## Part 2: The log posterior

### Introduction

This is the second part in a series of posts on MCMC-based Bayesian inference for a logistic regression model. If you are new to this series, please go back to Part 1.

In the previous post we looked at the basic modelling concepts, and how to fit the model using a variety of PPLs. In this post we will prepare for doing MCMC by considering the problem of computing the unnormalised log posterior for the model. We will then see how this posterior can be implemented in several different languages and libraries.

## Derivation

### Basic structure

In Bayesian inference the posterior distribution is just the conditional distribution of the model parameters given the data, and therefore proportional to the joint distribution of the model and data. We often write this as

$\displaystyle \pi(\theta|y) \propto \pi(\theta,y) = \pi(\theta)\pi(y|\theta).$

Taking logs we have

$\displaystyle \log \pi(\theta, y) = \log \pi(\theta) + \log \pi(y|\theta).$

So (up to an additive constant) the log posterior is just the sum of the log prior and log likelihood. There are many good (numerical) reasons why we try to work exclusively with the log posterior and try to avoid ever evaluating the raw posterior density.

For our example logistic regression model, the parameter vector $\theta$ is just the vector of regression coefficients, $\beta$. We assumed independent mean zero normal priors for the components of this vector, so the log prior is just the sum of logs of normal densities. Many scientific libraries will have built-in functions for returning the log-pdf of standard distributions, but if an explicit form is required, the log of the density of a $N(0,\sigma^2)$ at $x$ is just

$\displaystyle -\log(2\pi)/2 - \log|\sigma| - x^2/(2\sigma^2),$

and the initial constant term normalising the density can often be dropped.

### Log-likelihood (first attempt)

Information from the data comes into the log posterior via the log-likelihood. The typical way to derive the likelihood for problems of this type is to assume the usual binary encoding of the data (success 1, failure 0). Then, for a Bernoulli observation with probability $p_i,\ i=1,\ldots,n$, the likelihood associated with observation $y_i$ is

$\displaystyle f(y_i|p_i) = \left[ \hphantom{1-}p_i \quad :\ y_i=1 \atop 1-p_i \quad :\ y_i=0 \right. \quad = \quad p_i^{y_i}(1-p_i)^{1-y_i}.$

Taking logs and then switching to parameter $\eta_i=\text{logit}(p_i)$ we have

$\displaystyle \log f(y_i|\eta_i) = y_i\eta_i - \log(1+e^{\eta_i}),$

and summing over $n$ observations gives the log likelihood

$\displaystyle \log\pi(y|\eta) \equiv \ell(\eta;y) = y\cdot \eta - \mathbf{1}\cdot\log(\mathbf{1}+\exp{\eta}).$

In the context of logistic regression, $\eta$ is the linear predictor, so $\eta=X\beta$, giving

$\displaystyle \ell(\beta;y) = y^\textsf{T}X\beta - \mathbf{1}^\textsf{T}\log(\mathbf{1}+\exp[X\beta]).$

This is a perfectly good way of expressing the log-likelihood, and we will come back to it later when we want the gradient of the log-likelihood, but it turns out that there is a similar-but-different way of deriving it that results in an expression that is equivalent but slightly cheaper to evaluate.

### Log-likelihood (second attempt)

For our second attempt, we will assume that the data is coded in a different way. Instead of the usual binary encoding, we will assume that the observation $\tilde y_i$ is 1 for success and -1 for failure. This isn’t really a problem, since the two encodings are related by $\tilde y_i = 2y_i-1$. This new encoding is convenient in the context of a logit parameterisation since then

$\displaystyle f(y_i|\eta_i) = \left[ p_i \ :\ \tilde y_i=1\atop 1-p_i\ :\ \tilde y_i=-1 \right. \ = \ \left[ (1+e^{-\eta_i})^{-1} \ :\ \tilde y_i=1\atop (1+e^{\eta_i})^{-1} \ :\ \tilde y_i=-1 \right. \ = \ (1+e^{-\tilde y_i\eta_i})^{-1} ,$

and hence

$\displaystyle \log f(y_i|\eta_i) = -\log(1+e^{-\tilde y_i\eta_i}).$

Summing over observations gives

$\displaystyle \ell(\eta;\tilde y) = -\mathbf{1}\cdot \log(\mathbf{1}+\exp[-\tilde y\circ \eta]),$

where $\circ$ denotes the Hadamard product. Substituting $\eta=X\beta$ gives the log-likelihood

