampler So, the basic idea was to construct a Gibbs sampler for the bivariate distribution $f(x,y) = kx^2\exp\{-xy^2-y^2+2y-4x\},\qquad x>0,y\in\Bbb{R}$ with unknown normalising constant $k>0$ ensuring that the density integrates to one. Unfortunately, in the original post I dropped a factor of 2 constructing one of the full conditionals, which meant that none of the samplers actually had exactly the right target distribution (thanks to Sanjog Misra for bringing this to my attention). So actually, the correct full conditionals are $\displaystyle x|y \sim Ga(3,y^2+4)$ $\displaystyle y|x \sim N\left(\frac{1}{1+x},\frac{1}{2(1+x)}\right)$ Note the factor of two in the variance of the full conditional for $y$. Given the full conditionals, it is simple to alternately sample from them to construct a Gibbs sampler for the target distribution. We will run a Gibbs sampler with a thin of 1000 and obtain a final sample of 50000. ### Implementations #### R Let’s start with R again. The slightly modified version of the code from the old post is given below gibbs=function(N,thin) { mat=matrix(0,ncol=3,nrow=N) mat[,1]=1:N x=0 y=0 for (i in 1:N) { for (j in 1:thin) { x=rgamma(1,3,y*y+4) y=rnorm(1,1/(x+1),1/sqrt(2*x+2)) } mat[i,2:3]=c(x,y) } mat=data.frame(mat) names(mat)=c("Iter","x","y") mat } writegibbs=function(N=50000,thin=1000) { mat=gibbs(N,thin) write.table(mat,"data.tab",row.names=FALSE) } writegibbs()  I’ve just corrected the full conditional, and I’ve increased the sample size and thinning to 50k and 1k, respectively, to allow for more accurate timings (of the faster languages). This code can be run from the (Linux) command line with something like: time Rscript gibbs.R  I discuss timings in detail towards the end of the post, but this code is slow, taking over 7 minutes on my (very fast) laptop. Now, the above code is typical of the way code is often structured in R – doing as much as possible in memory, and writing to disk only if necessary. However, this can be a bad idea with large MCMC codes, and is less natural in other languages, anyway, so below is an alternative version of the code, written in more of a scripting language style. gibbs=function(N,thin) { x=0 y=0 cat(paste("Iter","x","y","\n")) for (i in 1:N) { for (j in 1:thin) { x=rgamma(1,3,y*y+4) y=rnorm(1,1/(x+1),1/sqrt(2*x+2)) } cat(paste(i,x,y,"\n")) } } gibbs(50000,1000)  This can be run with a command like time Rscript gibbs-script.R > data.tab  This code actually turns out to be a slightly slower than the in-memory version for this simple example, but for larger problems I would not expect that to be the case. I always analyse MCMC output using R, whatever language I use for running the algorithm, so for completeness, here is a bit of code to load up the data file, do some plots and compute summary statistics. fun=function(x,y) { x*x*exp(-x*y*y-y*y+2*y-4*x) } compare<-function(file="data.tab") { mat=read.table(file,header=TRUE) op=par(mfrow=c(2,1)) x=seq(0,3,0.1) y=seq(-1,3,0.1) z=outer(x,y,fun) contour(x,y,z,main="Contours of actual (unnormalised) distribution") require(KernSmooth) fit=bkde2D(as.matrix(mat[,2:3]),c(0.1,0.1)) contour(fit$x1,fit$x2,fit$fhat,main="Contours of empirical distribution")
par(op)
print(summary(mat[,2:3]))
}
compare()


#### Python

Another language I use a lot is Python. I don’t want to start any language wars, but I personally find python to be a better designed language than R, and generally much nicer for the development of large programs. A python script for this problem is given below

import random,math

def gibbs(N=50000,thin=1000):
x=0
y=0
print "Iter  x  y"
for i in range(N):
for j in range(thin):
x=random.gammavariate(3,1.0/(y*y+4))
y=random.gauss(1.0/(x+1),1.0/math.sqrt(2*x+2))
print i,x,y

gibbs()


It can be run with a command like

time python gibbs.py > data.tab


This code turns out to be noticeably faster than the R versions, taking around 4 minutes on my laptop (again, detailed timing information below). However, there is a project for python known as the PyPy project, which is concerned with compiling regular python code to very fast byte-code, giving significant speed-ups on certain problems. For this post, I downloaded and install version 1.5 of the 64-bit linux version of PyPy. Once installed, I can run the above code with the command

time pypy gibbs.py > data.tab


To my astonishment, this “just worked”, and gave very impressive speed-up over regular python, running in around 30 seconds. This actually makes python a much more realistic prospect for the development of MCMC codes than I imagined. However, I need to understand the limitations of PyPy better – for example, why doesn’t everyone always use PyPy for everything?! It certainly seems to make python look like a very good option for prototyping MCMC codes.

