Optimal scaling of the random walk Metropolis on elliptically symmetric unimodal targets

Transcription

1 Optimal scaling of the random walk Metropolis on ellipticall smmetric unimodal targets Chris Sherlock 1 and Gareth Roberts 2 1. Department of Mathematics and Statistics, Lancaster Universit, Lancaster, LA1 4YF, UK. 2. Department of Statistics, Universit of Warwick, Coventr, CV4 7AL, UK. * Correspondence should be addressed to Chris Sherlock. ( c.sherlock@lancaster.ac.uk). Summar Scaling of proposals for Metropolis algorithms is an important practical problem in MCMC implementation. Criteria for scaling based on empirical acceptance rates of algorithms have been found to work consistentl well across a broad range of problems. Essentiall, proposal jump sizes are increased when acceptance rates are high and decreased when rates are low. In recent ears, considerable theoretical support has been given for rules of this tpe which work on the basis that acceptance rates around should be preferred. This has been based on asmptotic results which approimate high dimensional algorithm trajectories b diffusions. In this paper, we develop a novel approach to understanding which avoids the need for diffusion limits. We derive eplicit formulae for algorithm efficienc and acceptance rates as functions of the scaling parameter. We appl these to the famil of ellipticall smmetric target densities, where further illuminating eplicit results are possible. Under suitable conditions, we verif the rule for a new class of 1

2 target densities. Moreover, we can characterise cases where fails to hold, either because the target densit is too diffuse in a sense we make precise, or because the eccentricit of the target densit is too severe, again in a sense we make precise. We provide numerical verifications of our results. Kewords: Random walk Metropolis; optimal scaling; optimal acceptance rate. 2

3 1 Introduction The Metropolis-Hastings updating scheme provides a ver general class of algorithms for obtaining a dependent sample from a target distribution, π( ). Given the current value X, a new value X is proposed from a pre-specified Lebesgue densit q ( ) and is then accepted with probabilit α(, ) = min (1,(π( ) q ( ))/(π() q ( ))). If the proposed value is accepted it becomes the net current value (X X ), otherwise the current value is left unchanged (X X). Consider the d-dimensional random-walk Metropolis (RWM) Metropolis et al. [7]: q ( ) = 1 λ d r ( λ ) = 1λ ( ) d r λ, (1) where := is the proposed jump, and r() = r( ) for all. In this case the acceptance probabilit simplifies to ( ) α(, ) = min 1, π( ) π(). (2) Now consider the behaviour of the RWM as a function of the scale of proposed jumps, λ, and some measure of the scale of variabilit of the target distribution, η. If λ << η then, although proposed jumps are often accepted, the chain moves slowl and eploration of the target distribution is relativel inefficient. If λ >> η then man proposed jumps are not accepted, the chain rarel moves and eploration is again inefficient. This suggests that given a particular target and form for the jump proposal distribution, there ma eist a finite scale parameter for the proposal with which the algorithm will eplore the target as efficientl as is possible. We are concerned with the definition and eistence of an optimal scaling, its asmptotic properties, and the process of finding it. We start with a brief review of current literature on the topic. 1.1 Eisting results for optimal scaling of the RWM Eisting literature on this problem has concentrated on obtaining a limiting diffusion process from a sequence of Metropolis algorithms with increasing dimension. The speed of this limiting diffusion is then maimised with respect to a transformation of the scale parameter to find the optimall-scaled 3

