## First steps with monads in Scala

### Introduction

In the previous post I gave a quick introduction to some important concepts in functional programming, such as HOFs, closures, currying and partial application, and hopefully gave some insight into why these concepts might be useful in the context of scientific computing. Another concept that is very important in modern functional programming is that of the monad. Monads are one of those concepts that turns out to be very simple and intuitive once you “get it”, but completely impenetrable until you do! Now, there zillions of monad tutorials out there, and I don’t think that I have anything particularly insightful to add to the discussion. That said, most of the tutorials focus on problems and examples that are some way removed from the interests of statisticians and scientific programmers. So in this post I want to try and give a very informal and intuitive introduction to the monad concept in a way that I hope will resonate with people from a more scientific computing background.

The term “monad” is borrowed from that of the corresponding concept in category theory. The connection between functional programming and category theory is strong and deep. I intend to expore this more in future posts, but for this post the connection is not important and no knowledge of category theory is assumed (or imparted!).

### Functors and Monads

#### Maps and Functors

All of the code used in this post in contained in the first-monads directory of my blog repo. The best way to follow this post is to copy-and-paste commands one-at-a-time from this post to a Scala REPL or sbt console. Note that only the numerical linear algebra examples later in this post require any non-standard dependencies.

The map method is one of the first concepts one meets when beginning functional programming. It is a higher order method on many (immutable) collection and other container types. Let’s start by looking at how map operates on Lists.

val x = (0 to 4).toList
// x: List[Int] = List(0, 1, 2, 3, 4)
val x2 = x map { x => x * 3 }
// x2: List[Int] = List(0, 3, 6, 9, 12)
val x3 = x map { _ * 3 }
// x3: List[Int] = List(0, 3, 6, 9, 12)
val x4 = x map { _ * 0.1 }
// x4: List[Double] = List(0.0, 0.1, 0.2, 0.30000000000000004, 0.4)


The last example shows that a List[T] can be converted to a List[S] if map is passed a function of type T => S. Of course there’s nothing particularly special about List here. It works with other collection types in the same way, as the following example with (immutable) Vector illustrates:

val xv = x.toVector
// xv: Vector[Int] = Vector(0, 1, 2, 3, 4)
val xv2 = xv map { _ * 0.2 }
// xv2: scala.collection.immutable.Vector[Double] = Vector(0.0, 0.2, 0.4, 0.6000000000000001, 0.8)
val xv3 = for (xi <- xv) yield (xi * 0.2)
// xv3: scala.collection.immutable.Vector[Double] = Vector(0.0, 0.2, 0.4, 0.6000000000000001, 0.8)


Note here that the for comprehension generating xv3 is exactly equivalent to the map call generating xv2 – the for-comprehension is just syntactic sugar for the map call. The benefit of this syntax will become apparent in the more complex examples we consider later.

Many collection and other container types have a map method that behaves this way. Any parametrised type that does have a map method like this is known as a Functor. Again, the name is due to category theory, but that doesn’t matter for this post. From a Scala-programmer perspective, a functor can be thought of as a trait, in pseudo-code as

trait F[T] {
def map(f: T => S): F[S]
}


with F representing the functor. In fact it turns out to be better to think of a functor as a type class, but that is yet another topic for a future post… Also note that to be a functor in the strict sense (from a category theory perspective), the map method must behave sensibly – that is, it must satisfy the functor laws. But again, I’m keeping things informal and intuitive for this post – there are plenty of other monad tutorials which emphasise the category theory connections.