$\displaystyle \ell(\beta;\tilde y) = -\mathbf{1}^\textsf{T} \log(\mathbf{1}+\exp[-\tilde y\circ X\beta]).$

This likelihood is a bit cheaper to evaluate that the one previously derived. If we prefer to write in terms of the original data encoding, we can obviously do so as

$\displaystyle \ell(\beta; y) = -\mathbf{1}^\textsf{T} \log(\mathbf{1}+\exp[-(2y-\mathbf{1})\circ (X\beta)]),$

and in practice, it is this version that is typically used. To be clear, as an algebraic function of $\beta$ and $y$ the two functions are different. But they coincide for binary vectors $y$ which is all that matters.

## Implementation

### R

In R we can create functions for evaluating the log-likelihood, log-prior and log-posterior as follows (assuming that X and y are in scope).

ll = function(beta)
sum(-log(1 + exp(-(2*y - 1)*(X %*% beta))))

lprior = function(beta)
dnorm(beta[1], 0, 10, log=TRUE) + sum(dnorm(beta[-1], 0, 1, log=TRUE))

lpost = function(beta) ll(beta) + lprior(beta)


### Python

In Python (with NumPy and SciPy) we can define equivalent functions with

def ll(beta):
return np.sum(-np.log(1 + np.exp(-(2*y - 1)*(X.dot(beta)))))

def lprior(beta):
return (sp.stats.norm.logpdf(beta[0], loc=0, scale=10) +
np.sum(sp.stats.norm.logpdf(beta[range(1,p)], loc=0, scale=1)))

def lpost(beta):
return ll(beta) + lprior(beta)


#### JAX

Python, like R, is a dynamic language, and relatively slow for MCMC algorithms. JAX is a tensor computation framework for Python that embeds a pure functional differentiable array processing language inside Python. JAX can JIT-compile high-performance code for both CPU and GPU, and has good support for parallelism. It is rapidly becoming the preferred way to develop high-performance sampling algorithms within the Python ecosystem. We can encode our log-posterior in JAX as follows.

@jit
def ll(beta):
return jnp.sum(-jnp.log(1 + jnp.exp(-(2*y - 1)*jnp.dot(X, beta))))

@jit
def lprior(beta):
return (jsp.stats.norm.logpdf(beta[0], loc=0, scale=10) +
jnp.sum(jsp.stats.norm.logpdf(beta[jnp.array(range(1,p))], loc=0, scale=1)))

@jit
def lpost(beta):
return ll(beta) + lprior(beta)



### Scala

JAX is a pure functional programming language embedded in Python. Pure functional programming languages are intrinsically more scalable and compositional than imperative languages such as R and Python, and are much better suited to exploit concurrency and parallelism. I’ve given a bunch of talks about this recently, so if you are interested in this, perhaps start with the materials for my Laplace’s Demon talk. Scala and Haskell are arguably the current best popular general purpose functional programming languages, so it is possibly interesting to consider the use of these languages for the development of scalable statistical inference codes. Since both languages are statically typed compiled functional languages with powerful type systems, they can be highly performant. However, neither is optimised for numerical (tensor) computation, so you should not expect that they will have performance comparable with optimised tensor computation frameworks such as JAX. We can encode our log-posterior in Scala (with Breeze) as follows:

  def ll(beta: DVD): Double =
sum(-log(ones + exp(-1.0*(2.0*y - ones)*:*(X * beta))))

def lprior(beta: DVD): Double =
Gaussian(0,10).logPdf(beta(0)) +
sum(beta(1 until p).map(Gaussian(0,1).logPdf(_)))

def lpost(beta: DVD): Double = ll(beta) + lprior(beta)


#### Spark

Apache Spark is a Scala library for distributed "big data" processing on clusters of machines. Despite fundamental differences, there is a sense in which Spark for Scala is a bit analogous to JAX for Python: both Spark and JAX are concerned with scalability, but they are targeting rather different aspects of scalability: JAX is concerned with getting regular sized data processing algorithms to run very fast (on GPUs), whereas Spark is concerned with running huge data processing tasks quickly by distributing work over clusters of machines. Despite obvious differences, the fundamental pure functional computational model adopted by both systems is interestingly similar: both systems are based on lazy transformations of immutable data structures using pure functions. This is a fundamental pattern for scalable data processing transcending any particular language, library or framework. We can encode our log posterior in Spark as follows.

