MCMC as a Stream

Introduction

This weekend I’ve been preparing some material for my upcoming Scala for statistical computing short course. As part of the course, I thought it would be useful to walk through how to think about and structure MCMC codes, and in particular, how to think about MCMC algorithms as infinite streams of state. This material is reasonably stand-alone, so it seems suitable for a blog post. Complete runnable code for the examples in this post are available from my blog repo.

A simple MH sampler

For this post I will just consider a trivial toy Metropolis algorithm using a Uniform random walk proposal to target a standard normal distribution. I’ve considered this problem before on my blog, so if you aren’t very familiar with Metropolis-Hastings algorithms, you might want to quickly review my post on Metropolis-Hastings MCMC algorithms in R before continuing. At the end of that post, I gave the following R code for the Metropolis sampler:

metrop3<-function(n=1000,eps=0.5) 
{
        vec=vector("numeric", n)
        x=0
        oldll=dnorm(x,log=TRUE)
        vec[1]=x
        for (i in 2:n) {
                can=x+runif(1,-eps,eps)
                loglik=dnorm(can,log=TRUE)
                loga=loglik-oldll
                if (log(runif(1)) < loga) { 
                        x=can
                        oldll=loglik
                        }
                vec[i]=x
        }
        vec
}

I will begin this post with a fairly direct translation of this algorithm into Scala:

def metrop1(n: Int = 1000, eps: Double = 0.5): DenseVector[Double] = {
    val vec = DenseVector.fill(n)(0.0)
    var x = 0.0
    var oldll = Gaussian(0.0, 1.0).logPdf(x)
    vec(0) = x
    (1 until n).foreach { i =>
      val can = x + Uniform(-eps, eps).draw
      val loglik = Gaussian(0.0, 1.0).logPdf(can)
      val loga = loglik - oldll
      if (math.log(Uniform(0.0, 1.0).draw) < loga) {
        x = can
        oldll = loglik
      }
      vec(i) = x
    }
    vec
}

This code works, and is reasonably fast and efficient, but there are several issues with it from a functional programmers perspective. One issue is that we have committed to storing all MCMC output in RAM in a DenseVector. This probably isn’t an issue here, but for some big problems we might prefer to not store the full set of states, but to just print the states to (say) the console, for possible re-direction to a file. It is easy enough to modify the code to do this:

def metrop2(n: Int = 1000, eps: Double = 0.5): Unit = {
    var x = 0.0
    var oldll = Gaussian(0.0, 1.0).logPdf(x)
    (1 to n).foreach { i =>
      val can = x + Uniform(-eps, eps).draw
      val loglik = Gaussian(0.0, 1.0).logPdf(can)
      val loga = loglik - oldll
      if (math.log(Uniform(0.0, 1.0).draw) < loga) {
        x = can
        oldll = loglik
      }
      println(x)
    }
}

But now we have two version of the algorithm. One for storing results locally, and one for streaming results to the console. This is clearly unsatisfactory, but we shall return to this issue shortly. Another issue that will jump out at functional programmers is the reliance on mutable variables for storing the state and old likelihood. Let’s fix that now by re-writing the algorithm as a tail-recursion.

@tailrec
def metrop3(n: Int = 1000, eps: Double = 0.5, x: Double = 0.0, oldll: Double = Double.MinValue): Unit = {
    if (n > 0) {
      println(x)
      val can = x + Uniform(-eps, eps).draw
      val loglik = Gaussian(0.0, 1.0).logPdf(can)
      val loga = loglik - oldll
      if (math.log(Uniform(0.0, 1.0).draw) < loga)
        metrop3(n - 1, eps, can, loglik)
      else
        metrop3(n - 1, eps, x, oldll)
    }
  }

This has eliminated the vars, and is just as fast and efficient as the previous version of the code. Note that the @tailrec annotation is optional – it just signals to the compiler that we want it to throw an error if for some reason it cannot eliminate the tail call. However, this is for the print-to-console version of the code. What if we actually want to keep the iterations in RAM for subsequent analysis? We can keep the values in an accumulator, as follows.

@tailrec
def metrop4(n: Int = 1000, eps: Double = 0.5, x: Double = 0.0, oldll: Double = Double.MinValue, acc: List[Double] = Nil): DenseVector[Double] = {
    if (n == 0)
      DenseVector(acc.reverse.toArray)
    else {
      val can = x + Uniform(-eps, eps).draw
      val loglik = Gaussian(0.0, 1.0).logPdf(can)
      val loga = loglik - oldll
      if (math.log(Uniform(0.0, 1.0).draw) < loga)
        metrop4(n - 1, eps, can, loglik, can :: acc)
      else
        metrop4(n - 1, eps, x, oldll, x :: acc)
    }
}

Factoring out the updating logic

This is all fine, but we haven’t yet addressed the issue of having different versions of the code depending on what we want to do with the output. The problem is that we have tied up the logic of advancing the Markov chain with what to do with the output. What we need to do is separate out the code for advancing the state. We can do this by defining a new function.

def newState(x: Double, oldll: Double, eps: Double): (Double, Double) = {
    val can = x + Uniform(-eps, eps).draw
    val loglik = Gaussian(0.0, 1.0).logPdf(can)
    val loga = loglik - oldll
    if (math.log(Uniform(0.0, 1.0).draw) < loga) (can, loglik) else (x, oldll)
}

This function takes as input a current state and associated log likelihood and returns a new state and log likelihood following the execution of one step of a MH algorithm. This separates the concern of state updating from the rest of the code. So now if we want to write code that prints the state, we can write it as

  @tailrec
  def metrop5(n: Int = 1000, eps: Double = 0.5, x: Double = 0.0, oldll: Double = Double.MinValue): Unit = {
    if (n > 0) {
      println(x)
      val ns = newState(x, oldll, eps)
      metrop5(n - 1, eps, ns._1, ns._2)
    }
  }

and if we want to accumulate the set of states visited, we can write that as

  @tailrec
  def metrop6(n: Int = 1000, eps: Double = 0.5, x: Double = 0.0, oldll: Double = Double.MinValue, acc: List[Double] = Nil): DenseVector[Double] = {
    if (n == 0) DenseVector(acc.reverse.toArray) else {
      val ns = newState(x, oldll, eps)
      metrop6(n - 1, eps, ns._1, ns._2, ns._1 :: acc)
    }
  }

Both of these functions call newState to do the real work, and concentrate on what to do with the sequence of states. However, both of these functions repeat the logic of how to iterate over the sequence of states.

MCMC as a stream

Ideally we would like to abstract out the details of how to do state iteration from the code as well. Most functional languages have some concept of a Stream, which represents a (potentially infinite) sequence of states. The Stream can embody the logic of how to perform state iteration, allowing us to abstract that away from our code, as well.

