Improving the acceptance rate of reversible jump MCMC proposals

Statistics & Probability Letters 69 (2004) 189–198

Improving the acceptance rate of reversible jump MCMCproposals

Fahimah Al-Awadhia, Merrilee Hurnb,*, Christopher Jennisonb

aDepartment of Statistics and OR, Kuwait University, P.O. Box 5969, Safat 13060, KuwaitbDepartment of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK

Received 17 June 2002; received in revised form 11 March 2004

Available online 06 July 2004

Abstract

Recent articles have commented on the difficulty of proposing efficient reversible jump moves withinMCMC. We suggest a new way to make proposals more acceptable using a secondary Markov chain tomodify proposed moves—at little extra programming cost.r 2004 Elsevier B.V. All rights reserved.

Keywords: Image analysis; Markov chain Monte Carlo; Object recognition; Reversible jump MCMC; Stochastic

simulation; Tempering

1. Introduction

Since 1970, when Hastings introduced Markov chain Monte Carlo algorithms into the statisticsliterature, their use has grown enormously and become an almost routine tool in some applicationareas (Hastings, 1970; Gilks et al., 1995). The appeal and applicability of these methods receivedanother boost with the extension to distributions defined over variable dimensions, reversiblejump MCMC (Green, 1995). Implementations of this RJMCMC methodology generally use aselection of move types, some of which explore the sample space within a fixed dimension whileothers make changes in dimensionality. Both types of move should be as efficient as possible inmoving around the sample space; one sign of inefficiency is an unacceptably high rejection rate for

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*Corresponding author. University of Bath, School of Mathematical Sciences, Claverton Down, Avon, Bath BA2

7AY, UK.

E-mail address: [email protected] (M. Hurn).

0167-7152/$ - see front matter r 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.spl.2004.06.025

the proposed changes. In fixed dimensions, small changes usually have higher acceptance ratesthan large ones, and proposal mechanisms can be scaled to achieve a desired acceptance rate.Brooks et al. (2003) point out that this option is not always available for moves betweendimensions as there may be no natural distance measure between states of different dimensions, forexample, these may correspond to competing non-nested models with different sets of parameters.In this article we consider highly structured problems where jumps must be made between

multi-modal distributions in different dimensional spaces. We focus on some of the severeproblems that occur in sampling image distributions, and suggest an approach which uses asecondary Markov chain to modify an initial proposal, allowing it to move from a low-probabilityregion of the new space towards a mode before comparing it with the starting state. The approachhas the benefit that existing code may often be used for the secondary chain. In Section 2, wedescribe the reversible jump framework, the problem of slow movement between dimensions, andexisting solutions. In Section 3, we describe our approach and the circumstances under which it islikely to be particularly beneficial. We present a simple example to illustrate the method’spotential in Section 4 and an object recognition example in Section 5.

2. The reversible jump framework

Suppose we wish to sample from a distribution pðxÞ in which the vector x is of variabledimension. Let pnðxÞ denote the sub-density with respect to Lebesgue measure on Rn for values ofx with dimension n: Suppose qmnðx; x0Þ is a proposal density on Rn; or a subspace of Rn;for moves from Rm to Rn and the reverse proposal for moving from Rn to Rm has densityqnmðx0; xÞ: (For mon; transitions from xARm to x0ARn are often defined such that x determines melements of x0 and qmnðx; x0Þ is a density on the remaining ðn � mÞ-dimensional subspace of Rn; inthis case, the reverse step is deterministic and the kernel qnmðx;0 xÞ degenerate.) Green (1995) givesthe formula

amnðx; x0Þ ¼ min 1;pnðx0Þqnmðx0;xÞpmðxÞqmnðx;x0Þ

� �ð1Þ

for the acceptance probability needed to maintain detailed balance with respect to p:In any MCMC sampler, the acceptance rate of proposals should be reasonably high. In fixed

dimension problems, a proposal of the form x0 ¼ x þ g; where g is a zero-mean random variable,can be calibrated by adjusting the variance of g to achieve a suitable acceptance rate.Unfortunately, there is often no simple parallel to this approach when x and x0 have differentdimensions.Suggestions have been made for improving acceptance rates in RJMCMC. Green and Mira

(2001) introduce a second stage to the proposal mechanism in their ‘‘delayed rejection sampler’’.Following rejection of any proposed move, a second proposal is made bearing in mind the firstrejection, and possibly making use of the rejected proposal value. The method offers the option ofproposing a large change in dimension and then switching to a smaller change in dimension if thisis rejected. The overall acceptance probability takes into account this more complicated structure.Brooks et al. (2003) return to the scaling idea by looking for canonical jump functions which

play the role of g ¼ 0 in tuning the acceptance rate, but now act across dimensions. For example,