    def ll(beta: DVD): Double =
df.map{row =>
val y = row.getAs[Double](0)
val x = BDV.vertcat(BDV(1.0),toBDV(row.getAs[DenseVector](1)))
-math.log(1.0 + math.exp(-1.0*(2.0*y-1.0)*(x.dot(beta))))}.reduce(_+_)
def lprior(beta: DVD): Double =
Gaussian(0,10).logPdf(beta(0)) +
sum(beta(1 until p).map(Gaussian(0,1).logPdf(_)))
def lpost(beta: DVD): Double =
ll(beta) + lprior(beta)



Haskell is an old, lazy pure functional programming language with an advanced type system, and remains the preferred language for the majority of functional programming language researchers. Hmatrix is the standard high performance numerical linear algebra library for Haskell, so we can use it to encode our log-posterior as follows.

ll :: Matrix Double -> Vector Double -> Vector Double -> Double
ll x y b = (negate) (vsum (cmap log (
(scalar 1) + (cmap exp (cmap (negate) (
(((scalar 2) * y) - (scalar 1)) * (x #> b)
)
)))))

pscale :: [Double] -- prior standard deviations
pscale = [10.0, 1, 1, 1, 1, 1, 1, 1]
lprior :: Vector Double -> Double
lprior b = sum $(\x -> logDensity (normalDistr 0.0 (snd x)) (fst x)) <$> (zip (toList b) pscale)

lpost :: Matrix Double -> Vector Double -> Vector Double -> Double
lpost x y b = (ll x y b) + (lprior b)


Again, a reminder that, here and elsewhere, there are various optimisations could be done that I’m not bothering with. This is all just proof-of-concept code.

### Dex

JAX proves that a pure functional DSL for tensor computation can be extremely powerful and useful. But embedding such a language in a dynamic imperative language like Python has a number of drawbacks. Dex is an experimental statically typed stand-alone DSL for differentiable array and tensor programming that attempts to combine some of the correctness and composability benefits of powerful statically typed functional languages like Scala and Haskell with the performance benefits of tensor computation systems like JAX. It is currently rather early its development, but seems very interesting, and is already quite useable. We can encode our log-posterior in Dex as follows.

def ll (b: (Fin 8)=>Float) : Float =
neg $sum (log (map (\ x. (exp x) + 1) ((map (\ yi. 1 - 2*yi) y)*(x **. b)))) pscale = [10.0, 1, 1, 1, 1, 1, 1, 1] -- prior SDs prscale = map (\ x. 1.0/x) pscale def lprior (b: (Fin 8)=>Float) : Float = bs = b*prscale neg$  sum ((log pscale) + (0.5 .* (bs*bs)))

def lpost (b: (Fin 8)=>Float) : Float =
(ll b) + (lprior b)


## Next steps

Now that we have a way of evaluating the log posterior, we can think about constructing Markov chains having the posterior as their equilibrium distribution. In the next post we will look at one of the simplest ways of doing this: the Metropolis algorithm.

Complete runnable scripts are available from this public github repo.

## Introduction

As discussed in the previous post, I’ve recently constructed and delivered a short course on statistical computing with Scala. Much of the course is concerned with writing statistical algorithms in Scala, typically making use of the scientific and numerical computing library, Breeze. Breeze has all of the essential tools necessary for building statistical algorithms, but doesn’t contain any higher level modelling functionality. As part of the course, I walked through how to build a small library for regression modelling on top of Breeze, including all of the usual regression diagnostics (such as standard errors, t-statistics, p-values, F-statistics, etc.). While preparing the course materials it occurred to me that it would be useful to package and document this code properly for general use. In advance of the course I packaged the code up into a bare-bones library, but since then I’ve fleshed it out, tidied it up and documented it properly, so it’s now ready for people to use.

The library covers PCA, linear regression modelling and simple one-parameter GLMs (including logistic and Poisson regression). The underlying algorithms are fairly efficient and numerically stable (eg. linear regression uses the QR decomposition of the model matrix, and the GLM fitting uses QR within each IRLS step), though they are optimised more for clarity than speed. The library also includes a few utility functions and procedures, including a pairs plot (scatter-plot matrix).