To do this, we will restructure our code slightly so that it more clearly maps old state to new state.

def nextState(eps: Double)(state: (Double, Double)): (Double, Double) = {
    val x = state._1
    val oldll = state._2
    val can = x + Uniform(-eps, eps).draw
    val loglik = Gaussian(0.0, 1.0).logPdf(can)
    val loga = loglik - oldll
    if (math.log(Uniform(0.0, 1.0).draw) < loga) (can, loglik) else (x, oldll)
}

The "real" state of the chain is just x, but if we want to avoid recalculation of the old likelihood, then we need to make this part of the chain’s state. We can use this nextState function in order to construct a Stream.

  def metrop7(eps: Double = 0.5, x: Double = 0.0, oldll: Double = Double.MinValue): Stream[Double] =
    Stream.iterate((x, oldll))(nextState(eps)) map (_._1)

The result of calling this is an infinite stream of states. Obviously it isn’t computed – that would require infinite computation, but it captures the logic of iteration and computation in a Stream, that can be thought of as a lazy List. We can get values out by converting the Stream to a regular collection, being careful to truncate the Stream to one of finite length beforehand! eg. metrop7().drop(1000).take(10000).toArray will do a burn-in of 1,000 iterations followed by a main monitoring run of length 10,000, capturing the results in an Array. Note that metrop7().drop(1000).take(10000) is a Stream, and so nothing is actually computed until the toArray is encountered. Conversely, if printing to console is required, just replace the .toArray with .foreach(println).

The above stream-based approach to MCMC iteration is clean and elegant, and deals nicely with issues like burn-in and thinning (which can be handled similarly). This is how I typically write MCMC codes these days. However, functional programming purists would still have issues with this approach, as it isn’t quite pure functional. The problem is that the code isn’t pure – it has a side-effect, which is to mutate the state of the under-pinning pseudo-random number generator. If the code was pure, calling nextState with the same inputs would always give the same result. Clearly this isn’t the case here, as we have specifically designed the function to be stochastic, returning a randomly sampled value from the desired probability distribution. So nextState represents a function for randomly sampling from a conditional probability distribution.

A pure functional approach

Now, ultimately all code has side-effects, or there would be no point in running it! But in functional programming the desire is to make as much of the code as possible pure, and to push side-effects to the very edges of the code. So it’s fine to have side-effects in your main method, but not buried deep in your code. Here the side-effect is at the very heart of the code, which is why it is potentially an issue.

To keep things as simple as possible, at this point we will stop worrying about carrying forward the old likelihood, and hard-code a value of eps. Generalisation is straightforward. We can make our code pure by instead defining a function which represents the conditional probability distribution itself. For this we use a probability monad, which in Breeze is called Rand. We can couple together such functions using monadic binds (flatMap in Scala), expressed most neatly using for-comprehensions. So we can write our transition kernel as

def kernel(x: Double): Rand[Double] = for {
    innov <- Uniform(-0.5, 0.5)
    can = x + innov
    oldll = Gaussian(0.0, 1.0).logPdf(x)
    loglik = Gaussian(0.0, 1.0).logPdf(can)
    loga = loglik - oldll
    u <- Uniform(0.0, 1.0)
} yield if (math.log(u) < loga) can else x

This is now pure – the same input x will always return the same probability distribution – the conditional distribution of the next state given the current state. We can draw random samples from this distribution if we must, but it’s probably better to work as long as possible with pure functions. So next we need to encapsulate the iteration logic. Breeze has a MarkovChain object which can take kernels of this form and return a stochastic Process object representing the iteration logic, as follows.

MarkovChain(0.0)(kernel).
  steps.
  drop(1000).
  take(10000).
  foreach(println)

The steps method contains the logic of how to advance the state of the chain. But again note that no computation actually takes place until the foreach method is encountered – this is when the sampling occurs and the side-effects happen.

Metropolis-Hastings is a common use-case for Markov chains, so Breeze actually has a helper method built-in that will construct a MH sampler directly from an initial state, a proposal kernel, and a (log) target.

MarkovChain.
  metropolisHastings(0.0, (x: Double) =>
  Uniform(x - 0.5, x + 0.5))(x =>
  Gaussian(0.0, 1.0).logPdf(x)).
  steps.
  drop(1000).
  take(10000).
  toArray

Note that if you are using the MH functionality in Breeze, it is important to make sure that you are using version 0.13 (or later), as I fixed a few issues with the MH code shortly prior to the 0.13 release.

Summary

Viewing MCMC algorithms as infinite streams of state is useful for writing elegant, generic, flexible code. Streams occur everywhere in programming, and so there are lots of libraries for working with them. In this post I used the simple Stream from the Scala standard library, but there are much more powerful and flexible stream libraries for Scala, including fs2 and Akka-streams. But whatever libraries you are using, the fundamental concepts are the same. The most straightforward approach to implementation is to define impure stochastic streams to consume. However, a pure functional approach is also possible, and the Breeze library defines some useful functions to facilitate this approach. I’m still a little bit ambivalent about whether the pure approach is worth the additional cognitive overhead, but it’s certainly very interesting and worth playing with and thinking about the pros and cons.

Complete runnable code for the examples in this post are available from my blog repo.

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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.

References

Parallel tempering and Metropolis coupled MCMC

Introduction

Parallel tempering is a method for getting Metropolis-Hastings based MCMC algorithms to work better on multi-modal distributions. Although the idea has been around for more than 20 years, and works well on many problems, it still isn’t routinely used in applications. I think this is partly because relatively few people understand how it works, and partly due to the perceived difficulty of implementation. I hope to show here that it is both very easy to understand and to implement. It is also rather easy to implement in parallel on multi-core systems, though I won’t get into that in this post.

Sampling a double-well potential

To illustrate the ideas, we need a toy multi-modal distribution to sample. There are obviously many possibilities here, but I rather like to use a double potential well distribution. The simplest version of this assumes a potential function of the form

U(x) = \gamma (x^2-1)^2

for some given potential barrier height \gamma. The potential function U(x) corresponds to the probability density function

\pi(x) \propto \exp\{-U(x)\}.

There is a physical explanation for the terminology, via Langevin diffusions, but that isn’t really important here. It is fine to just think of potentials as being a (negative) log-density scale. On this scale, high potential barrier heights correspond to regions of very low probability density. We can set up a double well potential and plot it for the case \gamma=4 in R with the following code

U=function(gam,x)
{
  gam*(x*x-1)*(x*x-1)
}

curried=function(gam)
{
  message(paste("Returning a function for gamma =",gam))
  function(x) U(gam,x)
}
U4=curried(4)

op=par(mfrow=c(2,1))
curve(U4(x),-2,2,main="Potential function, U(x)")
curve(exp(-U4(x)),-2,2,main="Unnormalised density function, exp(-U(x))")
par(op)

leading to the following plot
Double-well potential

Incidentally, the function curried(), which curries the potential function, did not include the message() statement when I first wrote it. It mostly worked fine, but some of the code below didn’t behave as I expected. I inserted the message() statement to figure out what was going on, and the code started behaving perfectly – a beautiful example of a Heisenbug! The reason is that the message statement is not redundant – it forces evaluation of the gam variable, which is necessary in some cases, due to the lazy evaluation model that R uses for function arguments. If you don’t like the message() statement, replacing it with a simple gam works just as well.

Anyway, the point is that we have defined a multi-modal density, and that a Metropolis-Hastings algorithm initialised in one of the modes will have a hard time jumping to the other mode, and the difficulty of this jump will increase as we increase the value of \gamma.

We can write a simple function for a Metropolis algorithm targeting a particular potential function as follows.

chain=function(target,tune=0.1,init=1)
{
  x=init
  xvec=numeric(iters)
  for (i in 1:iters) {
    can=x+rnorm(1,0,tune)
    logA=target(x)-target(can)
    if (log(runif(1))<logA)
      x=can
    xvec[i]=x
  }
  xvec
}

We can use this code to run some chains for a few different values of \gamma as follows.

temps=2^(0:3)
iters=1e5

mat=sapply(lapply(temps,curried),chain)
colnames(mat)=paste("gamma=",temps,sep="")

require(smfsb)
mcmcSummary(mat,rows=length(temps))

leading to the plot below.