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with nested models such as AR processes of unknown order, a canonical jump might be toincrease the order by one, with the newly introduced coefficient equal to zero and all othercoefficients unchanged. Once suitable correspondences have been identified, proposals can beconstructed as scaled perturbations around these central moves. A second idea suggested byBrooks et al. (2003) is that of using auxiliary variables to ‘‘pad out’’ the problem to a commondimension for all models under consideration. These auxiliary variables retain information abouta model even when that model is not the current one. This means that when a move back to thatmodel is proposed, the auxiliary variables may be able to guide the proposal to somewhere closeto where it last left.In another approach, Green (2003) tries to regain some of the simplicity of the random walk

Metropolis sampler with a view to automating implementation of RJMCMCmethods. A trial runis used to approximate the mean and variance of x in each dimension. These values are thenused in constructing appropriate dimension Gaussian proposals both within and betweendimensions.Stephens (2000) suggests an alternative to RJMCMC sampling of mixture distributions based

on spatial birth and death processes. Here, the mixture components correspond to individualmarked points in the point process and dimensionality changes are made using birth and deathprocesses to simulate the point process. Although this approach can be generalised to otherapplications, it does not necessarily offer a solution to mixing problems: Capp!e et al. (2003) notethe close relationship between RJMCMC and spatial birth and death processes, and warn thatproblems affecting the former are liable to arise in the latter.

3. Alternative proposal mechanism

When a target distribution is highly modal within each dimension, proposals which land awayfrom the modal region have virtually no chance of acceptance. If the locations of modes are notinitially known, designing a matched pair of dimension-changing proposals to produce jumpsfrom mode to mode requires much ingenuity, if indeed it is possible at all. Our approach is tomodify the value of a simpler RJMCMC proposal, moving it closer to a mode before the accept–reject decision is taken.Suppose the Markov chain sampler is currently in state xARm and we have chosen to consider a

jump to Rn: We first generate a proposal x0 in Rn from the density qmnðx;x0Þ; just as before. Butnow we follow this by k fixed-dimension MCMC steps through states x1;y;xk ¼ x� say, eachstep satisfying detailed balance with respect to a new distribution p�n ðxÞ: The final state x�ARn iseither accepted as the next state of the Markov chain or rejected, in which case the Markov chainremains at x: In the reverse of this move, a jump from x�ARn to Rm is proposed by taking k fixeddimension steps within Rn; each of which satisfies detailed balance with respect to p�n ðxÞ; ending atx0 say, then sampling from the density qnmðx0;xÞ to give the proposal x for possible acceptance.The role of p�n is to promote movement towards the modes of the original distribution; we willdiscuss the choice of p�n ðxÞ and k later.The acceptance probability must be calculated to ensure detailed balance with respect to the

overall target distribution p: Let P denote the transition kernel for the full sequence of k moves inRn from x0 to x�: Assuming equal probabilities of choosing the two types of move from Rm to Rn

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and from Rn to Rm; we requireZðx;x0;x�ÞAA�B�C

pmðxÞqmnðx;x0ÞPðx0;x�Þamnðx; x�Þdx dx0 dx�

¼Zðx;x0;x�ÞAA�B�C

pnðx�ÞPðx�;x0Þqnmðx0;xÞanmðx�;xÞdx dx0 dx�; ð2Þ

for all Borel sets AARm; BARn and CARn (or suitably defined sets BðxÞ when qmn and qnm aredefined on subspaces of Rn and Rm). Then, the probability under p of being in state x; moving tox0 and on to x� and accepting this proposal exactly balances the probability under p of being inx�; making the reverse transitions to x0 and onto x and accepting this proposal. One way to makethe integrals equal is to make the integrands in (2) equal by taking

amnðx; x�Þ ¼ min 1;pnðx�ÞPðx�;x0Þqnmðx0; xÞpmðxÞqmnðx;x0ÞPðx0; x�Þ

� �ð3Þ

and

anmðx�; xÞ ¼ min 1;pmðxÞqmnðx;x0ÞPðx0; x�Þpnðx�ÞPðx�;x0Þqnmðx0; xÞ

� �:

Since the k individual transitions comprising P satisfy detailed balance with respect to p�n ; so doesP; p�n ðx