#### FlatMap and Monads

Once we can map functions over elements of containers, we soon start mapping functions which themselves return values of the container type. eg. we can map a function returning a List over the elements of a List, as illustrated below.

val x5 = x map { x => List(x - 0.1, x + 0.1) }
// x5: List[List[Double]] = List(List(-0.1, 0.1), List(0.9, 1.1), List(1.9, 2.1), List(2.9, 3.1), List(3.9, 4.1))


Clearly this returns a list-of-lists. Sometimes this is what we want, but very often we actually want to flatten down to a single list so that, for example, we can subsequently map over all of the elements of the base type with a single map. We could take the list-of-lists and then flatten it, but this pattern is so common that the act of mapping and then flattening is often considered to be a basic operation, often known in Scala as flatMap. So for our toy example, we could carry out the flatMap as follows.

val x6 = x flatMap { x => List(x - 0.1, x + 0.1) }
// x6: List[Double] = List(-0.1, 0.1, 0.9, 1.1, 1.9, 2.1, 2.9, 3.1, 3.9, 4.1)


The ubiquity of this pattern becomes more apparent when we start thinking about iterating over multiple collections. For example, suppose now that we have two lists, x and y, and that we want to iterate over all pairs of elements consisting of one element from each list.

val y = (0 to 12 by 2).toList
// y: List[Int] = List(0, 2, 4, 6, 8, 10, 12)
val xy = x flatMap { xi => y map { yi => xi * yi } }
// xy: List[Int] = List(0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 6, 8, 10, 12, 0, 4, 8, 12, 16, 20, 24, 0, 6, 12, 18, 24, 30, 36, 0, 8, 16, 24, 32, 40, 48)


This pattern of having one or more nested flatMaps followed by a final map in order to iterate over multiple collections is very common. It is exactly this pattern that the for-comprehension is syntactic sugar for. So we can re-write the above using a for-comprehension

val xy2 = for {
xi <- x
yi <- y
} yield (xi * yi)
// xy2: List[Int] = List(0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 6, 8, 10, 12, 0, 4, 8, 12, 16, 20, 24, 0, 6, 12, 18, 24, 30, 36, 0, 8, 16, 24, 32, 40, 48)


This for-comprehension (usually called a for-expression in Scala) has an intuitive syntax reminiscent of the kind of thing one might write in an imperative language. But it is important to remember that <- is not actually an imperative assignment. The for-comprehension really does expand to the pure-functional nested flatMap and map call given above.

Recalling that a functor is a parameterised type with a map method, we can now say that a monad is just a functor which also has a flatMap method. We can write this in pseudo-code as

trait M[T] {
def map(f: T => S): M[S]
def flatMap(f: T => M[S]): M[S]
}


So far the functors and monads we have been working with have been collections, but not all monads are collections, and in fact collections are in some ways atypical examples of monads. Many monads are containers or wrappers, so it will be useful to see examples of monads which are not collections.

One of the first monads that many people encounter is the Option monad (referred to as the Maybe monad in Haskell, and Optional in Java 8). You can think of it as being a strange kind of “collection” that can contain at most one element. So it will either contain an element or it won’t, and so can be used to wrap the result of a computation which might fail. If the computation succeeds, the value computed can be wrapped in the Option (using the type Some), and if it fails, it will not contain a value of the required type, but simply be the value None. It provides a referentially transparent and type-safe alternative to raising exceptions or returning NULL references. We can transform Options using map.

val three = Option(3)
// three: Option[Int] = Some(3)
val twelve = three map (_ * 4)
// twelve: Option[Int] = Some(12)


But when we start combining the results of multiple computations that could fail, we run into exactly the same issues as before.

val four = Option(4)
// four: Option[Int] = Some(4)
val twelveB = three map (i => four map (i * _))
// twelveB: Option[Option[Int]] = Some(Some(12))


Here we have ended up with an Option wrapped in another Option, which is not what we want. But we now know the solution, which is to replace the first map with flatMap, or better still, use a for-comprehension.

val twelveC = three flatMap (i => four map (i * _))
// twelveC: Option[Int] = Some(12)
val twelveD = for {
i <- three
j <- four
} yield (i * j)
// twelveD: Option[Int] = Some(12)