Chains

We see that as \gamma increases, the chain jumps between modes less frequently. Indeed, for \gamma=8, the chain fails to jump to the second mode at all during this particular run of 100,000 iterations. That’s a problem if we are really interested in sampling from distributions like this. Of course, for this particular problem, there are all kinds of ways to fix this sampler, but the point is to try and develop methods that will work in high-dimensional situations where we cannot just “look” at what is going wrong.

Before we see how to couple the chains and improve the mixing, it is useful to think how to re-write this sampler. Above we sampled each chain in turn for different barrier heights. To couple the chains, we need to sample them together, sampling each iteration for all of the chains in turn. This is very easy to do. The code below isn’t especially efficient, but it is written to look very similar to the single chain code above.

chains=function(pot=U, tune=0.1, init=1)
{
  x=rep(init,length(temps))
  xmat=matrix(0,iters,length(temps))
  for (i in 1:iters) {
    can=x+rnorm(length(temps),0,tune)
    logA=unlist(Map(pot,temps,x))-unlist(Map(pot,temps,can))
    accept=(log(runif(length(temps)))<logA)
    x[accept]=can[accept]
    xmat[i,]=x
  }
  colnames(xmat)=paste("gamma=",temps,sep="")
  xmat
}

mcmcSummary(chains(),rows=length(temps))

This code should behave identically to the previous code, simulating independent parallel MCMC chains. However, the code is now in the form that is very easy to modify to couple the chains together in an attempt to improve mixing.

Coupling parallel chains

In the above example the chains we are simulating are all independent of one another. Some mix reasonably well, and some mix very badly. But the distributions are all related to one another, changing gradually as the barrier height changes. The idea behind coupling the chains is to try and swap states between the chains to use the chains which are mixing well to improve the mixing of the chains which aren’t. In particular, suppose interest is in the target of the worst mixing chain. The other chains can be constructed as “tempered” versions of the target of interest, here by raising it to a power between 0 and 1, with 0 corresponding to a complete flattening of the distribution, and 1 corresponding to the desired target. The use of parallel chains to improve mixing in this way is usually referred to as parallel tempering, but also sometimes as (\text{MC})^3. In the context of Bayesian inference, tempering using the “power posterior” can often be more natural and useful (to be discussed in a subsequent post).

So, how do we swap states between the chains without affecting the target distributions? As always, just use a Metropolis-Hastings update… To keep things simple, let’s just suppose that we have two (independent, parallel) chains, one with target f(x) and the other with target g(y). We can consider these chains to be evolving together, with joint target \pi(x,y)=f(x)g(y). The updates chosen to update the within-chain states will obviously preserve this joint target. Now we consider how to swap states between the two chains without destroying the target. We simply propose a swap of x and y. That is, we propose to move from (x,y) to (x^\star,y^\star), where x^\star=y and y^\star=x. We are proposing this move as a standard Metropolis-Hastings update of the joint chain. We may therefore wonder about exactly what the proposal density for this move is. In fact, it is a point mass at the swapped values, and therefore has density

q((x^\star,y^\star)|(x,y)) = \delta(x^\star-y)\delta(y^\star-x),

but it really doesn’t matter, as it is clearly a symmetric proposal, and hence will drop out of the M-H ratio. Our acceptance probability is therefore \min\{1,A\}, where

\displaystyle A = \frac{\pi(x^\star,y^\star)}{\pi(x,y)} = \frac{\pi(y,x)}{\pi(x,y)} = \frac{f(y)g(x)}{f(x)g(y)}.

So, if we use this acceptance probability whenever we propose a swap of the states between two chains, then we will preserve the joint target, and hence the marginal targets and asymptotic independence of the target. However, the chains themselves will no longer be independent of one another. They will be coupled – Metropolis coupled. This is very easy to implement. We can just add a few lines to our previous chains() function as follows

chains=function(pot=U, tune=0.1, init=1)
{
  x=rep(init,length(temps))
  xmat=matrix(0,iters,length(temps))
  for (i in 1:iters) {
    can=x+rnorm(length(temps),0,tune)
    logA=unlist(Map(pot,temps,x))-unlist(Map(pot,temps,can))
    accept=(log(runif(length(temps)))<logA)
    x[accept]=can[accept]
    # now the coupling update
    swap=sample(1:length(temps),2)
    logA=pot(temps[swap[1]],x[swap[1]])+pot(temps[swap[2]],x[swap[2]])-
            pot(temps[swap[1]],x[swap[2]])-pot(temps[swap[2]],x[swap[1]])
    if (log(runif(1))<logA)
      x[swap]=rev(x[swap])
    # end of the coupling update
    xmat[i,]=x
  }
  colnames(xmat)=paste("gamma=",temps,sep="")
  xmat
}

This can be used as before, but now gives results as illustrated in the following plots.

Metropolis coupled chains

We see immediately from the plots that whilst the individual target distributions remain unchanged, the mixing of the chains is greatly improved (though still far from perfect). Note that in the code above I just picked two chains at random to propose a state swap. In practice people typically only propose to swap states between chains which are adjacent – i.e. most similar, since proposed swaps between chains with very different targets are unlikely to be accepted. Since implementation of either strategy is very easy, I would recommend trying both to see which works best.

Parallel implementation

It should be clear that there are opportunities for parallelising the above algorithm to make effective use of modern multi-core hardware. An approach using OpenMP with C++ is discussed in this blog post. Also note that if the state space of the chains is large, it can be much more efficient to swap temperatures between the chains rather than states – so long as you are careful about keeping track of stuff – this is explored in detail in Altekar et al (’04).

References

Complete R script

For convenience, a complete R script to run all of the examples in this post is given below.

# temper.R
# functions for messing around with tempering MCMC

U=function(gam,x)
{
  gam*(x*x-1)*(x*x-1)
}

curried=function(gam)
{
  #gam
  message(paste("Returning a function for gamma =",gam))
  function(x) U(gam,x)
}
U4=curried(4)

op=par(mfrow=c(2,1))
curve(U4(x),-2,2,main="Potential function, U(x)")
curve(exp(-U4(x)),-2,2,main="Unnormalised density function, exp(-U(x))")
par(op)

# global settings
temps=2^(0:3)
iters=1e5

# First look at some independent chains
chain=function(target,tune=0.1,init=1)
{
  x=init
  xvec=numeric(iters)
  for (i in 1:iters) {
    can=x+rnorm(1,0,tune)
    logA=target(x)-target(can)
    if (log(runif(1))<logA)
      x=can
    xvec[i]=x
  }
  xvec
}

mat=sapply(lapply(temps,curried),chain)
colnames(mat)=paste("gamma=",temps,sep="")

require(smfsb)
mcmcSummary(mat,rows=length(temps))

# Next, let's generate 5 chains at once...
chains=function(pot=U, tune=0.1, init=1)
{
  x=rep(init,length(temps))
  xmat=matrix(0,iters,length(temps))
  for (i in 1:iters) {
    can=x+rnorm(length(temps),0,tune)
    logA=unlist(Map(pot,temps,x))-unlist(Map(pot,temps,can))
    accept=(log(runif(length(temps)))<logA)
    x[accept]=can[accept]
    xmat[i,]=x
  }
  colnames(xmat)=paste("gamma=",temps,sep="")
  xmat
}

mcmcSummary(chains(),rows=length(temps))