## A linear regression example

Plenty of documentation is available from the scala-glm github repo which I won’t repeat here. But to give a rough idea of how things work, I’ll run through an interactive session for the linear regression example.

First, download a dataset from the UCI ML Repository to disk for subsequent analysis (caching the file on disk is good practice, as it avoids unnecessary load on the UCI server, and allows running the code off-line):

import scalaglm._
import breeze.linalg._

val url = "http://archive.ics.uci.edu/ml/machine-learning-databases/00291/airfoil_self_noise.dat"
val fileName = "self-noise.csv"

val file = new java.io.File(fileName)
if (!file.exists) {
val s = new java.io.PrintWriter(file)
val data = scala.io.Source.fromURL(url).getLines
data.foreach(l => s.write(l.trim.
split('\t').filter(_ != "").
mkString("", ",", "\n")))
s.close
}


Once we have a CSV file on disk, we can load it up and look at it.

val mat = csvread(new java.io.File(fileName))
// mat: breeze.linalg.DenseMatrix[Double] =
// 800.0    0.0  0.3048  71.3  0.00266337  126.201
// 1000.0   0.0  0.3048  71.3  0.00266337  125.201
// 1250.0   0.0  0.3048  71.3  0.00266337  125.951
// ...
println("Dim: " + mat.rows + " " + mat.cols)
// Dim: 1503 6
val figp = Utils.pairs(mat, List("Freq", "Angle", "Chord", "Velo", "Thick", "Sound"))
// figp: breeze.plot.Figure = breeze.plot.Figure@37718125


We can then regress the response in the final column on the other variables.

val y = mat(::, 5) // response is the final column
// y: DenseVector[Double] = DenseVector(126.201, 125.201, ...
val X = mat(::, 0 to 4)
// X: breeze.linalg.DenseMatrix[Double] =
// 800.0    0.0  0.3048  71.3  0.00266337
// 1000.0   0.0  0.3048  71.3  0.00266337
// 1250.0   0.0  0.3048  71.3  0.00266337
// ...
val mod = Lm(y, X, List("Freq", "Angle", "Chord", "Velo", "Thick"))
// mod: scalaglm.Lm =
// Lm(DenseVector(126.201, 125.201, ...
mod.summary
// Estimate	 S.E.	 t-stat	p-value		Variable
// ---------------------------------------------------------
// 132.8338	 0.545	243.866	0.0000 *	(Intercept)
//  -0.0013	 0.000	-30.452	0.0000 *	Freq
//  -0.4219	 0.039	-10.847	0.0000 *	Angle
// -35.6880	 1.630	-21.889	0.0000 *	Chord
//   0.0999	 0.008	12.279	0.0000 *	Velo
// -147.3005	15.015	-9.810	0.0000 *	Thick
// Residual standard error:   4.8089 on 1497 degrees of freedom
// Multiple R-squared: 0.5157, Adjusted R-squared: 0.5141
// F-statistic: 318.8243 on 5 and 1497 DF, p-value: 0.00000
val fig = mod.plots
// fig: breeze.plot.Figure = breeze.plot.Figure@60d7ebb0


There is a .predict method for generating point predictions (and standard errors) given a new model matrix, and fitting GLMs is very similar – these things are covered in the quickstart guide for the library.

## Summary

scala-glm is a small Scala library built on top of the Breeze numerical library which enables simple and convenient regression modelling in Scala. It is reasonably well documented and usable in its current form, but I intend to gradually add additional features according to demand as time permits.

## Calling R from Scala sbt projects using rscala

### Overview

In the previous post I showed how the rscala package (which has replaced the jvmr package) can be used to call Scala code from within R. In this post I will show how to call R from Scala code. I have previously described how to do this using jvmr. This post is really just an update to show how things work with rscala.