Again, the for-comprehension is a little bit easier to understand than the chaining of calls to flatMap and map. Note that in the for-comprehension we don’t worry about whether or not the Options actually contain values – we just concentrate on the “happy path”, where they both do, safe in the knowledge that the Option monad will take care of the failure cases for us. Two of the possible failure cases are illustrated below.

val oops: Option[Int] = None
// oops: Option[Int] = None
val oopsB = for {
i <- three
j <- oops
} yield (i * j)
// oopsB: Option[Int] = None
val oopsC = for {
i <- oops
j <- four
} yield (i * j)
// oopsC: Option[Int] = None


This is a typical benefit of code written in a monadic style. We chain together multiple computations thinking only about the canonical case and trusting the monad to take care of any additional computational context for us.

#### IEEE floating point and NaN

Those with a background in scientific computing are probably already familiar with the NaN value in IEEE floating point. In many regards, this value and the rules around its behaviour mean that Float and Double types in IEEE compliant languages behave as an Option monad with NaN as the None value. This is simply illustrated below.

val nan = Double.NaN
3.0 * 4.0
// res0: Double = 12.0
3.0 * nan
// res1: Double = NaN
nan * 4.0
// res2: Double = NaN


The NaN value arises naturally when computations fail. eg.

val nanB = 0.0 / 0.0
// nanB: Double = NaN


This Option-like behaviour of Float and Double means that it is quite rare to see examples of Option[Float] or Option[Double] in Scala code. But there are some disadvantages of the IEEE approach, as discussed elsewhere. Also note that this only works for Floats and Doubles, and not for other types, including, say, Int.

val nanC=0/0
// This raises a runtime exception!


#### Option for matrix computations

Good practical examples of scientific computations which can fail crop up frequently in numerical linear algebra, so it’s useful to see how Option can simplify code in that context. Note that the code in this section requires the Breeze library, so should be run from an sbt console using the sbt build file associated with this post.

In statistical applications, one often needs to compute the Cholesky factorisation of a square symmetric matrix. This operation is built into Breeze as the function cholesky. However the factorisation will fail if the matrix provided is not positive semi-definite, and in this case the cholesky function will throw a runtime exception. We will use the built in cholesky function to create our own function, safeChol (using a monad called Try which is closely related to the Option monad) returning an Option of a matrix rather than a matrix. This function will not throw an exception, but instead return None in the case of failure, as illustrated below.

import breeze.linalg._
def safeChol(m: DenseMatrix[Double]): Option[DenseMatrix[Double]] = scala.util.Try(cholesky(m)).toOption
val m = DenseMatrix((2.0, 1.0), (1.0, 3.0))
val c = safeChol(m)
// c: Option[breeze.linalg.DenseMatrix[Double]] =
// Some(1.4142135623730951  0.0
// 0.7071067811865475  1.5811388300841898  )

val m2 = DenseMatrix((1.0, 2.0), (2.0, 3.0))
val c2 = safeChol(m2)
// c2: Option[breeze.linalg.DenseMatrix[Double]] = None


A Cholesky factorisation is often followed by a forward or backward solve. This operation may also fail, independently of whether the Cholesky factorisation fails. There doesn’t seem to be a forward solve function directly exposed in the Breeze API, but we can easily define one, which I call dangerousForwardSolve, as it will throw an exception if it fails, just like the cholesky function. But just as for the cholesky function, we can wrap up the dangerous function into a safe one that returns an Option.

import com.github.fommil.netlib.BLAS.{getInstance => blas}
def dangerousForwardSolve(A: DenseMatrix[Double], y: DenseVector[Double]): DenseVector[Double] = {
val yc = y.copy
blas.dtrsv("L", "N", "N", A.cols, A.toArray, A.rows, yc.data, 1)
yc
}
def safeForwardSolve(A: DenseMatrix[Double], y: DenseVector[Double]): Option[DenseVector[Double]] = scala.util.Try(dangerousForwardSolve(A, y)).toOption


Now we can write a very simple function which chains these two operations together, as follows.

def safeStd(A: DenseMatrix[Double], y: DenseVector[Double]): Option[DenseVector[Double]] = for {
L <- safeChol(A)
z <- safeForwardSolve(L, y)
} yield z

safeStd(m,DenseVector(1.0,2.0))
// res15: Option[breeze.linalg.DenseVector[Double]] = Some(DenseVector(0.7071067811865475, 0.9486832980505138))


Note how clean and simple this function is, concentrating purely on the “happy path” where both operations succeed and letting the Option monad worry about the three different cases where at least one of the operations fails.