# Next let's couple the chains...
chains=function(pot=U, tune=0.1, init=1)
{
  x=rep(init,length(temps))
  xmat=matrix(0,iters,length(temps))
  for (i in 1:iters) {
    can=x+rnorm(length(temps),0,tune)
    logA=unlist(Map(pot,temps,x))-unlist(Map(pot,temps,can))
    accept=(log(runif(length(temps)))<logA)
    x[accept]=can[accept]
    # now the coupling update
    swap=sample(1:length(temps),2)
    logA=pot(temps[swap[1]],x[swap[1]])+pot(temps[swap[2]],x[swap[2]])-
            pot(temps[swap[1]],x[swap[2]])-pot(temps[swap[2]],x[swap[1]])
    if (log(runif(1))<logA)
      x[swap]=rev(x[swap])
    # end of the coupling update
    xmat[i,]=x
  }
  colnames(xmat)=paste("gamma=",temps,sep="")
  xmat
}

mcmcSummary(chains(),rows=length(temps))

# eof

Metropolis Hastings MCMC when the proposal and target have differing support

Introduction

Very often it is desirable to use Metropolis Hastings MCMC for a target distribution which does not have full support (for example, it may correspond to a non-negative random variable), using a proposal distribution which does (for example, a Gaussian random walk proposal). This isn’t a problem at all, but on more than one occasion now I have come across students getting this wrong, so I thought it might be useful to have a brief post on how to do it right, see what people sometimes get wrong, and why, and then think about correcting the wrong method in order to make it right…

A simple example

For this post we will consider a simple Ga(2,1) target distribution, with density

\pi(x) = xe^{-x},\quad x\geq 0.

Of course this is a very simple distribution, and there are many straightforward ways to simulate it directly, but for this post we will use a random walk Metropolis-Hastings (MH) scheme with standard Gaussian innovations. So, if the current state of the chain is x, a proposed new value x^\star will be generated from

f(x^\star|x) = \phi(x^\star-x),

where \phi(\cdot) is the standard normal density. This proposed new value is accepted with probability \min\{1,A\}, where

\displaystyle A = \frac{\pi(x^\star)}{\pi(x)} \frac{f(x|x^\star)}{f(x^\star|x)} = \frac{\pi(x^\star)}{\pi(x)} \frac{\phi(x-x^\star)}{\phi(x^\star-x)} = \frac{\pi(x^\star)}{\pi(x)} ,

since the standard normal density is symmetric.

Correct implementation

We can easily implement this using R as follows:

met1=function(iters)
  {
    xvec=numeric(iters)
    x=1
    for (i in 1:iters) {
      xs=x+rnorm(1)
      A=dgamma(xs,2,1)/dgamma(x,2,1)
      if (runif(1)<A)
        x=xs
      xvec[i]=x
    }
    return(xvec)
  }

We can run it, plot the results and check it against the true target with the following commands.

iters=1000000
out=met1(iters)
hist(out,100,freq=FALSE,main="met1")
curve(dgamma(x,2,1),add=TRUE,col=2,lwd=2)

If you have a slow computer, you may prefer to use iters=100000. The above code uses R’s built-in gamma density. Alternatively, we can hard-code the density as follows.

met2=function(iters)
  {
    xvec=numeric(iters)
    x=1
    for (i in 1:iters) {
      xs=x+rnorm(1)
      A=xs*exp(-xs)/(x*exp(-x))
      if (runif(1)<A)
        x=xs
      xvec[i]=x
    }
    return(xvec)
  }

We can run this code using the following commands, to verify that it does work as expected.

out=met2(iters)
hist(out,100,freq=FALSE,main="met2")
curve(dgamma(x,2,1),add=TRUE,col=2,lwd=2)

However, there is a potential problem with the above code that we have got away with in this instance, which often catches people out. We have hard-coded the density for x>0 without checking the sign of x. Here we get away with it as a negative proposal will lead to a negative acceptance ratio that we will reject straight away. This is not always the case (consider, for example, a Ga(3,1) distribution). So really we should check the sign of x^\star and reject immediately if is not within the support of the target.

Although this problem often catches people out, it tends not to be a big issue in practice, as it typically leads to an obviously incorrect sampler, or a sampler which crashes, and is relatively simple to debug and fix.

An incorrect sampler

The problem I want to focus on here is more subtle, but closely related. It is clear that any x^\star<0 should be rejected. With the above code, such values are indeed rejected, and the sampler advances to the next iteration. However, in more complex samplers, where an update like this might be one tiny part of a massive sampler with a very high-dimensional state space, it seems like a bit of a "waste" of a MH move to just propose a negative value, throw it away, and move on. Evidently, it seems tempting, therefore, to keep on sampling x^\star values until a non-negative value is obtained, and then evaluate the acceptance ratio and decide whether or not to accept. We could code up this sampler as follows.

met3=function(iters)
  {
    xvec=numeric(iters)
    x=1
    for (i in 1:iters) {
      repeat {
        xs=x+rnorm(1)
        if (xs>0)
          break
      }
      A=xs*exp(-xs)/(x*exp(-x))
      if (runif(1)<A)
        x=xs
      xvec[i]=x
    }
    return(xvec)
  }

As reasonable as this idea may at first seem, it does not lead to a sampler having the desired target, as can be verified using the following commands.

out=met3(iters)
hist(out,100,freq=FALSE,main="met3")
curve(dgamma(x,2,1),add=TRUE,col=2,lwd=2)

So, this sampler seems to be sampling something close to the desired target, but not the same. This raises a couple of questions. First and most important, can we fix this sampler so that it does sample the correct target (yes), and second, can we figure out what target density the incorrect sampler is actually sampling (again, yes)? Let’s start with the issue of how to fix the sampler, as this will also help us to understand what the incorrect sampler is doing.

Fixing the truncated sampler

By repeatedly sampling from the proposal until we obtain a non-negative value, we are actually implementing a rejection sampler for sampling from the proposal distribution truncated at zero. This is a perfectly reasonable proposal distribution, so we can use it provided that we use the correct MH acceptance ratio. Now, the truncated density has the same density as the untruncated density, apart from the differing support and a normalising constant. Indeed, this may be why people often assume this method will work, because normalising constants often don’t matter in MH schemes. However, the normalising constant only doesn’t matter if it is independent of the state, and here it is not… Explicitly, we have

f(x^\star|x) \propto \phi(x^\star-x),\quad x^\star>0.

Including the normalising constant we have

\displaystyle f(x^\star|x) = \frac{\phi(x^\star-x)}{\Phi(x)},\quad x^\star>0,

where \Phi(x) is the standard normal CDF. Consequently, the correct acceptance ratio to use with this proposal is

\displaystyle A = \frac{\pi(x^\star)}{\pi(x)} \frac{\phi(x-x^\star)}{\phi(x^\star-x)}\frac{\Phi(x)}{\Phi(x^\star)} =   \frac{\pi(x^\star)}{\pi(x)}\frac{\Phi(x)}{\Phi(x^\star)},

where we see that the normalising constants do not cancel out. We can modify the previous sampler to use the correct acceptance ratio as follows.

met4=function(iters)
  {
    xvec=numeric(iters)
    x=1
    for (i in 1:iters) {
      repeat {
        xs=x+rnorm(1)
        if (xs>0)
          break
      }
      A=xs*exp(-xs)/(x*exp(-x))
      A=A*pnorm(x)/pnorm(xs)
      if (runif(1)<A)
        x=xs
      xvec[i]=x
    }
    return(xvec)
  }

We can verify that this sampler gives leads to the correct target with the following commands.

out=met4(iters)
hist(out,100,freq=FALSE,main="met4")
curve(dgamma(x,2,1),add=TRUE,col=2,lwd=2)

So, truncating the proposal at zero is fine, provided that you modify the acceptance ratio accordingly.