#### C

Traditionally, I have mainly written MCMC codes in C, using the GSL. C is a fast, efficient, statically typed language, which compiles to native code. In many ways it represents the “gold standard” for speed. So, here is the C code for this problem.

#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>

void main()
{
int N=50000;
int thin=1000;
int i,j;
gsl_rng *r = gsl_rng_alloc(gsl_rng_mt19937);
double x=0;
double y=0;
printf("Iter x y\n");
for (i=0;i<N;i++) {
for (j=0;j<thin;j++) {
x=gsl_ran_gamma(r,3.0,1.0/(y*y+4));
y=1.0/(x+1)+gsl_ran_gaussian(r,1.0/sqrt(2*x+2));
}
printf("%d %f %f\n",i,x,y);
}
}


It can be compiled and run with command like

gcc -O4 gibbs.c -lgsl -lgslcblas -lm -o gibbs
time ./gibbs > datac.tab


This runs faster than anything else I consider in this post, taking around 8 seconds.

#### Java

I’ve recently been experimenting with Java for MCMC codes, in conjunction with Parallel COLT. Java is a statically typed object-oriented (O-O) language, but is usually compiled to byte-code to run on a virtual machine (known as the JVM). Java compilers and virtual machines are very fast these days, giving “close to C” performance, but with a nicer programming language, and advantages associated with virtual machines. Portability is a huge advantage of Java. For example, I can easily get my Java code to run on almost any University Condor pool, on both Windows and Linux clusters – they all have a recent JVM installed, and I can easily bundle any required libraries with my code. Suffice to say that getting GSL/C code to run on generic Condor pools is typically much less straightforward. Here is the Java code:

import java.util.*;
import cern.jet.random.tdouble.*;
import cern.jet.random.tdouble.engine.*;

class Gibbs
{

public static void main(String[] arg)
{
int N=50000;
int thin=1000;
DoubleRandomEngine rngEngine=new DoubleMersenneTwister(new Date());
Normal rngN=new Normal(0.0,1.0,rngEngine);
Gamma rngG=new Gamma(1.0,1.0,rngEngine);
double x=0;
double y=0;
System.out.println("Iter x y");
for (int i=0;i<N;i++) {
for (int j=0;j<thin;j++) {
x=rngG.nextDouble(3.0,y*y+4);
y=rngN.nextDouble(1.0/(x+1),1.0/Math.sqrt(2*x+2));
}
System.out.println(i+" "+x+" "+y);
}
}

}


It can be compiled and run with

javac Gibbs.java
time java Gibbs > data.tab


This takes around 11.6s seconds on my laptop. This is well within a factor of 2 of the C version, and around 3 times faster than even the PyPy python version. It is around 40 times faster than R. Java looks like a good choice for implementing MCMC codes that would be messy to implement in C, or that need to run places where it would be fiddly to get native codes to run.

#### Scala

Another language I’ve been taking some interest in recently is Scala. Scala is a statically typed O-O/functional language which compiles to byte-code that runs on the JVM. Since it uses Java technology, it can seamlessly integrate with Java libraries, and can run anywhere that Java code can run. It is a much nicer language to program in than Java, and feels more like a dynamic language such as python. In fact, it is almost as nice to program in as python (and in some ways nicer), and will run in a lot more places than PyPy python code. Here is the scala code (which calls Parallel COLT for random number generation):

object GibbsSc {

import cern.jet.random.tdouble.engine.DoubleMersenneTwister
import cern.jet.random.tdouble.Normal
import cern.jet.random.tdouble.Gamma
import Math.sqrt
import java.util.Date

def main(args: Array[String]) {
val N=50000
val thin=1000
val rngEngine=new DoubleMersenneTwister(new Date)
val rngN=new Normal(0.0,1.0,rngEngine)
val rngG=new Gamma(1.0,1.0,rngEngine)
var x=0.0
var y=0.0
println("Iter x y")
for (i <- 0 until N) {
for (j <- 0 until thin) {
x=rngG.nextDouble(3.0,y*y+4)
y=rngN.nextDouble(1.0/(x+1),1.0/sqrt(2*x+2))
}
println(i+" "+x+" "+y)
}
}

}


It can be compiled and run with

scalac GibbsSc.scala
time scala GibbsSc > data.tab


This code takes around 11.8s on my laptop – almost as fast as the Java code! So, on the basis of this very simple and superficial example, it looks like scala may offer the best of all worlds – a nice, elegant, terse programming language, functional and O-O programming styles, the safety of static typing, the ability to call on Java libraries, great speed and efficiency, and the portability of Java! Very interesting.