#### The Future monad

Let’s finish with a monad for parallel and asynchronous computation, the Future monad. The Future monad is used for wrapping up slow computations and dispatching them to another thread for completion. The call to Future returns immediately, allowing the calling thread to continue while the additional thread processes the slow work. So at that stage, the Future will not have completed, and will not contain a value, but at some (unpredictable) time in the future it (hopefully) will (hence the name). In the following code snippet I construct two Futures that will each take at least 10 seconds to complete. On the main thread I then use a for-comprehension to chain the two computations together. Again, this will return immediately returning another Future that at some point in the future will contain the result of the derived computation. Then, purely for illustration, I force the main thread to stop and wait for that third future (f3) to complete, printing the result to the console.

import scala.concurrent.duration._
import scala.concurrent.{Future,ExecutionContext,Await}
import ExecutionContext.Implicits.global
val f1=Future{
1 }
val f2=Future{
2 }
val f3=for {
v1 <- f1
v2 <- f2
} yield (v1+v2)
println(Await.result(f3,30.second))


When you paste this into your console you should observe that you get the result in 10 seconds, as f1 and f2 execute in parallel on separate threads. So the Future monad is one (of many) ways to get started with parallel and async programming in Scala.

## Introduction to the particle Gibbs sampler

### Introduction

Particle MCMC (the use of approximate SMC proposals within exact MCMC algorithms) is arguably one of the most important developments in computational Bayesian inference of the 21st Century. The key concepts underlying these methods are described in a famously impenetrable “read paper” by Andrieu et al (2010). Probably the most generally useful method outlined in that paper is the particle marginal Metropolis-Hastings (PMMH) algorithm that I have described previously – that post is required preparatory reading for this one.

In this post I want to discuss some of the other topics covered in the pMCMC paper, leading up to a description of the particle Gibbs sampler. The basic particle Gibbs algorithm is arguably less powerful than PMMH for a few reasons, some of which I will elaborate on. But there is still a lot of active research concerning particle Gibbs-type algorithms, which are attempting to address some of the deficiencies of the basic approach. Clearly, in order to understand and appreciate the recent developments it is first necessary to understand the basic principles, and so that is what I will concentrate on here. I’ll then finish with some pointers to more recent work in this area.

### PIMH

I will adopt the same approach and notation as for my post on the PMMH algorithm, using a simple bootstrap particle filter for a state space model as the SMC proposal. It is simplest to understand particle Gibbs first in the context of known static parameters, and so it is helpful to first reconsider the special case of the PMMH algorithm where there are no unknown parameters and only the state path, $x$ of the process is being updated. That is, we target $p(x|y)$ (for known, fixed, $\theta$) rather than $p(\theta,x|y)$. This special case is known as the particle independent Metropolis-Hastings (PIMH) sampler.

Here we envisage proposing a new path $x_{0:T}^\star$ using a bootstrap filter, and then accepting the proposal with probability $\min\{1,A\}$, where $A$ is the Metropolis-Hastings ratio

$\displaystyle A = \frac{\hat{p}(y_{1:T})^\star}{\hat{p}(y_{1:T})},$

where $\hat{p}(y_{1:T})^\star$ is the bootstrap filter’s estimate of marginal likelihood for the new path, and $\hat{p}(y_{1:T})$ is the estimate associated with the current path. Again using notation from the previous post it is clear that this ratio targets a distribution on the joint space of all simulated random variables proportional to

$\displaystyle \hat{p}(y_{1:T})\tilde{q}(\mathbf{x}_0,\ldots,\mathbf{x}_T,\mathbf{a}_0,\ldots,\mathbf{a}_{T-1})$

and that in this case the marginal distribution of the accepted path is exactly $p(x_{0:T}|y_{1:T})$. Again, be sure to see the previous post for the explanation.