Since I’m focusing here on Scala sbt projects, I’m assuming that sbt is installed, in addition to rscala (described in the previous post). The only “trick” required for calling back to R from Scala is telling sbt where the rscala jar file is located. You can find the location from the R console as illustrated by the following session:

> library(rscala)
> rscala::rscalaJar("2.11")
[1] "/home/ndjw1/R/x86_64-pc-linux-gnu-library/3.2/rscala/java/rscala_2.11-1.0.6.jar"


This location (which will obviously be different for you) can then be added in to your sbt classpath by adding the following line to your build.sbt file:

unmanagedJars in Compile += file("/home/ndjw1/R/x86_64-pc-linux-gnu-library/3.2/rscala/java/rscala_2.11-1.0.6.jar")


Once this is done, calling out to R from your Scala sbt project can be carried out as described in the rscala documentation. For completeness, a working example is given below.

### Example

In this example I will use Scala to simulate some data consistent with a Poisson regression model, and then push the data to R to fit it using the R function glm(), and then pull back the fitted regression coefficients into Scala. This is obviously a very artificial example, but the point is to show how it is possible to call back to R for some statistical procedure that may be “missing” from Scala.

The dependencies for this project are described in the file build.sbt

name := "rscala test"

version := "0.1"

scalacOptions ++= Seq("-unchecked", "-deprecation", "-feature")

libraryDependencies  ++= Seq(
"org.scalanlp" %% "breeze" % "0.10",
"org.scalanlp" %% "breeze-natives" % "0.10"
)

resolvers ++= Seq(
"Sonatype Snapshots" at "https://oss.sonatype.org/content/repositories/snapshots/",
"Sonatype Releases" at "https://oss.sonatype.org/content/repositories/releases/"
)

unmanagedJars in Compile += file("/home/ndjw1/R/x86_64-pc-linux-gnu-library/3.2/rscala/java/rscala_2.11-1.0.6.jar")

scalaVersion := "2.11.6"


The complete Scala program is contained in the file PoisReg.scala

import org.ddahl.rscala.callback._
import breeze.stats.distributions._
import breeze.linalg._

object ScalaToRTest {

def main(args: Array[String]) = {

// first simulate some data consistent with a Poisson regression model
val x = Uniform(50,60).sample(1000)
val eta = x map { xi => (xi * 0.1) - 3 }
val mu = eta map { math.exp(_) }
val y = mu map { Poisson(_).draw }