#### Groovy

James Durbin has kindly sent me a Groovy version of the code, which he has also discussed in his own blog post. Groovy is a dynamic O-O language for the JVM, which, like Scala, can integrate nicely with Java applications. It isn’t a language I have examined closely, but it seems quite nice. The code is given below:

import cern.jet.random.tdouble.engine.*;
import cern.jet.random.tdouble.*;

N=50000;
thin=1000;
rngEngine= new DoubleMersenneTwister(new Date());
rngN=new Normal(0.0,1.0,rngEngine);
rngG=new Gamma(1.0,1.0,rngEngine);
x=0.0;
y=0.0;
println("Iter x y");
for(i in 1..N){
for(j in 1..thin){
x=rngG.nextDouble(3.0,y*y+4);
y=rngN.nextDouble(1.0/(x+1),1.0/Math.sqrt(2*x+2));
}
println("$i$x \$y");
}


It can be run with a command like:

time groovy Gibbs.gv > data.tab


Again, rather amazingly, this code runs in around 35 seconds – very similar to the speed of PyPy. This makes Groovy also seem like a potential very attractive environment for prototyping MCMC codes, especially if I’m thinking about ultimately porting to Java.

### Timings

The laptop I’m running everything on is a Dell Precision M4500 with an Intel i7 Quad core (x940@2.13Ghz) CPU, running the 64-bit version of Ubuntu 11.04. I’m running stuff from the Ubuntu (Unity) desktop, and running several terminals and applications, but the machine is not loaded at the time each job runs. I’m running each job 3 times and taking the arithmetic mean real elapsed time. All timings are in seconds.

 R 2.12.1 (in memory) 435 R 2.12.1 (script) 450.2 Python 2.7.1+ 233.5 PyPy 1.5 32.2 Groovy 1.7.4 35.4 Java 1.6.0 11.6 Scala 2.7.7 11.8 C (gcc 4.5.2) 8.1

If we look at speed-up relative to the R code (in-memory version), we get:

 R (in memory) 1 R (script) 0.97 Python 1.86 PyPy 13.51 Groovy 12.3 Java 37.5 Scala 36.86 C 53.7

Alternatively, we can look at slow-down relative to the C version, to get:

 R (in memory) 53.7 R (script) 55.6 Python 28.8 PyPy 4 Groovy 4.4 Java 1.4 Scala 1.5 C 1

### Discussion

The findings here are generally consistent with those of the old post, but consideration of PyPy, Groovy and Scala does throw up some new issues. I was pretty stunned by PyPy. First, I didn’t expect that it would “just work” – I thought I would either have to spend time messing around with my configuration settings, or possibly even have to modify my code slightly. Nope. Running python code with pypy appears to be more than 10 times faster than R, and only 4 times slower than C. I find it quite amazing that it is possible to get python code to run just 4 times slower than C, and if that is indicative of more substantial examples, it really does open up the possibility of using python for “real” problems, although library coverage is currently a problem. It certainly solves my “prototyping problem”. I often like to prototype algorithms in very high level dynamic languages like R and python before porting to a more efficient language. However, I have found that this doesn’t always work well with complex MCMC codes, as they just run too slowly in the dynamic languages to develop, test and debug conveniently. But it looks now as though PyPy should be fast enough at least for prototyping purposes, and may even be fast enough for production code in some circumstances. But then again, exactly the same goes for Groovy, which runs on the JVM, and can access any existing Java library… I haven’t yet looked into Groovy in detail, but it appears that it could be a very nice language for prototyping algorithms that I intend to port to Java.

The results also confirm my previous findings that Java is now “fast enough” that one shouldn’t worry too much about the difference in speed between it and native code written in C (or C++). The Java language is much nicer than C or C++, and the JVM platform is very attractive in many situations. However, the Scala results were also very surprising for me. Scala is a really elegant language (certainly on a par with python), comes with all of the advantages of Java, and appears to be almost as fast as Java. I’m really struggling to come up with reasons not to use Scala for everything!

#### Speeding up R

MCMC codes are used by a range of different scientists for a range of different problems. However, they are very (most?) often used by Bayesian statisticians who use the algorithms to target a Bayesian posterior distribution. For various (good) reasons, many statisticians are heavily invested in R, like to use R as much as possible, and do as much as possible from within the R environment. These results show why R is not a good language in which to implement MCMC algorithms, so what is an R-dependent statistician supposed to do? One possibility would be to byte-code compile R code in an analogous way to python and pypy. The very latest versions of R support such functionality, but the post by Dirk Eddelbuettel suggests that the current version of cmpfun will only give a 40% speedup on this problem, which is still slower than regular python code. Short of a dramatic improvement in this technology, the only way forward seems to be to extend R using code from another language. It is easy to extend R using C, C++ and Java. I have shown in previous posts how to do this using Java and using C, and the recent post by Dirk shows how to extend using C++. Although interesting, this doesn’t really have much bearing on the current discussion. If you extend using Java you get Java-like speedups, and if you extend using C you get C-like speedups. However, in case people are interested, I intend to gather up these examples into one post and include detailed timing information in a subsequent post.