### Conditional SMC update

So far we have just recapped the previous post in the case of known parameters, but it gives us insight in how to proceed. A general issue with Metropolis independence samplers in high dimensions is that they often exhibit “sticky” behaviour, whereby an unusually “good” accepted path is hard to displace. This motivates consideration of a block-Gibbs-style algorithm where updates are used that are always accepted. It is clear that simply running a bootstrap filter will target the particle filter distribution

$\tilde{q}(\mathbf{x}_0,\ldots,\mathbf{x}_T,\mathbf{a}_0,\ldots,\mathbf{a}_{T-1})$

and so the marginal distribution of the accepted path will be the approximate $\hat{p}(x_{0:T}|y_{1:T})$ rather than the exact conditional distribution $p(x_{0:T}|y_{1:T})$. However, we know from consideration of the PIMH algorithm that what we really want to do is target the slightly modified distribution proportional to

$\displaystyle \hat{p}(y_{1:T})\tilde{q}(\mathbf{x}_0,\ldots,\mathbf{x}_T,\mathbf{a}_0,\ldots,\mathbf{a}_{T-1})$,

as this will lead to accepted paths with the exact marginal distribution. For the PIMH this modification is achieved using a Metropolis-Hastings correction, but we now try to avoid this by instead conditioning on the previously accepted path. For this target the accepted paths have exactly the required marginal distribution, so we now write the target as the product of the marginal for the current path times a conditional for all of the remaining variables.

$\displaystyle \frac{p(x_{0:T}^k|y_{1:T})}{M^T} \times \frac{M^T}{p(x_{0:T}^k|y_{1:T})} \hat{p}(y_{1:T})\tilde{q}(\mathbf{x}_0,\ldots,\mathbf{x}_T,\mathbf{a}_0,\ldots,\mathbf{a}_{T-1})$

where in addition to the correct marginal for $x$ we assume iid uniform ancestor indices. The important thing to note here is that the conditional distribution of the remaining variables simplifies to

$\displaystyle \frac{\tilde{q}(\mathbf{x}_0,\ldots,\mathbf{x}_T,\mathbf{a}_0,\ldots,\mathbf{a}_{T-1})} {\displaystyle p(x_0^{b_0^k})\left[\prod_{t=0}^{T-1} \pi_t^{b_t^k}p\left(x_{t+1}^{b_{t+1}^k}|x_t^{b_t^k}\right)\right]}$.

The terms in the denominator are precisely the terms in the numerator corresponding to the current path, and hence “cancel out” the current path terms in the numerator. It is therefore clear that we can sample directly from this conditional distribution by running a bootstrap particle filter that includes the current path and which leaves the current path fixed. This is the conditional SMC (CSMC) update, which here is just a conditional bootstrap particle filter update. It is clear from the form of the conditional density how this filter must be constructed, but for completeness it is described below.

The bootstrap filter is run conditional on one trajectory. This is usually the trajectory sampled at the last run of the particle filter. The idea is that you do not sample new state or ancestor values for that one trajectory. Note that this guarantees that the conditioned on trajectory survives the filter right through to the final sweep of the filter at which point a new trajectory is picked from the current selection of $M$ paths, of which the conditioned-on trajectory is one.