�ÞPðx�; x0Þ ¼ p�n ðx0ÞPðx0;x�Þ; and (3) simplifies to

amnðx; x�Þ ¼ min 1;pnðx�Þp�n ðx

0Þqnmðx0;xÞpmðxÞp�n ðx�Þqmnðx; x0Þ

� �: ð4Þ

There is an asymmetry in transitions in the two directions: in moves from Rm to Rn the jumpacross dimensions occurs first, then the MCMC steps under p�n ; moves in the other direction startby taking steps within Rn under p�n and then jump to the proposal in Rm: With mon; this may beappropriate if there is only a need to improve initial proposals in the higher dimensional space. Ifit is also desirable to modify the value reached on jumping to Rm a spell of MCMC samplingunder suitably modified target distributions could be included in both Rm and Rn; either side ofthe reversible jump step, with a consequent change in the acceptance probability formula.It is instructive to compare likely values of amnðx; x0Þ given by (1) for the original method and of

amnðx;x�Þ in (4) for the new method. Since the current state x has arisen in a Markov chain withequilibrium distribution p; we can hope x will take a typical value for the sub-distribution pm:Assuming it is relatively easy to define a suitable transition to a lower dimension, we supposeqnmðx0; xÞ will be comparable with pmðxÞ: Difficulty arises in generating the additional elements inx0 in such a way that qmnðx;x0Þ places weight on important parts of pnðx0Þ: if this is not achievedand x0 is a typical sample from qmnðx;x0Þ but not from pn; we have pnðx0Þ5qmnðx; x0Þ and theacceptance probability amnðx;x0Þ in (1) will be extremely small. The difference in our new methodis that the ratio pnðx0Þ=qmnðx;x0Þ in the old acceptance probability (1) is replaced in the newamnðx;x�Þ given by (4) by

pnðx�Þp�n ðx�Þ

�p�n ðx

0Þqmnðx;x0Þ

: ð5Þ

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These two factors are still liable to be less than 1 but a good choice of p�n ; intermediate in somesense between qmn and pn; can ensure that (4) is substantially larger than (1). Distributions definedin terms of a temperature g could be employed, where p� ¼ pg is a ‘‘tempered’’ version of thetarget distribution p: Although not the only option, tempering is a natural way to create a flatterversion of p for the transition process to pass through, but one which retains the same modes.There is a compromise here, between making p� easy to sample efficiently, and ensuring that theresulting x� do actually tend to be in modal regions of p: Sampling from a tempered version of pcan also be based on the same fixed-dimensional transitions used in sampling from p; hence littleadditional programming is needed beyond that required to implement the original reversible jumpmethod. (For more discussion of tempering, see Neal, 1996.)

4. A simple illustrative example

The following example demonstrates the scale of improvement our proposed method offers.Suppose p is defined on R1 and R2 with p1 equal to 0.5 times a N (0,1) density on R1 and p2equal to 0.5 times a bivariate normal density on R2 in which X1 and X2 are independentwith X1BNð0; 1Þ and X2BNð0; s2Þ: In proposing transitions from xAR1 to x0AR2 we set x0

1 ¼ x1

and generate x02 from a Nð0; x2Þ distribution where x2bs2; thus, this is an example of a

proposal kernel q12 which is poorly matched to p2: For transitions from x0AR2 to xAR1 we simplytake x1 ¼ x0

1:Consider first the standard reversible jump method when the typical value x0

2 ¼ x is proposed.(A more thorough treatment would integrate over the Nð0; x2Þ distribution of x0

2 but this simplecase provides a simple, and compelling, illustration.) The factor in p2ðx0Þ associated with x0

1 isp1ðx0

1Þ and the remaining factor is a Nð0; s2Þ density evaluated at x02 ¼ x: By definition, q21ðx0; xÞ ¼

1: In the denominator of (1) p1ðxÞ ¼ p1ðx01Þ and q12ðx0;xÞ is a Nð0; x2Þ density evaluated at x:

Writing fvðwÞ to denote a Nð0; vÞ density evaluated at w; the acceptance probability of x0 is

a12ðx; x0Þ ¼min 1;p2ðx0Þq21ðx0;xÞp1ðxÞq12ðx;x0Þ

� �¼

f1ðx1Þfs2ðx02Þ

f1ðx1Þfx2ðx02Þ

¼fs2ðxÞfx2ðxÞ

¼xsexp �

1

2

x2

s2� 1

� �� ð6Þ

and for x2bs2 this probability can be very small.Now consider our method with p�2 defined as a bivariate density in which X1 and X2 are

independent, X1BNð0; 1Þ and X2BNð0;f2Þ with s2of2ox2: After an initial move to x0 wherex01 ¼ x1 and x0