## Java math libraries and Monte Carlo simulation codes

### Java libraries for (non-uniform) random number simulation

Anyone writing serious Monte Carlo (and MCMC) codes relies on having a very good and fast (uniform) random number generator and associated functions for generation of non-uniform random quantities, such as Gaussian, Poisson, Gamma, etc. In a previous post I showed how to write a simple Gibbs sampler in four different languages. In C (and C++) random number generation is easy for most scientists, as the (excellent) GNU Scientific Library (GSL) provides exactly what most people need. But it wasn’t always that way… I remember the days before the GSL, when it was necessary to hunt around on the net for bits of C code to implement different algorithms. Worse, it was often necessary to hunt around for a bit of free FORTRAN code, and compile that with an F77 compiler and figure out how to call it from C. Even in the early Alpha days of the GSL, coverage was patchy, and the API changed often. Bad old days… But those days are long gone, and C programmers no longer have to worry about the problem of random variate generation – they can safely concentrate on developing their interesting new algorithm, and leave the rest to the GSL. Unfortunately for Java programmers, there isn’t yet anything quite comparable to the GSL in Java world.

I pretty much ignored Java until Java 5. Before then, the language was too limited, and the compilers and JVMs were too primitive to really take seriously for numerical work. But since the launch of Java 5 I’ve been starting to pay more interest. The language is now a perfectly reasonable O-O language, and the compilers and JVMs are pretty good. On a lot of benchmarks, Java is really quite comparable to C/C++, and Java is nicer to code, and has a lot of impressive associated technology. So if there was a math library comparable to the GSL, I’d be quite tempted to jump ship to the Java world and start writing all of my Monte Carlo codes in Java. But there isn’t. At least not yet.

When I first started to take Java seriously, the only good math library with good support for non-uniform random number generation was COLT. COLT was, and still is, pretty good. The code is generally well-written, and fast, and the documentation for it is reasonable. However, the structure of the library is very idiosyncratic, the coverage is a bit patchy, and there doesn’t ever seem to have been a proper development community behind it. It seems very much to have been a one-man project, which has long since stagnated. Unsurprisingly then, COLT has been forked. There is now a Parallel COLT project. This project is continuing the development of COLT, adding new features that were missing from COLT, and, as the name suggests, adding concurrency support. Parallel COLT is also good, and is the main library I currently use for random number generation in Java. However, it has obviously inherited all of the idiosyncrasies that COLT had, and still doesn’t seem to have a large and active development community associated with it. There is no doubt that it is an incredibly useful software library, but it still doesn’t really compare to the GSL.

I have watched the emergence of the Apache Commons Math project with great interest (not to be confused with Uncommons Math – another one-man project). I think this project probably has the greatest potential for providing the Java community with their own GSL equivalent. The Commons project has a lot of momentum, the Commons Math project seems to have an active development community, and the structure of the library is more intuitive than that of (Parallel) COLT. However, it is early days, and the library still has patchy coverage and is a bit rough around the edges. It reminds me a lot of the GSL back in its Alpha days. I’d not bothered to even download it until recently, as the random number generation component didn’t include the generation of gamma random quantities – an absolutely essential requirement for me. However, I noticed recently that the latest release (2.2) did include gamma generation, so I decided to download it and try it out. It works, but the generation of gamma random quantities is very slow (around 50 times slower than Parallel COLT). This isn’t a fundamental design flaw of the whole library – generating Gaussian random quantities is quite comparable with other libraries. It’s just that an inversion method has been used for gamma generation. All efficient gamma generators use a neat rejection scheme. In case anyone would like to investigate for themselves, here is a complete program for gamma generation designed to be linked against Parallel COLT:

import java.util.*;
import cern.jet.random.tdouble.*;
import cern.jet.random.tdouble.engine.*;

class GammaPC
{

public static void main(String[] arg)
{
DoubleRandomEngine rngEngine=new DoubleMersenneTwister();
Gamma rngG=new Gamma(1.0,1.0,rngEngine);
long N=10000;
double x=0.0;
for (int i=0;i<N;i++) {
for (int j=0;j<1000;j++) {
x=rngG.nextDouble(3.0,25.0);
}
System.out.println(x);
}
}