4 algorithm. Roberts et al. [9] first follow this program for densities of the form d π () = f( i ) (3) i=1 using Gaussian jump proposals, Y (d) N(0,σd 2I d). Here and throughout this article I d denotes the d-dimensional identit matri. For high dimensional targets which satisf certain moment conditions it is shown that the optimal value of the scale parameter satisfies d 1/2ˆλd = l, for some fied l which is dependent on the roughness of the target. Particularl appealing however, from a practical perspective, is the following distribution-free interpretation of the optimal scaling for the class of distributions given b (3). It is the scaling which leads to the proportion of proposed moves being accepted. Empiricall this rule has been observed to be approimatel right much more generall. Etensions and generalisations of this result can be found in Roberts and Rosenthal [10], which also provides an accessible review of the area, and Bedard [2], Breer and Roberts [3], Roberts [8]. The focus of much of this work is in tring to characterise when the rule holds and to eplain how and wh it breaks down in other situations. One major disadvantage of the diffusion limit work is its reliance on asmptotics in the dimensionalit of the problem. Although it is often empiricall observed that the limiting behaviour can be seen in rather small dimensional problems, see for eample Gelman et al. [5], it is difficult to quantif this in an general wa. In this paper we adopt a finite dimensional approach, deriving and working with eplicit solutions for algorithm efficienc and overall acceptance rates. 1.2 Efficienc and epected acceptance rate In order to consider the problem of optimising the algorithm, an optimisation criterion needs to be chosen. Unfortunatel this is far from unique. In practical MCMC, interest ma lie in the estimation of a collection of epected functionals. For an one of these functionals, f sa, a plausible criterion 4

5 to minimise is the stationar integrated autocorrelation time for f given b τ f = Cor(f(X 0 ),f(x i )). i=1 Under appropriate conditions, the MCMC central limit theorem for {f(x i )} gives a Monte Carlo variance proportional to τ f. This approach has two major disadvantages. Firstl, estimation of τ f is notoriousl difficult, and secondl this optimisation criterion gives a different solution for the optimal chain for different functionals f. In the diffusion limit, the problem of non-uniqueness of the optimal chain is avoided since in all cases τ f is proportional to the inverse of the diffusion speed. This suggests that plausible criteria might be based on optimising properties of single increments of the chain. The most general target distributions that we shall eamine here possess elliptical smmetr. If a d-dimensional target distribution has elliptical contours then there is a simple invertible linear transformation T : R d R d which produces a sphericall smmetric target. To fi it (up to an arbitrar rotation) we define T to be the transformation that produces a sphericall smmetric target with unit scale parameter. Here the eact meaning of unit scale parameter ma be decided arbitraril or b convention. The scale parameter β i along the i th principal ais of the ellipse is the i th eigenvalue of T 1. Let X and X be consecutive elements of a stationar chain eploring a d-dimensional target distribution. A natural efficienc measure for elliptical targets is Mahalanobis distance e.g. Krzanowski [6]: [ Sd 2 := E X X [ ] d 2 1 := E β β 2 i=1 i ( X i X i ) 2 ] = E [ d 1 β 2 i=1 i Y 2 i ], (4) where X i and X i are the components of X and X along the i th principal ais and Y i are components of the realised jump Y = X X. We refer to this as the epected square jump distance, or ESJD. We will relate ESJD to epected acceptance rate (EAR) which we define as α d := E[α(X,X )], where the epectation is with respect to the joint law for the current value X and the proposed value X. Note that we are not interested in the value of the ESJD itself but onl in the scaling and EAR at which the maimum ESJD is attained. 5

6 1.3 Outline of this paper The bod of this paper investigates the RWM algorithm on sphericall and then ellipticall smmetric unimodal targets. Section 2 considers finite dimensional algorithms on sphericall smmetric unimodal targets and derives eplicit formulae for ESJD and EAR in terms of the scale parameter associated with the proposed jumps (Theorem 1). Several eample algorithms are then introduced and the forms of α d (λ) and Sd 2 (λ) are derived for specific values of d either analticall or b numerical integration. Numerical results for the relationship between the optimal acceptance rate and dimension are then described; in most of these eamples the limiting optimal acceptance rate appears to be less than The eplicit formulae in Theorem 1 involve the target s marginal one-dimensional distribution function. Theorem 2 of Section 3 provides a limiting form for the marginal one-dimensional distribution function of a sphericall smmetric random variable as d and Theorem 3 combines this with a result from measure theor to provide limiting forms for EAR and ESJD as d. A natural net step would be to use the limiting ESJD to estimate a limiting optimal scale parameter rather than directl eamining the limit of the optimal scale parameters of the finite dimensional ESJDs. It is shown that this process is sometimes invalid when the target contains a miture of scales which produce local maima in ESJD and whose ratio increases without bound. Eact criteria are provided in Lemma 2 and are related to the numerical eamples. Man standard sequences of distributions satisf the condition that as d the probabilit mass becomes concentrated in a spherical shell which itself becomes infinitesimall thin relative to its radius. Thus the random walk on a rescaling of the target is, in the limit, effectivel confined to the surface of this shell. Theorem 4 considers RWM algorithms on sequences of sphericall smmetric unimodal targets where the sequence of proposal distributions satisf this shell condition. It is shown that if the target sequence also satisfies the shell condition then the limiting optimal EAR is 0.234, however if the target mass does not converge to an infinitesimall thin shell then the limiting optimal EAR (if it eists) is strictl less than Rescalings of both the target and proposal are usuall required in order to stabilise the radius of the shell, whether or not it becomes infinitesimall thin. These influence the form of the optimal scale parameter so that in general it is not proportional to d 1/2 ; Corollar 4 provides an eplicit formula which is consistent with the 6