What does the incorrect sampler target?

Now that we understand why the naive truncated sampler was wrong and how to fix it, we can, out of curiosity, wonder what distribution that sampler actually targets. Now we understand what proposal we are actually using, we can re-write the acceptance ratio as

\displaystyle A = \frac{\pi(x^\star)\Phi(x^\star)}{\pi(x)\Phi(x)}\frac{\frac{\phi(x-x^\star)}{\Phi(x^\star)}}{\frac{\phi(x^\star-x)}{\Phi(x)}},

from which it is clear that the actual target of this chain is

\tilde\pi(x) \propto \pi(x)\Phi(x),

or

\tilde\pi(x)\propto xe^{-x}\Phi(x),\quad x\geq 0.

The constant of proportionality is not immediately obvious, but is tractable, and turns out to be a nice undergraduate exercise in integration by parts, leading to

\displaystyle \tilde\pi(x) = \frac{2\sqrt{2\pi}}{2+\sqrt{2\pi}}xe^{-x}\Phi(x),\quad x\geq 0.

We can verify this using the following commands.

out=met3(iters)
hist(out,100,freq=FALSE,main="met3")
curve(dgamma(x,2,1)*pnorm(x)*2*sqrt(2*pi)/(sqrt(2*pi)+2),add=TRUE,col=3,lwd=2)

Now we know the actual target of the incorrect sampler, we can compare it with the correct target as follows.

curve(dgamma(x,2,1),0,10,col=2,lwd=2,main="Densities")
curve(dgamma(x,2,1)*pnorm(x)*2*sqrt(2*pi)/(sqrt(2*pi)+2),add=TRUE,col=3,lwd=2)

So we see that the distributions are different, but not so different that one would immediate suspect an error on the basis of a sample of output. This makes it a difficult bug to track down.

Summary

There is no problem in principle using a proposal with full support for a target with limited support in MH algorithms. However, it is important to check whether a proposed value is within the support of the target and reject the proposed move if it is not. If you are concerned that such a scheme might be inefficient, it is possible to use a truncated proposal provided that you modify the MH acceptance ratio to include the relevant normalisation constants. If you don’t modify the acceptance probability, you will get a sampler which targets the wrong distribution, but it will often be quite similar to the correct target, making it a difficult bug to spot and track down.

The particle marginal Metropolis-Hastings (PMMH) particle MCMC algorithm

Introduction

In the previous post I explained how one can use an unbiased estimate of marginal likelihood derived from a particle filter within a Metropolis-Hastings MCMC algorithm in order to construct an exact pseudo-marginal MCMC scheme for the posterior distribution of the model parameters given some time course data. This idea is closely related to that of the particle marginal Metropolis-Hastings (PMMH) algorithm of Andreiu et al (2010), but not really exactly the same. This is because for a Bayesian model with parameters \theta, latent variables x and data y, of the form

\displaystyle  p(\theta,x,y) = p(\theta)p(x|\theta)p(y|x,\theta),

the pseudo-marginal algorithm which exploits the fact that the particle filter’s estimate of likelihood is unbiased is an MCMC algorithm which directly targets the marginal posterior distribution p(\theta|y). On the other hand, the PMMH algorithm is an MCMC algorithm which targets the full joint posterior distribution p(\theta,x|y). Now, the PMMH scheme does reduce to the pseudo-marginal scheme if samples of x are not generated and stored in the state of the Markov chain, and it certainly is the case that the pseudo-marginal algorithm gives some insight into why the PMMH algorithm works. However, the PMMH algorithm is much more powerful, as it solves the “smoothing” and parameter estimation problem simultaneously and exactly, including the “initial value” problem (computing the posterior distribution of the initial state, x_0). Below I will describe the algorithm and explain why it works, but first it is necessary to understand the relationship between marginal, joint and “likelihood-free” MCMC updating schemes for such latent variable models.

MCMC for latent variable models

Marginal approach

If we want to target p(\theta|y) directly, we can use a Metropolis-Hastings scheme with a fairly arbitrary proposal distribution for exploring \theta, where a new \theta^\star is proposed from f(\theta^\star|\theta) and accepted with probability \min\{1,A\}, where

\displaystyle  A = \frac{p(\theta^\star)}{p(\theta)} \times  \frac{f(\theta|\theta^\star)}{f(\theta^\star|\theta)} \times \frac{p({y}|\theta^\star)}{p({y}|\theta)}.

As previously discussed, the problem with this scheme is that the marginal likelihood p(y|\theta) required in the acceptance ratio is often difficult to compute.

Likelihood-free MCMC

A simple “likelihood-free” scheme targets the full joint posterior distribution p(\theta,x|y). It works by exploiting the fact that we can often simulate from the model for the latent variables p(x|\theta) even when we can’t evaluate it, or marginalise x out of the problem. Here the Metropolis-Hastings proposal is constructed in two stages. First, a proposed new \theta^\star is sampled from f(\theta^\star|\theta) and then a corresponding x^\star is simulated from the model p(x^\star|\theta^\star). The pair (\theta^\star,x^\star) is then jointly accepted with ratio

\displaystyle  A = \frac{p(\theta^\star)}{p(\theta)} \times  \frac{f(\theta|\theta^\star)}{f(\theta^\star|\theta)} \times \frac{p(y|{x}^\star,\theta^\star)}{p(y|{x},\theta)}.

The proposal mechanism ensures that the proposed x^\star is consistent with the proposed \theta^\star, and so the procedure can work provided that the dimension of the data y is low. However, in order to work well more generally, we would want the proposed latent variables to be consistent with the data as well as the model parameters.

Ideal joint update

Motivated by the likelihood-free scheme, we would really like to target the joint posterior p(\theta,x|y) by first proposing \theta^\star from f(\theta^\star|\theta) and then a corresponding x^\star from the conditional distribution p(x^\star|\theta^\star,y). The pair (\theta^\star,x^\star) is then jointly accepted with ratio

\displaystyle  A = \frac{p(\theta^\star)}{p(\theta)}   \frac{p({x}^\star|\theta^\star)}{p({x}|\theta)}   \frac{f(\theta|\theta^\star)}{f(\theta^\star|\theta)}   \frac{p(y|{x}^\star,\theta^\star)}{p(y|{x},\theta)}  \frac{p({x}|y,\theta)}{p({x}^\star|y,\theta^\star)}\\  \qquad = \frac{p(\theta^\star)}{p(\theta)}  \frac{p(y|\theta^\star)}{p(y|\theta)} \frac{f(\theta|\theta^\star)}{f(\theta^\star|\theta)}.

Notice how the acceptance ratio simplifies, using the basic marginal likelihood identity (BMI) of Chib (1995), and x drops out of the ratio completely in order to give exactly the ratio used for the marginal updating scheme. Thus, the “ideal” joint updating scheme reduces to the marginal updating scheme if x is not sampled and stored as a component of the Markov chain.