// call to R to fit the Poission regression model
val R = RClient() // initialise an R interpreter
R.x=x.toArray // send x to R
R.y=y.toArray // send y to R
R.eval("mod <- glm(y~x,family=poisson())") // fit the model in R
// pull the fitted coefficents back into scala
val beta = DenseVector[Double](R.evalD1("mod$coefficients")) // print the fitted coefficents println(beta) } }  If these two files are put in an empty directory, the code can be compiled and run by typing sbt run from the command prompt in the relevant directory. The commented code should be self-explanatory, but see the rscala documentation for further details. In particular, the rscala scaladoc is useful. ## Statistical computing languages at the RSS On Friday the Royal Statistical Society hosted a meeting on Statistical computing languages, organised by my colleague Colin Gillespie. Four languages were presented at the meeting: Python, Scala, Matlab and Julia. I presented the talk on Scala. The slides I presented are available, in addition to the code examples and instructions on how to run them, in a public github repo I have created. The talk mainly covered material I have discussed in various previous posts on this blog. What was new was the emphasis on how easy it is to use and run Scala code, and the inclusion of complete examples and instructions on how to run them on any platform with a JVM installed. I also discussed some of the current limitations of Scala as an environment for interactive statistical data analysis and visualisation, and how these limitations could be overcome with a little directed effort. Colin suggested that all of the speakers covered a couple of examples (linear regression and a Monte Carlo integral) in “their” languages, and he provided an R solution. It is interesting to see the examples in the five different languages side by side for comparison purposes. Colin is collecting together all of the material relating to the meeting in a combined github repo, which should fill out over the next few days. For other Scala posts on this blog, see all of my posts with a “scala” tag. ## Summary stats for ABC #### Introduction In the previous post I gave a very brief introduction to ABC, including a simple example for inferring the parameters of a Markov process given some time series observations. Towards the end of the post I observed that there were (at least!) two potential problems with scaling up the simple approach described, one relating to the dimension of the data and the other relating to the dimension of the parameter space. Before moving on to the (to me, more interesting) problem of the dimension of the parameter space, I will briefly discuss the data dimension problem in this post, and provide a couple of references for further reading. #### Summary stats Recall that the simple rejection sampling approach to ABC involves first sampling a candidate parameter $\theta^\star$ from the prior and then sampling a corresponding data set $x^\star$ from the model. This simulated data set is compared with the true data $x$ using some (pseudo-)norm, $\Vert\cdot\Vert$, and accepting $\theta^\star$ if the simulated data set is sufficiently close to the true data, $\Vert x^\star - x\Vert <\epsilon$. It should be clear that if we are using a proper norm then as $\epsilon$ tends to zero the distribution of the accepted values tends to the desired posterior distribution of the parameters given the data. However, smaller choices of $\epsilon$ will lead to higher rejection rates. This will be a particular problem in the context of high-dimensional $x$, where it is often unrealistic to expect a close match between all components of $x$ and the simulated data $x^\star$, even for a good choice of $\theta^\star$. In this case, it makes more sense to look for good agreement between particular aspects of $x$, such as the mean, or variance, or auto-correlation, depending on the exact problem and context. If we can find a finite set of sufficient statistics, $s(x)$ for $\theta$, then it should be clear that replacing the acceptance criterion with $\Vert s(x^\star) - s(x)\Vert <\epsilon$ will also lead to a scheme tending to the true posterior as $\epsilon$ tends to zero (assuming a proper norm on the space of sufficient statistics), and will typically be better than the naive method, since the sufficient statistics will be of lower dimension and less “noisy” that the raw data, leading to higher acceptance rates with no loss of information. Unfortunately for most problems of practical interest it is not possible to find low-dimensional sufficient statistics, and so people in practice use domain knowledge and heuristics to come up with a set of summary statistics, $s(x)$ which they hope will closely approximate sufficient statistics. There is still a question as to how these statistics should be weighted or transformed to give a particular norm. This can be done using theory or heuristics, and some relevant references for this problem are given at the end of the post. #### Implementation in R Let’s now look at the problem from the previous post. Here, instead of directly computing the Euclidean distance between the real and simulated data, we will look at the Euclidean distance between some (normalised) summary statistics. First we will load some packages and set some parameters. require(smfsb) require(parallel) options(mc.cores=4) data(LVdata) N=1e7 bs=1e5 batches=N/bs message(paste("N =",N," | bs =",bs," | batches =",batches))  Next we will define some summary stats for a univariate time series – the mean, the (log) variance, and the first two auto-correlations. ssinit <- function(vec) { ac23=as.vector(acf(vec,lag.max=2,plot=FALSE)$acf)[2:3]
c(mean(vec),log(var(vec)+1),ac23)
}


Once we have this, we can define some stats for a bivariate time series by combining the stats for the two component series, along with the cross-correlation between them.

ssi <- function(ts)
{
c(ssinit(ts[,1]),ssinit(ts[,2]),cor(ts[,1],ts[,2]))
}


This gives a set of summary stats, but these individual statistics are potentially on very different scales. They can be transformed and re-weighted in a variety of ways, usually on the basis of a pilot run which gives some information about the distribution of the summary stats. Here we will do the simplest possible thing, which is to normalise the variance of the stats on the basis of a pilot run. This is not at all optimal – see the references at the end of the post for a description of better methods.

message("Batch 0: Pilot run batch")
prior=cbind(th1=exp(runif(bs,-6,2)),th2=exp(runif(bs,-6,2)),th3=exp(runif(bs,-6,2)))
rows=lapply(1:bs,function(i){prior[i,]})
samples=mclapply(rows,function(th){simTs(c(50,100),0,30,2,stepLVc,th)})
sumstats=mclapply(samples,ssi)
sds=apply(sapply(sumstats,c),1,sd)
print(sds)

# now define a standardised distance
ss<-function(ts)
{
ssi(ts)/sds
}

ss0=ss(LVperfect)

distance <- function(ts)
{
diff=ss(ts)-ss0
sum(diff*diff)
}


Now we have a normalised distance function defined, we can proceed exactly as before to obtain an ABC posterior via rejection sampling.