7 numerical eamples. Section 4 etends the results for finite-dimensional random walks to all ellipticall smmetric targets. Limit results are etended through Theorem 5 to sequences of ellipticall smmetric targets for which the ellipses do not become too eccentric. The article concludes in Section 5 with a discussion. 2 Eact results for finite dimension In this section we derive Theorem 1 which provides eact formulae for ESJD and EAR for a random walk Metropolis algorithm acting on a unimodal sphericall smmetric target. The formulae in Theorem 1 refer to the target s marginal one-dimensional distribution function; these are then converted to use the more intuitive marginal radial distribution function. Several eample targets are introduced and results from eact calculations of ESJD and EAR are presented. We adopt the notation outlined in Section 1.2. All distributions (target and proposal) are assumed to have densities with respect to Lebesgue measure, and we consider the chain to be stationar, so that the marginal densities of both X and X are π( ). We also assume that the space of possible values for element of a d-dimensional chain is R d. We consider onl target densities with a single mode, however the densit need not decrease with strict monotonicit and ma have a series of plateau. We refer to random variables with such densities as unimodal. In this section and the section that follows we further restrict our choice of target to include onl random variables where the densit has spherical contour lines. Such random variables are termed isotropic or sphericall smmetric. ESJD is as defined in (4) where the epectation is taken with respect to the joint law for the current position and the realised jump. For a spherical target β i = β i, the ESJD is proportional to the epected squared Euclidean distance, and both are maimised b the same scaling ˆλ. Since the constant of proportionalit, β, derives from an arbitrar definition of unit scale parameter, we simpl set it to 1 for sphericall smmetric random variables. Denote the one-dimensional marginal distribution function of a general d-dimensional target X (d) along unit vector ŷ as F 1 d (). When X (d) is sphericall smmetric, this is independent of ŷ, and 7

8 we simpl refer to it as the one-dimensional marginal distribution function of X (d). The following is proved in Appendi A.1. Theorem 1 Consider a stationar random walk Metropolis algorithm on a sphericall smmetric unimodal target which has marginal one-dimensional distribution function F 1 d (). Let jumps be proposed from a smmetric densit as defined in (1). In this case the epected acceptance rate and the epected square jump distance are α d (λ) = 2E [F 1 d ( 12 )] λ Y and (5) Sd 2 (λ) = 2λ2 E [ Y 2 F 1 d ( 12 )] λ Y, (6) where the epectation is taken with respect to measure r( ). The marginal distribution function F 1 d ( λ Y /2) is bounded and decreasing in λ. Also lim F 1 d ( ) = 0 and b smmetr, provided F 1 d ( ) is continuous at the origin, lim 0 F 1 d ( ) = 0.5. Appling the bounded convergence theorem to (5) we therefore obtain the following intuitive result: Corollar 1 Let λ be the scaling parameter for an RWM algorithm on a unimodal isotropic target Lebesgue densit. In this situation the EAR at stationarit α d (λ) decreases with increasing λ, with lim λ 0 α d (λ) = 1 and lim λ α d (λ) = 0. In our search for an optimal scaling there is an implicit assumption that such a scaling eists. This was justified intuitivel in Section 1 but the eistence of an optimal scaling has previousl onl been proven for the limiting diffusion process as d e.g. Roberts et al. [9]. Starting from Theorem 1 the following is relativel straightforward to prove e.g. Sherlock [11], and starts to justif a search for an optimal scaling for a finite dimensional random walk algorithm rather than a limit process. Corollar 2 Consider a sphericall smmetric unimodal d-dimensional target Lebesgue densit π(). Let π( ) be eplored via a RWM algorithm with proposal Lebesgue densit 1 λ d r(/λ). If E π [ X 2] < and E r [ Y 2] < then the ESJD of the Markov chain at stationarit attains its maimum at a finite non-zero value (or values) of λ. 8