Understanding the relationship between these schemes is useful for understanding the PMMH algorithm. Indeed, we will see that the “ideal” joint updating scheme (and the marginal scheme) corresponds to PMMH using infinitely many particles in the particle filter, and that the likelihood-free scheme corresponds to PMMH using exactly one particle in the particle filter. For an intermediate number of particles, the PMMH scheme is a compromise between the “ideal” scheme and the “blind” likelihood-free scheme, but is always likelihood-free (when used with a bootstrap particle filter) and always has an acceptance ratio leaving the exact posterior invariant.

The PMMH algorithm

The algorithm

The PMMH algorithm is an MCMC algorithm for state space models jointly updating \theta and x_{0:T}, as the algorithms above. First, a proposed new \theta^\star is generated from a proposal f(\theta^\star|\theta), and then a corresponding x_{0:T}^\star is generated by running a bootstrap particle filter (as described in the previous post, and below) using the proposed new model parameters, \theta^\star, and selecting a single trajectory by sampling once from the final set of particles using the final set of weights. This proposed pair (\theta^\star,x_{0:T}^\star) is accepted using the Metropolis-Hastings ratio

\displaystyle  A = \frac{\hat{p}_{\theta^\star}(y_{1:T})p(\theta^\star)q(\theta|\theta^\star)}{\hat{p}_{\theta}(y_{1:T})p(\theta)q(\theta^\star|\theta)},

where \hat{p}_{\theta^\star}(y_{1:T}) is the particle filter’s (unbiased) estimate of marginal likelihood, described in the previous post, and below. Note that this approach tends to the perfect joint/marginal updating scheme as the number of particles used in the filter tends to infinity. Note also that for a single particle, the particle filter just blindly forward simulates from p_\theta(x^\star_{0:T}) and that the filter’s estimate of marginal likelihood is just the observed data likelihood p_\theta(y_{1:T}|x^\star_{0:T}) leading precisely to the simple likelihood-free scheme. To understand for an arbitrary finite number of particles, M, one needs to think carefully about the structure of the particle filter.

Why it works

To understand why PMMH works, it is necessary to think about the joint distribution of all random variables used in the bootstrap particle filter. To this end, it is helpful to re-visit the particle filter, thinking carefully about the resampling and propagation steps.

First introduce notation for the “particle cloud”: \mathbf{x}_t=\{x_t^k|k=1,\ldots,M\}, \boldsymbol{\pi}_t=\{\pi_t^k|k=1,\ldots,M\}, \tilde{\mathbf{x}}_t=\{(x_t^k,\pi_t^k)|k=1,\ldots,M\}. Initialise the particle filter with \tilde{\mathbf{x}}_0, where x_0^k\sim p(x_0) and \pi_0^k=1/M (note that w_0^k is undefined). Now suppose at time t we have a sample from p(x_t|y_{1:t}): \tilde{\mathbf{x}}_t. First resample by sampling a_t^k \sim \mathcal{F}(a_t^k|\boldsymbol{\pi}_t), k=1,\ldots,M. Here we use \mathcal{F}(\cdot|\boldsymbol{\pi}) for the discrete distribution on 1:M with probability mass function \boldsymbol{\pi}. Next sample x_{t+1}^k\sim p(x_{t+1}^k|x_t^{a_t^k}). Set w_{t+1}^k=p(y_{t+1}|x_{t+1}^k) and \pi_{t+1}^k=w_{t+1}^k/\sum_{i=1}^M w_{t+1}^i. Finally, propagate \tilde{\mathbf{x}}_{t+1} to the next step… We define the filter’s estimate of likelihood as \hat{p}(y_t|y_{1:t-1})=\frac{1}{M}\sum_{i=1}^M w_t^i and \hat{p}(y_{1:T})=\prod_{i=1}^T \hat{p}(y_t|y_{1:t-1}). See Doucet et al (2001) for further theoretical background on particle filters and SMC more generally.

Describing the filter carefully as above allows us to write down the joint density of all random variables in the filter as

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

For PMMH we also sample a final index k' from \mathcal{F}(k'|\boldsymbol{\pi}_T) giving the joint density

\displaystyle  \tilde{q}(\mathbf{x}_0,\ldots,\mathbf{x}_T,\mathbf{a}_0,\ldots,\mathbf{a}_{T-1})\pi_T^{k'}

We write the final selected trajectory as

\displaystyle  x_{0:T}^{k'}=(x_0^{b_0^{k'}},\ldots,x_T^{b_T^{k'}}),

where b_t^{k'}=a_t^{b_{t+1}^{k'}}, and b_T^{k'}=k'. If we now think about the structure of the PMMH algorithm, our proposal on the space of all random variables in the problem is in fact

\displaystyle  f(\theta^\star|\theta)\tilde{q}_{\theta^\star}(\mathbf{x}_0^\star,\ldots,\mathbf{x}_T^\star,\mathbf{a}_0^\star,\ldots,\mathbf{a}_{T-1}^\star)\pi_T^{{k'}^\star}

and by considering the proposal and the acceptance ratio, it is clear that detailed balance for the chain is satisfied by the target with density proportional to

\displaystyle  p(\theta)\hat{p}_\theta(y_{1:T})  \tilde{q}_\theta(\mathbf{x}_0,\ldots,\mathbf{x}_T,\mathbf{a}_0,\ldots,\mathbf{a}_{T-1})  \pi_T^{k'}

We want to show that this target marginalises down to the correct posterior p(\theta,x_{0:T}|y_{1:T}) when we consider just the parameters and the selected trajectory. But if we consider the terms in the joint distribution of the proposal corresponding to the trajectory selected by k', this is given by

\displaystyle  p_\theta(x_0^{b_0^{k'}})\left[\prod_{t=0}^{T-1} \pi_t^{b_t^{k'}}    p_\theta(x_{t+1}^{b_{t+1}^{k'}}|x_t^{b_t^{k'}})\right]\pi_T^{k'}  =  p_\theta(x_{0:T}^{k'})\prod_{t=0}^T \pi_t^{b_t^{k'}}

which, by expanding the \pi_t^{b_t^{k'}} in terms of the unnormalised weights, simplifies to

\displaystyle  \frac{p_\theta(x_{0:T}^{k'})p_\theta(y_{1:T}|x_{0:T}^{k'})}{M^{T+1}\hat{p}_\theta(y_{1:T})}

It is worth dwelling on this result, as this is the key insight required to understand why the PMMH algorithm works. The whole point is that the terms in the joint density of the proposal corresponding to the selected trajectory exactly represent the required joint distribution modulo a couple of normalising constants, one of which is the particle filter’s estimate of marginal likelihood. Thus, by including \hat{p}_\theta(y_{1:T}) in the acceptance ratio, we knock out the normalising constant, allowing all of the other terms in the proposal to be marginalised away. In other words, the target of the chain can be written as proportional to

\displaystyle  \frac{p(\theta)p_\theta(x_{0:T}^{k'},y_{1:T})}{M^{T+1}} \times  \text{(Other terms...)}

The other terms are all probabilities of random variables which do not occur elsewhere in the target, and hence can all be marginalised away to leave the correct posterior

\displaystyle  p(\theta,x_{0:T}|y_{1:T})

Thus the PMMH algorithm targets the correct posterior for any number of particles, M. Also note the implied uniform distribution on the selected indices in the target.