Let $x_{1:T} = (x_1^{b_1},x_2^{b_2},\ldots,x_T^{b_T})$ be the path that is to be conditioned on, with ancestral lineage $b_{1:T}$. Then, for $k\not= b_1$, sample $x_0^k \sim p(x_0)$ and set $\pi_0^k=1/M$. Now suppose that at time $t$ we have a weighted sample from $p(x_t|y_{1:t})$. First resample by sampling $a_t^k\sim \mathcal{F}(a_t^k|\boldsymbol{\pi}_t),\ \forall k\not= b_t$. Next sample $x_{t+1}^k\sim p(x_{t+1}^k|x_t^{a_t^k}),\ \forall k\not=b_t$. Then for all $k$ set $w_{t+1}^k=p(y_{t+1}|x_{t+1}^k)$ and normalise with $\pi_{t+1}^k=w_{t+1}^k/\sum_{i=1}^M w_{t+1}^i$. Propagate this weighted set of particles to the next time point. At time $T$ select a single trajectory by sampling $k'\sim \mathcal{F}(k'|\boldsymbol{\pi}_T)$.

This defines a block Gibbs sampler which updates $2(M-1)T+1$ of the $2MT+1$ random variables in the augmented state space at each iteration. Since the block of variables to be updated is random, this defines an ergodic sampler for $M\geq2$ particles, and we have explained why the marginal distribution of the selected trajectory is the exact conditional distribution.

Before going on to consider the introduction of unknown parameters, it is worth considering the limitations of this method. One of the main motivations for considering a Gibbs-style update was concern about the “stickiness” of a Metropolis independence sampler. However, it is clear that conditional SMC updates also have the potential to stick. For a large number of time points, particle filter genealogies coalesce, or degenerate, to a single path. Since here we are conditioning on the current path, if there is coalescence, it is guaranteed to be to the previous path. So although the conditional SMC updates are always accepted, it is likely that much of the new path will be identical to the previous path, which is just another kind of “sticking” of the sampler. This problem with conditional SMC and particle Gibbs more generally is well recognised, and quite a bit of recent research activity in this area is directed at alleviating this sticking problem. The most obvious strategy to use is “backward sampling” (Godsill et al, 2004), which has been used in this context by Lindsten and Schon (2012), Whiteley et al (2010), and Chopin and Singh (2013), among others. Another related idea is “ancestor sampling” (Lindsten et al, 2014), which can be done in a single forward pass. Both of these techniques work well, but both rely on the tractability of the transition kernel of the state space model, which can be problematic in certain applications.

### Particle Gibbs sampling

As we are working in the context of Gibbs-style updates, the introduction of static parameters, $\theta$, into the problem is relatively straightforward. It turns out to be correct to do the obvious thing, which is to alternate between sampling $\theta$ given $y$ and the currently sampled path, $x$, and sampling a new path using a conditional SMC update, conditional on the previous path in addition to $\theta$ and $y$. Although this is the obvious thing to do, understanding exactly why it works is a little delicate, due to the augmented state space and conditional SMC update. However, it is reasonably clear that this strategy defines a “collapsed Gibbs sampler” (Lui, 1994), and so actually everything is fine. This particular collapsed Gibbs sampler is relatively easy to understand as a marginal sampler which integrates out the augmented variables, but then nevertheless samples the augmented variables at each iteration conditional on everything else.

Note that the Gibbs update of $\theta$ may be problematic in the context of a state space model with intractable transition kernel.

In a subsequent post I’ll show how to code up the particle Gibbs and other pMCMC algorithms in a reasonably efficient way.

## 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.

## Hypercubes in R (getting started with programming in R)

On the train home from Manchester the other day (no wifi!), I was thinking about exploring high-dimensional constrained parameter spaces (possibly sad, but something that statisticians do!), and it occurred to me that I didn’t have very good intuition about the structure of hypercubes in dimensions d>4. Since I was sat with my netbook in front of me listening to music and too tired from 2 days of meetings to do any real work, I decided to write a bit of code to build, rotate and plot 2-D projections of d-D hypercubes for arbitrary integer d>1. In principle there are many languages I could have done this in, but I knew it would be very quick and easy using R.