9 For the remainder of this section we eamine the behaviour of real, finite dimensional eamples of random walk algorithms. As well as being of interest in its own right, this will motivate Section 3 where Theorem 1 will provide the basis from which properties of EAR and ESJD are obtained as dimension d. To render Theorem 1 of more use for practical calculation, we first convert it to involve the more intuitive marginal radial distribution rather than the marginal one-dimensional distribution function. We introduce some further notation; write F d ( ) and f d ( ) for the marginal radial distribution and densit functions of d-dimensional sphericall smmetric target X (d) ; these are the distribution and densit functions of X (d). The densit of Y (when λ = 1) is denoted rd ( ). We start with a form for the one-dimensional marginal distribution function of a sphericall smmetric random variable in terms of its marginal radial distribution function. Derivation of this result from first principles is straightforward e.g. Sherlock [11]. Lemma 1 For an d-dimensional sphericall smmetric random variable with continuous marginal radial distribution function F d (r) with F d (0) = 0, the 1-dimensional marginal distribution function along an ais is ( ( )]) F 1 d ( 1 ) = sign( 1 ) E 2 X (d) [G 2 d X (d) 2 (d 1), (7) where sign() = 1 for 0 and sign() = 1 for < 0, and G d ( ) is the distribution function of U d, with U 1 = 1 and ( 1 U d Beta 2, d 1 ) 2 (d > 1). For the RWM we are concerned onl with targets with Lebesgue densities. In this case both the marginal one-dimensional and radial distribution functions are continuous, and F d (0) = 0 as there can be no point mass at the origin (or anwhere else). Substituting (7) into (5) and (6) gives ( )] λ Y α d (λ) = E Y,X (d) [K d 2 X (d) ( )] and Sd 2 (λ) = λ2 E Y,X (d) [ Y 2 λ Y K d 2 X (d), 9

10 where Y is a random variable with densit r( ) and K d () := 1 G d ( 2 ). The epectations depend on X and Y onl through their modulii, thus allowing epressions for EAR and ESJD in terms of simple double integrals involving the marginal radial densities of X and Y. For unimodal sphericall smmetric targets we therefore obtain: ( ) λ α d (λ) = d d r d ()f 1 d () K d (8) 0 2 λ 2 ( ) λ Sd 2 (λ) = λ2 d d r d ()f d () 2 K d. (9) λ 2 Since X (d) is sphericall smmetric f d ( ) = a d d 1 π(), where a d := 2π d/2 /Γ(d/2) [e.g. 1, Chapter 15]. In the eamples below we also consider onl sphericall smmetric proposals so that r d ( ) = a d d 1 r d (). 2.1 Eplicit and computational results Using (8) and (9) we first eamine the dependenc of EAR and ESJD on λ for an given dimension. We then eamine the behaviour of the optimal scaling and optimal acceptance rate as dimension d increases. Now K 1 (u) = 1 for u < 1 and K 1 (u) = 0 otherwise, and so for one dimensional RWM algorithms the integrals in (8) and (9) ma sometimes be evaluated eactl. For eample with a Gaussian target and Gaussian proposal (8) and (9) give α 1 (λ) = S 2 1 (λ) = 2 π tan 1 ( 2 λ 2λ2 π ) and ( ( ) 2 tan 1 2λ ) λ λ (10) Maimising (10) numericall gives an optimal scaling of ˆλ 2.43 which corresponds to an optimal EAR of With both target and proposal following a double eponential distribution (8) and (9) produce α 1 (λ) = 2 λ + 2 and S 2 1(λ) = 16λ2 (λ + 2) 3. 10