post=NULL
for (i in 1:batches) {
message(paste("batch",i,"of",batches))
prior=cbind(th1=exp(runif(bs,-6,2)),th2=exp(runif(bs,-6,2)),th3=exp(runif(bs,-6,2)))
rows=lapply(1:bs,function(i){prior[i,]})
samples=mclapply(rows,function(th){simTs(c(50,100),0,30,2,stepLVc,th)})
dist=mclapply(samples,distance)
dist=sapply(dist,c)
cutoff=quantile(dist,1000/N,na.rm=TRUE)
post=rbind(post,prior[dist<cutoff,])
}
message(paste("Finished. Kept",dim(post)[1],"simulations"))


Having obtained the posterior, we can use the following code to plot the results.

th=c(th1 = 1, th2 = 0.005, th3 = 0.6)
op=par(mfrow=c(2,3))
for (i in 1:3) {
hist(post[,i],30,col=5,main=paste("Posterior for theta[",i,"]",sep=""))
abline(v=th[i],lwd=2,col=2)
}
for (i in 1:3) {
hist(log(post[,i]),30,col=5,main=paste("Posterior for log(theta[",i,"])",sep=""))
abline(v=log(th[i]),lwd=2,col=2)
}
par(op)


This gives the plot shown below. From this we can see that the ABC posterior obtained here is very similar to that obtained in the previous post using the full data. Here the dimension reduction is not that great – reducing from 32 data points to 9 summary statistics – and so the improvement in performance is not that noticable. But in higher dimensional problems reducing the dimension of the data is practically essential.

#### Summary and References

As before, I recommend the wikipedia article on approximate Bayesian computation for further information and a comprehensive set of references for further reading. Here I just want to highlight two references particularly relevant to the issue of summary statistics. It is quite difficult to give much practical advice on how to construct good summary statistics, but how to transform a set of summary stats in a “good” way is a problem that is reasonably well understood. In this post I did something rather naive (normalising the variance), but the following two papers describe much better approaches.

I still haven’t addressed the issue of a high-dimensional parameter space – that will be the topic of a subsequent post.

#### The complete R script

require(smfsb)
require(parallel)
options(mc.cores=4)
data(LVdata)

N=1e6
bs=1e5
batches=N/bs
message(paste("N =",N," | bs =",bs," | batches =",batches))

ssinit <- function(vec)
{
ac23=as.vector(acf(vec,lag.max=2,plot=FALSE)\$acf)[2:3]
c(mean(vec),log(var(vec)+1),ac23)
}

ssi <- function(ts)
{
c(ssinit(ts[,1]),ssinit(ts[,2]),cor(ts[,1],ts[,2]))
}

message("Batch 0: Pilot run batch")
prior=cbind(th1=exp(runif(bs,-6,2)),th2=exp(runif(bs,-6,2)),th3=exp(runif(bs,-6,2)))
rows=lapply(1:bs,function(i){prior[i,]})
samples=mclapply(rows,function(th){simTs(c(50,100),0,30,2,stepLVc,th)})
sumstats=mclapply(samples,ssi)
sds=apply(sapply(sumstats,c),1,sd)
print(sds)

# now define a standardised distance
ss<-function(ts)
{
ssi(ts)/sds
}

ss0=ss(LVperfect)

distance <- function(ts)
{
diff=ss(ts)-ss0
sum(diff*diff)
}

post=NULL
for (i in 1:batches) {
message(paste("batch",i,"of",batches))
prior=cbind(th1=exp(runif(bs,-6,2)),th2=exp(runif(bs,-6,2)),th3=exp(runif(bs,-6,2)))
rows=lapply(1:bs,function(i){prior[i,]})
samples=mclapply(rows,function(th){simTs(c(50,100),0,30,2,stepLVc,th)})
dist=mclapply(samples,distance)
dist=sapply(dist,c)
cutoff=quantile(dist,1000/N,na.rm=TRUE)
post=rbind(post,prior[dist<cutoff,])
}
message(paste("Finished. Kept",dim(post)[1],"simulations"))

# plot the results
th=c(th1 = 1, th2 = 0.005, th3 = 0.6)
op=par(mfrow=c(2,3))
for (i in 1:3) {
hist(post[,i],30,col=5,main=paste("Posterior for theta[",i,"]",sep=""))
abline(v=th[i],lwd=2,col=2)
}
for (i in 1:3) {
hist(log(post[,i]),30,col=5,main=paste("Posterior for log(theta[",i,"])",sep=""))
abline(v=log(th[i]),lwd=2,col=2)
}
par(op)