I will give some code examples in a future post.

The pseudo-marginal approach to “exact approximate” MCMC algorithms

Motivation and background

In this post I will try and explain an important idea behind some recent developments in MCMC theory. First, let me give some motivation. Suppose you are trying to implement a Metropolis-Hastings algorithm, as discussed in a previous post (required reading!), but a key likelihood term needed for the acceptance ratio is difficult to evaluate (most likely it is a marginal likelihood of some sort). If it is possible to obtain a Monte-Carlo estimate for that likelihood term (which it sometimes is, for example, using importance sampling), one could obviously just plug it in to the acceptance ratio and hope for the best. What is not at all obvious is that if your Monte-Carlo estimate satisfies some fairly weak property then the equilibrium distribution of the Markov chain will remain exactly as it would be if the exact likelihood had been available. Furthermore, it is exact even if the Monte-Carlo estimate is very noisy and imprecise (though the mixing of the chain will be poorer in this case). This is the “exact approximate” pseudo-marginal MCMC approach. To give credit where it is due, the idea was first introduced by Mark Beaumont in Beaumont (2003), where he was using an importance sampling based approximate likelihood in the context of a statistical genetics example. This was later picked up by Christophe Andrieu and Gareth Roberts, who studied the technical properties of the approach in Andrieu and Roberts (2009). The idea is turning out to be useful in several contexts, and in particular, underpins the interesting new Particle MCMC algorithms of Andrieu et al (2010), which I will discuss in a future post. I’ve heard Mark, Christophe, Gareth and probably others present this concept, but this post is most strongly inspired by a talk that Christophe gave at the IMS 2010 meeting in Gothenburg this summer.

The pseudo-marginal Metropolis-Hastings algorithm

Let’s focus on the simplest version of the problem, where we want to sample from a target p(x) using a proposal q(x'|x). As explained previously, the required Metropolis-Hastings acceptance ratio will take the form

\displaystyle A=\frac{p(x')q(x|x')}{p(x)q(x'|x)}.

Here we are assuming that p(x) is difficult to evaluate (usually because it is a marginalised version of some higher-dimensional distribution), but that a Monte-Carlo estimate of p(x), which we shall denote r(x), can be computed. We can obviously just substitute this estimate into the acceptance ratio to get

\displaystyle A=\frac{r(x')q(x|x')}{r(x)q(x'|x)},

but it is not immediately clear that in many cases this will lead to the Markov chain having an equilibrium distribution that is exactly p(x). It turns out that it is sufficient that the likelihood estimate, r(x) is non-negative, and unbiased, in the sense that E(r(x))=p(x), where the expectation is with respect to the Monte-Carlo error for a given fixed value of x. In fact, as we shall see, this condition is actually a bit stronger than is really required.

Put W=r(x)/p(x), representing the noise in the Monte-Carlo estimate of p(x), and suppose that W \sim p(w|x) (note that in an abuse of notation, the function p(w|x) is unrelated to p(x)). The main condition we will assume is that E(W|x)=c, where c>0 is a constant independent of x. In the case of c=1, we have the (typical) special case of E(r(x))=p(x). For now, we will also assume that W\geq 0, but we will consider relaxing this constraint later.

The key to understanding the pseudo-marginal approach is to realise that at each iteration of the MCMC algorithm a new value of W is being proposed in addition to a new value for x. If we regard the proposal mechanism as a joint update of x and w, it is clear that the proposal generates (x',w') from the density q(x'|x)p(w'|x'), and we can re-write our “approximate” acceptance ratio as

\displaystyle A=\frac{w'p(x')p(w'|x')q(x|x')p(w|x)}{wp(x)p(w|x)q(x'|x)p(w'|x')}.

Inspection of this acceptance ratio reveals that the target of the chain must be (proportional to)

p(x)wp(w|x).

This is a joint density for (x,w), but the marginal for x can be obtained by integrating over the range of W with respect to w. Using the fact that E(W|x)=c, this then clearly gives a density proportional to p(x), and this is precisely the target that we require. Note that for this to work, we must keep the old value of w from one iteration to the next – that is, we must keep and re-use our noisy r(x) value to include in the acceptance ratio for our next MCMC iteration – we should not compute a new Monte-Carlo estimate for the likelihood of the old state of the chain.

Examples

We will consider again the example from the previous post – simulation of a chain with a N(0,1) target using uniform innovations. Using R, the main MCMC loop takes the form

pmmcmc<-function(n=1000,alpha=0.5) 
{
        vec=vector("numeric", n)
        x=0
        oldlik=noisydnorm(x)
        vec[1]=x
        for (i in 2:n) {
                innov=runif(1,-alpha,alpha)
                can=x+innov
                lik=noisydnorm(can)
                aprob=lik/oldlik
                u=runif(1)
                if (u < aprob) { 
                        x=can
                        oldlik=lik
			}
                vec[i]=x
        }
        vec
}

Here we are assuming that we are unable to compute dnorm exactly, but instead only a Monte-Carlo estimate called noisydnorm. We can start with the following implementation

noisydnorm<-function(z)
{
	dnorm(z)*rexp(1,1)
}

Each time this function is called, it will return a non-negative random quantity whose expectation is dnorm(z). We can now run this code as follows.

plot.mcmc<-function(mcmc.out)
{
	op=par(mfrow=c(2,2))
	plot(ts(mcmc.out),col=2)
	hist(mcmc.out,30,col=3)
	qqnorm(mcmc.out,col=4)
	abline(0,1,col=2)
	acf(mcmc.out,col=2,lag.max=100)
	par(op)
}

metrop.out<-pmmcmc(10000,1)
plot.mcmc(metrop.out)
MCMC output and convergence diagnostics

So we see that we have exactly the right N(0,1) target, despite using the (very) noisy noisydnorm function in place of the dnorm function. This noisy likelihood function is clearly unbiased. However, as already discussed, a constant bias in the noise is also acceptable, as the following function shows.

noisydnorm<-function(z)
{
	dnorm(z)*rexp(1,2)
}

Re-running the code with this function also leads to the correct equilibrium distribution for the chain. However, it really does matter that the bias is independent of the state of the chain, as the following function shows.

noisydnorm<-function(z)
{
	dnorm(z)*rexp(1,0.1+10*z*z)
}

Running with this function leads to the wrong equilibrium. However, it is OK for the distribution of the noise to depend on the state, as long as its expectation does not. The following function illustrates this.

noisydnorm<-function(z)
{
	dnorm(z)*rgamma(1,0.1+10*z*z,0.1+10*z*z)
}

This works just fine. So far we have been assuming that our noisy likelihood estimates are non-negative. This is clearly desirable, as otherwise we could wind up with negative Metropolis-Hasting ratios. However, as long as we are careful about exactly what we mean, even this non-negativity condition may be relaxed. The following function illustrates the point.

noisydnorm<-function(z)
{
	dnorm(z)*rnorm(1,1)
}

Despite the fact that this function will often produce negative values, the equilibrium distribution of the chain still seems to be correct! An even more spectacular example follows.

noisydnorm<-function(z)
{
	dnorm(z)*rnorm(1,0)
}

Astonishingly, this one works too, despite having an expected value of zero! However, the following doesn’t work, even though it too has a constant expectation of zero.

noisydnorm<-function(z)
{
	dnorm(z)*rnorm(1,0,0.1+10*z*z)
}

I don’t have time to explain exactly what is going on here, so the precise details are left as an exercise for the reader. Suffice to say that the key requirement is that it is the distribution of W conditioned to be non-negative which must have the constant bias property – something clearly violated by the final noisydnorm example.