2 takes the typical value x; the k MCMC steps will lead to a new value x�1 of x1 and avalue x�2 sampled from p�2 : Using the typical value f for x�2 ; we obtain

a12ðx; x�Þ ¼min 1;p2ðx�Þp�2 ðx

0Þq21ðx0; xÞp1ðxÞp�2 ðx

�Þq12ðx; x0Þ

� �¼

f1ðx�1 Þfs2ðx�2 Þf1ðx1Þff2ðx0

2Þ

f1ðx1Þf1ðx�1 Þff2ðx�2 Þfx2ðx02Þ

¼fs2ðfÞff2ðxÞ

ff2ðfÞfx2ðxÞ¼

xsexp �

1

2

x2

f2þf2

s2� 2

� �� : ð7Þ

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This acceptance probability (7) is greater than that given by (6) for all s2of2ox2; and ismaximised over f2 by f2 ¼ xs which we shall assume to be used. In this case

a12ðx; x�Þ ¼xsexp �

xs� 1

� �� : ð8Þ

Since only the square root of the ratio x2=s2 now appears in the exponential term, this acceptanceprobability decreases much more slowly than (6) as x2=s2 increases. A few numerical values of (6)and (8) are tabulated below:

x2=s2 a12ðx;x0Þ a12ðx; x�Þ

10 0.035 0.36425 0.00003 0.09250 1:6� 10�10 0.016100 3:2� 10�21 0.0012

Further analysis of this example shows that there can be additional gains from introducing asuccession of target distributions intermediate between q12 and p2: with r � 1 suitably chosenintermediate distributions, the rth root of x2=s2 appears in the exponential in a12ðx; x0Þ: The clearconclusion is that our method can lead to a practically useful acceptance rate in situations wherethe original method is hopelessly slow.The price of the improved acceptance rate for jumps between dimensions is the computational

cost of the k intermediate MCMC steps. In this particular example, an independence sampler canbe used for the intermediate steps and k ¼ 1 suffices. In other situations, more steps may benecessary to reach states typical of p�n : Then, the choice of k will be a compromise, large enoughfor the intermediate sampler to move towards a mode but small enough to remain inexpensivecomputationally.

5. An example in object recognition

We consider an example which first motivated our search for an improved proposal mechanism.The data Y displayed in Fig. 1 were obtained by confocal microscopy. Our aim is to identify thenumber of the cells, N; in the scene and approximate their shapes by a set of ellipses X : Using aBayesian approach, inference is based on the posterior distribution of N and X given the dataY ¼ y: In line with our previous notation, we denote this posterior distribution pðxÞ and let pnðxÞ

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Fig. 1. The image data.


denote the sub-density of the set of parameters for scenes with n cells—now with respect toLebesgue measure on R6n since each cell has 6 parameters. Thus,

pnðxÞpLðyjn; xÞpnðxÞ; ð9Þ

where Lðyjn;xÞ is the likelihood of the observed data given N and X ; and pn is the densityassigned to this scene in the prior model. Following the marked pointed process approach ofBaddeley and van Lieshout (1993), we take the prior model to be a hard core interaction pointprocess which prevents cells overlapping. The data Y are assumed to be Gaussian with differentmeans and variances assigned to observations from different cells and from the backgroundregion; for more details see, for example, Rue and Hurn (1999) or Al-Awadhi et al. (2004).In MCMC sampling from p; several types of proposal are defined, including fixed dimensional

moves which alter the size, shape or location of a single cell, as well as reversible jump moveswhich alter the dimension of x by adding or removing a cell. Other very useful proposals are tosplit one cell into two or, conversely, to merge two nearby cells into one. Such split and mergetransitions are essential when there is uncertainty whether a feature in the data is one large cell ortwo smaller ones because moving between these two configurations by birth and death moveswould require the chain to pass through intermediate states of very low probability. It is difficultto create rules for splitting or merging cells expressed in terms of the various shape, orientationand location parameters which manage to be both efficient and inexpensive to compute. Crudeproposals are generally available, but these tend to suffer high rejection rates since the currentconfiguration has had ample opportunity to match the data well while an initial proposal ofmerged or split cells has not. Incorporating a fit to the data into the proposal itself is hard becauseof the computational cost of evaluating the posterior over a range of values for the proposal andof evaluating the reverse step to compute the acceptance probability. Rue and Hurn (1999)demonstrate one approach for such ‘‘data-matching’’ suitable for work with deformablepolygonal templates, but the restriction to ellipses prevents us from using their approach here.Fig. 2 displays some simple split and merge proposals for elliptical templates. These proposals

are easily generated and use the common RJMCMC idea of averaging pairs of parameters whenreducing dimension, and perturbing a parameter by a random amount in opposite directions toincrease dimension. It is only sensible to attempt to merge cells which are close together and acondition formalising this is included in the proposal (and thus in the acceptance ratio).