}


and here is a complete program designed to be linked against Commons Math:

import java.util.*;
import org.apache.commons.math.*;
import org.apache.commons.math.random.*;

class GammaACM
{

public static void main(String[] arg) throws MathException
{
RandomDataImpl rng=new RandomDataImpl();
long N=10000;
double x=0.0;
for (int i=0;i<N;i++) {
for (int j=0;j<1000;j++) {
x=rng.nextGamma(3.0,1.0/25.0);
}
System.out.println(x);
}
}

}


The two codes do the same thing (note that they parameterise the gamma distribution differently). Both programs work (they generate variates from the same, correct, distribution), and the Commons Math interface is slightly nicer, but the code is much slower to execute. I’m still optimistic that Commons Math will one day be Java’s GSL, but I’m not giving up on Parallel COLT (or C, for that matter!) just yet…

## Parallel Monte Carlo with an Intel i7 Quad Core

I’ve recently acquired a new laptop with an Intel i7 Quad Core CPU – an i7-940XM to be precise, and I’m interested in the possibility of running parallel MCMC codes on this machine (a Dell Precision M4500) in order to speed things up a bit. I’m running the AMD64 version of Ubuntu Linux 10.10 on it, as it has 8GB of (1333MHz DDR2 dual channel) RAM. It also contains an NVIDIA 1GB Quadro FX graphics card, which I could probably also use for GPU-style speed-up using CUDA, but I don’t really want that kind of hassle if I can avoid it. In a previous post I gave an Introduction to parallel MCMC, which explains how to set up an MPI-based parallel computing environment with Ubuntu. In this post I’m just going to look at a very simple embarrassingly parallel Monte Carlo code, based on the code monte-carlo.c from the previous post. The slightly modified version of the code is given below.

#include <math.h>
#include <mpi.h>
#include <gsl/gsl_rng.h>
#include "gsl-sprng.h"

int main(int argc,char *argv[])
{
int i,k,N; long Iters; double u,ksum,Nsum; gsl_rng *r;
MPI_Init(&argc,&argv);
MPI_Comm_size(MPI_COMM_WORLD,&N);
MPI_Comm_rank(MPI_COMM_WORLD,&k);
Iters=1e9;
r=gsl_rng_alloc(gsl_rng_sprng20);
for (i=0;i<(Iters/N);i++) {
u = gsl_rng_uniform(r);
ksum += exp(-u*u);
}
MPI_Reduce(&ksum,&Nsum,1,MPI_DOUBLE,MPI_SUM,0,MPI_COMM_WORLD);
if (k == 0) {
printf("Monte carlo estimate is %f\n", Nsum/Iters );
}
MPI_Finalize();
exit(EXIT_SUCCESS);
}


This code does 10^9 iterations of a Monte Carlo integral, dividing them equally among the available processes. This code can be compiled with something like:

mpicc -I/usr/local/src/sprng2.0/include -L/usr/local/src/sprng2.0/lib -o monte-carlo monte-carlo.c -lsprng -lgmp -lgsl -lgslcblas


and run with (say) 4 processes with a command like:

time mpirun -np 4 monte-carlo


How many processes should one run on this machine? This is an interesting question. There is only one CPU in this laptop, but as the name suggests, it has 4 cores. Furthermore, each of those cores is hyper-threaded, so the linux kernel presents it to the user as 8 processors (4 cores, 8 siblings), as a quick cat /proc/cpuinfo will confirm. Does this really mean that it is the equivalent of 8 independent CPUs? The short answer is “no”, but running 8 processes of an MPI (or OpenMP) job could still be optimal. I investigated this issue empirically by varying the number of processes for the MPI job from 1 to 10, doing 3 repeats for each to get some kind of idea of variability (I’ve been working in a lab recently, so I know that all good biologists always do 3 repeats of everything!). The conditions were by no means optimal, but probably quite typical – I was running the Ubuntu window manager, several terminal windows, had Firefox open with several tabs in another workspace, and Xemacs open with several buffers, etc. The raw timings (in seconds) are given below.

NP	T1	T2	T3
1	62.046	62.042	61.900
2	35.652	34.737	36.116
3	29.048	28.238	28.567
4	23.273	24.184	22.207
5	24.418	24.735	24.580
6	21.279	21.184	22.379
7	20.072	19.758	19.836
8	17.858	17.836	18.330
9	20.392	21.290	21.279
10	22.342	19.685	19.309


A quick scan of the numbers in the table makes sense – the time taken decreases steadily as the number of processes increases up to 8 processes, and then increases again as the number goes above 8. This is exactly the kind of qualitative pattern one would hope to see, but the quantitative pattern is a bit more interesting. First let’s look at a plot of the timings.