Before finishing this post, it is worth re-emphasising the issue of re-using old Monte-Carlo estimates. The following function will not work (exactly), though in the case of good Monte-Carlo estimates will often work tolerably well.

approxmcmc<-function(n=1000,alpha=0.5) 
{
        vec=vector("numeric", n)
        x=0
        vec[1]=x
        for (i in 2:n) {
                innov=runif(1,-alpha,alpha)
                can=x+innov
                lik=noisydnorm(can)
                oldlik=noisydnorm(x)
                aprob=lik/oldlik
                u=runif(1)
                if (u < aprob) { 
                        x=can
			}
                vec[i]=x
        }
        vec
}

In a subsequent post I will show how these ideas can be put into practice in the context of a Bayesian inference example.

Metropolis-Hastings MCMC algorithms

Background

Over the next few months I’m intending to have a series of posts on recent developments in MCMC and algorithms of Metropolis-Hastings type. These posts will assume a basic familiarity with stochastic simulation and R. For reference, I have some old notes on stochastic simulation and MCMC from a course I used to teach. There were a set of computer practicals associated with the course which focused on understanding the algorithms, mainly using R. Indeed, this post is going to make use of the example from this old practical. If anything about the rest of this post doesn’t make sense, you might need to review some of this background material. My favourite introductory text book on this subject is Gamerman’s Markov Chain Monte Carlo. In this post I’m going to briefly recap the key idea behind the Metropolis-Hastings algorithm, and illustrate how these kinds of algorithms are typically implemented. In order to keep things as simple as possible, I’m going to use R for implementation, but as discussed in a previous post, that probably won’t be a good idea for non-trivial applications.

Metropolis-Hastings

The motivation is a need to understand a complex probability distribution, p(x). In applications to Bayesian statistics, this distribution is usually a posterior from a Bayesian analysis. A good way to understand an intractable distribution is to simulate realisations from it and study those. However, this often isn’t easy, either. The idea behind MCMC is to simulate a Markov chain whose equilibrium distribution is p(x). Metropolis-Hastings (M-H) provides a way to correct a fairly arbitrary transition kernel q(x’|x) (which typically won’t have p(x) as its equilibrium) to give a chain which does have the desired target. In M-H, the transition kernel is used to generate a proposed new value, x’ for the chain, which is then accepted as the new state at random with a particular probability a(x’|x)=min(1,A), where A = p(x’)q(x|x’)/[p(x)q(x’|x)].

If the value is accepted, then the chain moves to the proposed new state, x’. If the value is not accepted, the chain still advances to the next step, but the new state is given by the previous state of the chain, x.

It is clear that this algorithm is also a Markov chain, and that for x’ != x, the transition kernel of this chain is q(x’|x)a(x’|x). A couple of lines of algebra will then confirm that this “corrected” Markov chain satisfies detailed balance for the target p(x). Therefore, under some regularity conditions, this chain will have p(x) as its equilibrium, and this gives a convenient way of simulating values from this target.

Special case: the Metropolis method

It is clear from the form of the acceptance probability that a considerable simplification arises in the case where the proposal kernel is symmetric, having q(x’|x)=q(x|x’). This is useful, as a convenient updating strategy is often to update the current value of x by adding an innovation sampled from a distribution that is symmetric about zero. This clearly gives a symmetric kernel, which then drops out of the acceptance ratio, to give A=p(x’)/p(x).

Example

The example we will use for illustration is a Metropolis method for the standard normal distribution using innovations from a U(-eps,eps) distribution. Below is a simple R implementation of the algorithm

metrop1=function(n=1000,eps=0.5) 
{
        vec=vector("numeric", n)
        x=0
        vec[1]=x
        for (i in 2:n) {
                innov=runif(1,-eps,eps)
                can=x+innov
                aprob=min(1,dnorm(can)/dnorm(x))
                u=runif(1)
                if (u < aprob) 
                        x=can
                vec[i]=x
        }
        vec
}

First a candidate value (can) is constructed by perturbing the current state of the chain with the uniform innovation. Then the acceptance probability is computed, and the chain is updated appropriately depending on whether the proposed new value is accepted. Note the standard trick of picking an event with probability p by checking to see if u<p, where u is a draw from a U(0,1).

We can execute this code and plot the results with the following peice of code:

plot.mcmc<-function(mcmc.out)
{
	op=par(mfrow=c(2,2))
	plot(ts(mcmc.out),col=2)
	hist(mcmc.out,30,col=3)
	qqnorm(mcmc.out,col=4)
	abline(0,1,col=2)
	acf(mcmc.out,col=2,lag.max=100)
	par(op)
}

metrop.out<-metrop1(10000,1)
plot.mcmc(metrop.out)
Plots showing convergence of the Markov chain to a N(0,1) equilibrium distribution

From these plots we see that the algorithm seems to be working as expected. Before finishing this post, it is worth explaining how to improve this code to get something which looks a bit more like the kind of code that people would actually write in practice. The next version of the code (below) makes use of the fact that min(1,A) is redundant if you are just going to compare to a uniform (since values of the ratio larger than 1 will be accepted with probability 1, as they should), and also that it is unnecessary to recalculate the likelihood of the current value of the chain at every iteration – better to store and re-use the value. That obviously doesn’t make much difference here, but for real problems likelihood computations are often the main computational bottleneck.

metrop2=function(n=1000,eps=0.5) 
{
        vec=vector("numeric", n)
        x=0
        oldlik=dnorm(x)
        vec[1]=x
        for (i in 2:n) {
                innov=runif(1,-eps,eps)
                can=x+innov
                lik=dnorm(can)
                a=lik/oldlik
                u=runif(1)
                if (u < a) { 
                        x=can
                        oldlik=lik
                        }
                vec[i]=x
        }
        vec
}

However, this code still has a very serious flaw. It computes likelihoods! For this problem it isn’t a major issue, but in general likelihoods are the product of a very large number of small numbers, and numerical underflow is a constant hazard. For this reason (and others), no-one ever computes likelihoods in code if they can possibly avoid it, but instead log-likelihoods (which are the sum of a large number of reasonably-sized numbers, and therefore numerically very stable). We can use these log-likelihoods to calculate a log-acceptance ratio, which can then be compared to the log of a uniform for the accept-reject decision. We end up with the code below, which now doesn’t look too different to the kind of code one might actually write…

metrop3=function(n=1000,eps=0.5) 
{
        vec=vector("numeric", n)
        x=0
        oldll=dnorm(x,log=TRUE)
        vec[1]=x
        for (i in 2:n) {
                can=x+runif(1,-eps,eps)
                loglik=dnorm(can,log=TRUE)
                loga=loglik-oldll
                if (log(runif(1)) < loga) { 
                        x=can
                        oldll=loglik
                        }
                vec[i]=x
        }
        vec
}

It is easy to verify that all 3 of these implementations behave identically. Incidentally, most statisticians would consider all 3 of the above codes to represent exactly the same algorithm. They are just slightly different implementations of the same algorithm. I mention this because I’ve noticed that computing scientists often have a slightly lower-level view of what constitutes an algorithm, and would sometimes consider the above 3 algorithms to be different.