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lA lC lB

lA

lA

lA

lClC

lC

lB

lB

lB

Fig. 2. Some possible proposed split and merge moves.


To illustrate how poorly the Markov chain can mix, we note results from several runs of 20,000steps of the sampler incorporating all move types: these runs produced between 1 and 5 acceptedsplits and no merges.In this problem, we know that the high failure rate of proposals is due to their relative lack of fit

to the rather strong data; our suggestion is to improve the data matching of a proposal beforemaking the accept/reject decision. We apply tempering to the contribution to p from thelikelihood term to define the modified distribution

p�n ðxÞpLgðyjn;xÞpnðxÞ: ð10Þ

Here gAð0; 1Þ and a good choice of g should give sufficient weight to the data to improve the data-fit of the final proposal x� while limiting the effect of the low probability under p of the initialproposal x0 on the final acceptance probability (4). In modifying x0 we use the usual fixeddimensional proposal types. Note too that only the cells affected by the RJ proposal are changed,all others are held fixed.Selecting the tempering parameter g and the number of steps k is a tuning problem; we have

repeated experiments running 20,000 iterations of reversible jump Markov chains using variouscombinations of the tempering parameter g and the number of steps, k: Results, averaged over 3replicates, are listed in Table 1 and show an increase in the number of successful splits and mergesacross parameter combinations. In particular, there have been successful merge moves, somethingwhich was not achieved before. There is no significant evidence for improvement when k isincreased from 100 to 300 suggesting that adequate data-matching occurs quite quickly and a lowvalue of k ¼ 50 or k ¼ 100 is sufficient. Obviously, this is important because the cost of themodified proposals increases with k (although at less than a factor of k since the variabledimension part of the move is more costly than the fixed dimension part). Note also that theoverall cost of the sampler increases by an even smaller factor since only the merge and splitmoves become more expensive.There is also welcome robustness to the choice of the tempering parameter g: values between 0.1

and 0.001 seem effective. The success at very small values of g is perhaps surprising but can beattributed to the strong signal in this problem which leads to a highly modal target distribution p:Some pairs of image configurations during the RJMCMC runs are shown in Fig. 3 to illustrate thescale of the changes; in each pair, the arrow indicates the area about to be affected. The modifiedmethod has succeeded in improving a sampler which was really not mixing at all. Although theacceptance rates remain painfully low, we now nave greater confidence that the sampler is capable

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Table 1

The average number of acceptances for different values of g and k

g k ¼ 50 k ¼ 100 k ¼ 300

Merges Splits Merges Splits Merges Splits

0.1 6 14 5 9 5 7

0.01 3 3 10 5 10 4

0.001 2 3 8 6 10 4

0.0001 1 2 4 5 4 7


of changing dimension through these essential routes. For this extremely hard sampling problem,this is a significant progress.In this example, p and p� have the same support. In consequence, any proposal x0 with zero

probability under p (as a result of cell overlaps, for example) can be rejected straight away as weknow Pðx�;x0Þ will be equal to zero in the hypothetical return route from a legal x�: It maysometimes be beneficial to extend the support of p�; thereby allowing initial jumps to an illegal x0

to revert to legitimacy before reaching the final proposal x�: In order to encourage x0 to be in thestate space of p; relatively little probability should be assigned to the extension of the state space.Similar ideas have been explored in a different context by Hurn et al. (1999).In conclusion, the approach suggested in this article is one way in which low acceptance

rates for RJMCMC samplers could be improved. It is particularly suitable for problemswhere it is hard to design very efficient proposals directly, but where the modality of thetarget distribution gives clues as to why many proposals are rejected. As another application,consider mixture modelling where observations are allocated to an unknown number ofcomponents, proposing to split a component may only be acceptable if the observations arereasonably well allocated to the two new components. A crude split might allocate theobservations at random to the two new components; our modification would allow reallocationproposals before accepting or rejecting this split. Although some extra running costs are incurredin the modified proposals, there is little additional programming outlay, as the existing code maybe used.

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Fig. 3. Several pairs of output configurations using the modified proposal technique (the areas affected by successful

split and merge moves are pointed out using arrows, the left panel is before and the right panel is after the move).


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Improving the acceptance rate of reversible jump MCMC proposals