You will probably need to click on the image to be able to read the axes. The black line gives the average time, and the grey envelope covers all 3 timings. Again, this is broadly what I would have expected – time decreasing steadily to 4 processes, then diminishing returns from 4 to 8 processes, and a penalty for attempting to run more than 8 processes in parallel. Now lets look at the speed up (speed relative to the average time of the 1 processor version).

Here again, the qualitative pattern is as expected. However, inspection of the numbers on the y-axis is a little disappointing. Remeber that this not some cleverly parallelised MCMC chain with lots of inter-process communication – this is an embarrassingly parallel Monte Carlo job – one would expect to see close to 8x speedup on 8 independent CPUs. Here we see that for 4 processes, we get a little over 2.5x speedup, and if we crank things all the way up to 8 processes, we get nearly 3.5x speedup. This isn’t mind-blowing, but it is a lot better than nothing, and this is a fairly powerful processor, so even the 1 processor code is pretty quick…

So, in answer to the question of how many processes to run, the answer seems to be that if the code is very parallel, running 8 processes will probably be quickest, but running 4 processes probably won’t be much slower, and will leave some spare capacity for web browsing, editing, etc, while waiting for the MPI job to run. As ever, YMMV…

## Getting started with parallel MCMC

### Introduction to parallel MCMC for Bayesian inference, using C, MPI, the GSL and SPRNG

#### Introduction

This post is aimed at people who already know how to code up Markov Chain Monte Carlo (MCMC) algorithms in C, but are interested in how to parallelise their code to run on multi-core machines and HPC clusters. I discussed different languages for coding MCMC algorithms in a previous post. The advantage of C is that it is fast, standard and has excellent scientific library support. Ultimately, people pursuing this route will be interested in running their code on large clusters of fast servers, but for the purposes of development and testing, this really isn’t necessary. A single dual-core laptop, or similar, is absolutely fine. I develop and test on a dual-core laptop running Ubuntu linux, so that is what I will assume for the rest of this post.

There are several possible environments for parallel computing, but I will focus on the Message-Passing Interface (MPI). This is a well-established standard for parallel computing, there are many implementations, and it is by far the most commonly used high performance computing (HPC) framework today. Even if you are ultimately interested in writing code for novel architectures such as GPUs, learning the basics of parallel computation using MPI will be time well spent.

#### MPI

The whole point of MPI is that it is a standard, so code written for one implementation should run fine with any other. There are many implementations. On Linux platforms, the most popular are OpenMPI, LAM, and MPICH. There are various pros and cons associated with each implementation, and if installing on a powerful HPC cluster, serious consideration should be given to which will be the most beneficial. For basic development and testing, however, it really doesn’t matter which is used. I use OpenMPI on my Ubuntu laptop, which can be installed with a simple:

sudo apt-get install openmpi-bin libopenmpi-dev


That’s it! You’re ready to go… You can test your installation with a simple “Hello world” program such as:

#include <stdio.h>
#include <mpi.h>

int main (int argc,char **argv)
{
int rank, size;
MPI_Init (&argc, &argv);
MPI_Comm_rank (MPI_COMM_WORLD, &rank);
MPI_Comm_size (MPI_COMM_WORLD, &size);
printf( "Hello world from process %d of %d\n", rank, size );
MPI_Finalize();
return 0;
}


You should be able to compile this with

mpicc -o helloworld helloworld.c


and run (on 2 processors) with

mpirun -np 2 helloworld


#### GSL

If you are writing non-trivial MCMC codes, you are almost certainly going to need to use a sophisticated math library and associated random number generation (RNG) routines. I typically use the GSL. On Ubuntu, the GSL can be installed with:

sudo apt-get install gsl-bin libgsl0-dev


A simple script to generate some non-uniform random numbers is given below.

#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>

int main(void)
{
int i; double z; gsl_rng *r;
r = gsl_rng_alloc(gsl_rng_mt19937);
gsl_rng_set(r,0);
for (i=0;i<10;i++) {
z = gsl_ran_gaussian(r,1.0);
printf("z(%d) = %f\n",i,z);
}
exit(EXIT_SUCCESS);
}


This can be compiled with a command like:

gcc gsl_ran_demo.c -o gsl_ran_demo -lgsl -lgslcblas


and run with

./gsl_ran_demo


#### SPRNG

When writing parallel Monte Carlo codes, it is important to be able to use independent streams of random numbers on each processor. Although it is tempting to “fudge” things by using a random number generator with a different seed on each processor, this does not guarantee independence of the streams, and an unfortunate choice of seeds could potentially lead to bad behaviour of your algorithm. The solution to this problem is to use a parallel random number generator (PRNG), designed specifically to give independent streams on different processors. Unfortunately the GSL does not have native support for such parallel random number generators, so an external library should be used. SPRNG 2.0 is a popular choice. SPRNG is designed so that it can be used with MPI, but also that it does not have to be. This is an issue, as the standard binary packages distributed with Ubuntu (libsprng2, libsprng2-dev) are compiled without MPI support. If you are going to be using SPRNG with MPI, things are simpler with MPI support, so it makes sense to download sprng2.0b.tar.gz from the SPRNG web site, and build it from source, carefully following the instructions for including MPI support. If you are not familiar with building libraries from source, you may need help from someone who is.