R is the de-facto standard language for statistical computing, data analysis and plotting, and is rapidly becoming the standard language for statistical bioinformatics and systems biology. It is free, interactive, and has a mind-boggling array of packages implementing a huge variety of statistical methods, including most state-of-the-art techniques. Further, via the BioConductoR project, it now has an impressive range of packages specifically targeting bioinformatics applications. The language itself is relatively simple and easy-to-learn, and is well-suited to programming with data. That said, it is not the most elegant of programming languages, and (rather confusingly) it has two somewhat incompatible object models, but that is perhaps a topic for another day. There are many available introductions to R, so I’m not going to repeat that here, though I do have an introductory tutorial on an old web page that is still worth a look.

So I wrote the code (it’s only around 50 lines, and took around 30 minutes to write and debug), and running it produces plots like these:

I’ve appended the code below, as it illustrates a lot of basic R programming concepts. Essentially, the idea is to use vectors and matrices as much as possible, but not to be really obsessed about avoiding for loops if performance isn’t critical. The code isn’t commented, but it’s highly structured and easy to understand. The function makevertices creates a matrix whose columns are the vertex locations of a hypercube. The function makeedges creates a matrix whose columns give pairs of vertex indices representing the edges of the hypercube (pairs of vertices a distance of 1 apart). The function rotate picks a pair of coordinate axes at random and rotates the hypercube in that plane. The rest are obvious, and the file finishes with some example ways to call the code. As well as illustrating the basic programming concepts such as functions, vectors, matrices and matrix operations and looping, the code also illustrates several other useful concepts such as integer division and modular arithmetic, conditional execution, optional arguments, dynamically growing a matrix, randomly sampling without replacement, plotting lines, plot options, anonymous function arguments, etc.

# hypercube.R
# code to rotate arbitrary d-dimensional hypercubes

# function definitions
makevertices=function(d)
{
numv=2^d
vertices=matrix(0,nrow=d,ncol=numv)
for (j in 1:numv) {
count=j-1
for (i in 1:d) {
vertices[i,j]=count%%2
count=count%/%2
}
}
vertices
}

makeedges=function(vertices)
{
d=dim(vertices)[1]
numv=dim(vertices)[2]
edges=NULL
for (j in 1:numv) {
for (i in j:numv) {
if (sum(abs((vertices[,i]-vertices[,j])))==1) {
edges=cbind(edges,c(i,j))
}
}
}
edges
}

rotate=function(vertices,angle=0.2)
{
d=dim(vertices)[1]
axes=sort(sample(1:d,2))
rotmat=diag(rep(1,d))
rotmat[axes[1],axes[1]]=cos(angle)
rotmat[axes[2],axes[1]]=sin(angle)
rotmat[axes[1],axes[2]]=-sin(angle)
rotmat[axes[2],axes[2]]=cos(angle)
vertices=rotmat %*% vertices
vertices
}

plotcube=function(vertices,edges,col=2,lwd=2)
{
d=dim(vertices)[1]
plot(NULL,xlim=c(-2,2),ylim=c(-2,2),main=paste(d,"-d hypercube",sep=""),xlab="x(1)",ylab="x(2)")
for (i in 1:dim(edges)[2]) {
lines(c(vertices[1,edges[1,i]],vertices[1,edges[2,i]]),c(vertices[2,edges[1,i]],vertices[2,edges[2,i]]),col=col,lwd=lwd)
}
}

rotationplot=function(vertices,edges,rotations=20,...)
{
for (count in 1:rotations) {
vertices=rotate(vertices,...)
plotcube(vertices,edges)
}
}

hypercube=function(d=4,...)
{
vertices=makevertices(d)
edges=makeedges(vertices)
vertices=2*vertices
vertices=vertices-1
rotationplot(vertices,edges,...)
}

# examples

# hypercube()
# hypercube(3)
# hypercube(4,angle=0.1)
# hypercube(5,rotations=30)

# eof