11 S1 2 and α 1 are thus related b the simple analtical epression S1 2 = 8α 1(1 α 1 ) 2, and the ESJD attains a maimum at an EAR of 1/3, for which ˆλ = 4. We now consider two eample targets with d = 10, firstl a simple Gaussian (π d () e ), and secondl a miture of Gaussians: π d () (1 p d ) e p d 1 d de 1 2d 2 2 (d 2), (11) with p d = 1/d 2. Both targets are eplored using sphericall smmetric Gaussian proposals; results are shown in Figure 1. As with the previous two eamples, increasing λ from 0 to decreases the EAR from 1 to 0, as deduced in Corollar 1. Further in all four eamples, as noted in Corollar 2, ESJD achieves a global maimum at finite, strictl positive values of λ. In the first three eamples ESJD as a function of the scaling shows a single maimum, however in the miture eample similar high ESJDs are achieved with two ver different scale parameters (approimatel 0.8 and 7.6). The acceptance rates at these maima are 0.26 and respectivel. The values ˆλ = 0.8, ˆα = 0.26 are almost identical to the optimal values for eploring a standard ten-dimensional Gaussian and so are ideal for eploring the first component of the miture. Optimal eploration of the second component is clearl to be achieved b increasing the scale parameter b a factor of 10, however the second component has a miture weight of 0.01 and so the acceptance rate for such proposals is reduced accordingl. The miture weighting of the second component, 1/d 2, is just sufficient to balance the increase in optimal jump size for that component, with the result that the two peaks in ESJD are of equal heights. We net eamine the behaviour of the optimal scaling and the corresponding EAR as d increases. Calculations are performed for eight different targets: 1. Gaussian densit: π d () e ; 2. eponential densit: π d () e ; 3. target with a Gaussian marginal radial densit: π d () d+1 e ; 4. target with an eponential marginal radial densit: π d () d+1 e ; 11

12 S Gaussian λ Gaussian S Gaussian miture λ Gaussian miture S 10 2 α λ Gaussian α S 10 2 α λ Gaussian miture α Figure 1: Plots for a Gaussian target (left) and the Gaussian miture target of (11) with p d = 1/d 2 (right), both at d = 10 and with a Gaussian jump proposal. Panels from top to bottom are (i) ESJD against scaling, (ii) EAR against scaling, and (iii) ESJD against EAR. 5. lognormal densit altered so as to be unimodal: π d () 1 { e (d 1) } + e 1 2 (log +(d 1))2 1 { >e (d 1) } ; 6. the miture of Gaussians given b (11) with p d = 0.2; 7. the miture of Gaussians given b (11) with p d = 1/d; 8. the miture of Gaussians given b (11) with p d = 1/d 3. Proposals are generated from a Gaussian densit. For each combination of target and proposal simple numerical routines are emploed to find the scaling ˆλ that produces the largest ESJD. Substitution into (8) gives the corresponding optimal EAR ˆα. Figure 2 shows plots of optimal EAR against dimension for Eample Targets 1-4. The first of these is entirel consistent with Figure 4 in Roberts and Rosenthal [10], which shows optimal acceptance rates obtained through repeated runs of the RWM algorithm. The first two are consistent with a conjecture that the optimal EAR approaches as d, however for Eamples Targets 3 and 4, the optimal EAR appears to approach limits of approimatel 0.10 and 0.06 respectivel. For Target 5 with d = 1, 2 or 3, plots of ESJD against scale parameter, EAR against scale parameter, and ESJD against EAR (not shown) are heuristicall similar to those for the standard 12

13 Target 1 Target 2 ˆαd ˆαd d d Target 3 Target 4 ˆαd ˆαd d d Figure 2: Plots of the optimal EAR ˆα against dimension for Eample Targets 1-4 using a Gaussian jump proposal. The horizontal dotted line approimates the apparent asmptoticall optimal acceptance rate of in the first two plots and 0.10 and 0.06 in the third and fourth plots respectivel. Gaussian target in Figure 1. However for d = 1, 2 and 3 the optimal EARs are approimatel 0.111, 0.010, and respectivel, and appear to be approaching a limiting optimal acceptance rate of 0. Figure 3 shows plots of EAR against dimension for the three miture Targets (6-8). Here the asmptoticall optimal EAR appears to be approimatel 0.234/5, 0, and respectivel. The limiting behaviour of each of these eamples is eplained in the net section. Target 6 Target 7 Target 8 ˆαd ˆαd ˆαd d d d Figure 3: Plots of the optimal EAR ˆα against dimension for Eample Targets 6-8 using a Gaussian jump proposal. The horizontal dotted line in each plot represents the apparent asmptoticall optimal acceptance rate of 0.234/5, 0, and respectivel. 13