Once you have compiled SPRNG with MPI support, you can test it with the following code:

#include <stdio.h>
#include <stdlib.h>
#include <mpi.h>
#define SIMPLE_SPRNG
#define USE_MPI
#include "sprng.h"

int main(int argc,char *argv[])
{
double rn; int i,k;
MPI_Init(&argc,&argv);
MPI_Comm_rank(MPI_COMM_WORLD,&k);
init_sprng(DEFAULT_RNG_TYPE,0,SPRNG_DEFAULT);
for (i=0;i<10;i++)
{
rn = sprng();
printf("Process %d, random number %d: %f\n", k, i+1, rn);
}
MPI_Finalize();
exit(EXIT_SUCCESS);
}


which can be compiled with a command like:

mpicc -I/usr/local/src/sprng2.0/include -L/usr/local/src/sprng2.0/lib -o sprng_demo sprng_demo.c -lsprng -lgmp


You will need to edit paths here to match your installation. If if builds, it can be run on 2 processors with a command like:

mpirun -np 2 sprng_demo


If it doesn’t build, there are many possible reasons. Check the error messages carefully. However, if the compilation fails at the linking stage with obscure messages about not being able to find certain SPRNG MPI functions, one possibility is that the SPRNG library has not been compiled with MPI support.

The problem with SPRNG is that it only provides a uniform random number generator. Of course we would really like to be able to use the SPRNG generator in conjunction with all of the sophisticated GSL routines for non-uniform random number generation. Many years ago I wrote a small piece of code to accomplish this, gsl-sprng.h. Download this and put it in your include path for the following example:

#include <mpi.h>
#include <gsl/gsl_rng.h>
#include "gsl-sprng.h"
#include <gsl/gsl_randist.h>

int main(int argc,char *argv[])
{
int i,k,po; gsl_rng *r;
MPI_Init(&argc,&argv);
MPI_Comm_rank(MPI_COMM_WORLD,&k);
r=gsl_rng_alloc(gsl_rng_sprng20);
for (i=0;i<10;i++)
{
po = gsl_ran_poisson(r,2.0);
printf("Process %d, random number %d: %d\n", k, i+1, po);
}
MPI_Finalize();
exit(EXIT_SUCCESS);
}


A new GSL RNG, gsl_rng_sprng20 is created, by including gsl-sprng.h immediately after gsl_rng.h. If you want to set a seed, do so in the usual GSL way, but make sure to set it to be the same on each processor. I have had several emails recently from people who claim that gsl-sprng.h “doesn’t work”. All I can say is that it still works for me! I suspect the problem is that people are attempting to use it with a version of SPRNG without MPI support. That won’t work… Check that the previous SPRNG example works, first.

I can compile and run the above code with

mpicc -I/usr/local/src/sprng2.0/include -L/usr/local/src/sprng2.0/lib -o gsl-sprng_demo gsl-sprng_demo.c -lsprng -lgmp -lgsl -lgslcblas
mpirun -np 2 gsl-sprng_demo


#### Parallel Monte Carlo

Now we have parallel random number streams, we can think about how to do parallel Monte Carlo simulations. Here is a simple example:

#include <math.h>
#include <mpi.h>
#include <gsl/gsl_rng.h>
#include "gsl-sprng.h"

int main(int argc,char *argv[])
{
int i,k,N; double u,ksum,Nsum; gsl_rng *r;
MPI_Init(&argc,&argv);
MPI_Comm_size(MPI_COMM_WORLD,&N);
MPI_Comm_rank(MPI_COMM_WORLD,&k);
r=gsl_rng_alloc(gsl_rng_sprng20);
for (i=0;i<10000;i++) {
u = gsl_rng_uniform(r);
ksum += exp(-u*u);
}
MPI_Reduce(&ksum,&Nsum,1,MPI_DOUBLE,MPI_SUM,0,MPI_COMM_WORLD);
if (k == 0) {
printf("Monte carlo estimate is %f\n", (Nsum/10000)/N );
}
MPI_Finalize();
exit(EXIT_SUCCESS);
}


which calculates a Monte Carlo estimate of the integral

$\displaystyle I=\int_0^1 \exp(-u^2)du$

using 10k variates on each available processor. The MPI command MPI_Reduce is used to summarise the values obtained independently in each process. I compile and run with

mpicc -I/usr/local/src/sprng2.0/include -L/usr/local/src/sprng2.0/lib -o monte-carlo monte-carlo.c -lsprng -lgmp -lgsl -lgslcblas
mpirun -np 2 monte-carlo


#### Parallel chains MCMC

Once parallel Monte Carlo has been mastered, it is time to move on to parallel MCMC. First it makes sense to understand how to run parallel MCMC chains in an MPI environment. I will illustrate this with a simple Metropolis-Hastings algorithm to sample a standard normal using uniform proposals, as discussed in a previous post. Here an independent chain is run on each processor, and the results are written into separate files.

#include <gsl/gsl_rng.h>
#include "gsl-sprng.h"
#include <gsl/gsl_randist.h>
#include <mpi.h>

int main(int argc,char *argv[])
{
int k,i,iters; double x,can,a,alpha; gsl_rng *r;
FILE *s; char filename[15];
MPI_Init(&argc,&argv);
MPI_Comm_rank(MPI_COMM_WORLD,&k);
if ((argc != 3)) {
if (k == 0)
fprintf(stderr,"Usage: %s <iters> <alpha>\n",argv[0]);
MPI_Finalize(); return(EXIT_FAILURE);
}
iters=atoi(argv[1]); alpha=atof(argv[2]);
r=gsl_rng_alloc(gsl_rng_sprng20);
sprintf(filename,"chain-%03d.tab",k);
s=fopen(filename,"w");
if (s==NULL) {
perror("Failed open");
MPI_Finalize(); return(EXIT_FAILURE);
}
x = gsl_ran_flat(r,-20,20);
fprintf(s,"Iter X\n");
for (i=0;i<iters;i++) {
can = x + gsl_ran_flat(r,-alpha,alpha);
a = gsl_ran_ugaussian_pdf(can) / gsl_ran_ugaussian_pdf(x);
if (gsl_rng_uniform(r) < a)
x = can;
fprintf(s,"%d %f\n",i,x);
}
fclose(s);
MPI_Finalize(); return(EXIT_SUCCESS);
}


I can compile and run this with the following commands

mpicc -I/usr/local/src/sprng2.0/include -L/usr/local/src/sprng2.0/lib -o mcmc mcmc.c -lsprng -lgmp -lgsl -lgslcblas
mpirun -np 2 mcmc 100000 1


#### Parallelising a single MCMC chain

The parallel chains approach turns out to be surprisingly effective in practice. Obviously the disadvantage of that approach is that “burn in” has to be repeated on every processor, which limits how much efficiency gain can be acheived by running across many processors. Consequently it is often desirable to try and parallelise a single MCMC chain. As MCMC algorithms are inherently sequential, parallelisation is not completely trivial, and most (but not all) approaches to parallelising a single MCMC chain focus on the parallelisation of each iteration. In order for this to be worthwhile, it is necessary that the problem being considered is non-trivial, having a large state space. The strategy is then to divide the state space into “chunks” which can be updated in parallel. I don’t have time to go through a real example in detail in this blog post, but fortunately I wrote a book chapter on this topic almost 10 years ago which is still valid and relevant today. The citation details are:

Wilkinson, D. J. (2005) Parallel Bayesian Computation, Chapter 16 in E. J. Kontoghiorghes (ed.) Handbook of Parallel Computing and Statistics, Marcel Dekker/CRC Press, 481-512.

The book was eventually published in 2005 after a long delay. The publisher which originally commisioned the handbook (Marcel Dekker) was taken over by CRC Press before publication, and the project lay dormant for a couple of years until the new publisher picked it up again and decided to proceed with publication. I have a draft of my original submission in PDF which I recommend reading for further information. The code examples used are also available for download, including several of the examples used in this post, as well as an extended case study on parallelisation of a single chain for Bayesian inference in a stochastic volatility model. Although the chapter is nearly 10 years old, the issues discussed are all still remarkably up-to-date, and the code examples all still work. I think that is a testament to the stability of the technology adopted (C, MPI, GSL). Some of the other handbook chapters have not stood the test of time so well.

For basic information on getting started with MPI and key MPI commands for implementing parallel MCMC algorithms, the above mentioned book chapter is a reasonable place to start. Read it all through carefully, run the examples, and carefully study the code for the parallel stochastic volatility example. Once that is understood, you should find it possible to start writing your own parallel MCMC algorithms. For further information about more sophisticated MPI usage and additional commands, I find the annotated specification: MPI – The complete reference to be as good a source as any.