14 3 Limit results for sphericall smmetric distributions Theorem 1 provides eact analtical forms for the EAR and ESJD of a RWM algorithm on a unimodal sphericall smmetric target in terms of the target s marginal one-dimensional distribution function. In this section we investigate the behaviour of EAR and ESJD in the limit as dimension d. As groundwork for this investigation we must first eamine the possible limiting forms of the marginal one-dimensional distribution function of a sphericall smmetric random variable. We adopt the following notation: convergence in distribution, is denoted b D ; convergence in probabilit is denoted p and convergence in mean square b m.s.. Convergence of the sequence of characteristic functions of a sequence of d-dimensional isotropic random variables (indeed b d) to that of a miture of normals is proved as Theorem 2.21 of Fang et al. [4]. Thus the limiting marginal distribution along an given ais ma be written as X 1 = RZ with Z a standard Gaussian and R the miing distribution. Sherlock [11] proves from first principles the following etension. Theorem 2 Let X (d) be a sequence of d-dimensional sphericall smmetric random variables. If there is a k d such that X (d) /kd D R then the sequence of marginal one-dimensional distributions of X (d) satisfies ( ) kd [ ( 1 )] F 1 d d 1/2 1 Θ( 1 ) := E R Φ, R where Φ( ) is the standard Gaussian distribution function. X (d) possesses a Lebesgue densit and therefore no point mass at the origin; however the rescaled limit R ma possess such a point mass. Provided R has no point mass at 0, the limiting marginal one-dimensional distribution function Θ( 1 ) as defined in Theorem 2 is therefore continuous for all R. This continuit implies that the limit in Theorem 2 is approached uniforml in 1, and for this reason the lack of a radial point mass at 0 is an essential requirement in Theorem 3. The condition of convergence of the rescaled modulus to 1 or to random variable R will turn out to be the ke factor in determining the behaviour of the optimal EAR as d ; we now eamine this limiting convergence behaviour in more detail. For man standard sequences of densit functions there is a k d such that X (d) /kd 14 p 1. This

15 includes Eample Targets 1 and 2 from Section 2.1, and more generall an densit of the form π d () a e c. An intuitive understanding of target sequences satisfing this condition is that, as d the probabilit mass becomes concentrated in a spherical shell which itself becomes infinitesimall thin relative to its radius. The random walk on a rescaling of the target is, in the limit, effectivel confined to the surface of this shell. Eample Targets 3 and 4 have marginal radial distributions which are alwas respectivel a positive unit Gaussian and a unit eponential. The first term in the densit of Eample Target 5 simpl ensures unimodalit and becomes increasingl unimportant as d increases. Trivial algebraic rearrangement of the second component shows that its marginal radial distribution has the same lognormal form whatever the dimension. Eample Targets 6-8 are eamined in detail in Section A limit theorem for EAR and ESJD We now return to the RWM and derive limiting forms for ESJD and EAR on unimodal sphericall smmetric targets as d. Henceforth it is assumed that the radial distribution of the target, rescaled b a suitable quantit k (d), converges weakl to some continuous limiting distribution, that of a random variable R. From Theorem 2, the limiting marginal distribution function Θ( ) is in general a scaled miture of Gaussian distribution functions but in the special case that R is a point mass at 1 the scaled miture of Gaussians clearl reduces to the standard Gaussian cumulative ( ) distribution function Φ( ); F kd 1 d d 1/2 1 Φ ( 1 ). Consider a sequence of jump proposal random variables {Y (d) } with unit scale parameter. If there eist k (d) such that Y (d) /k (d) converges (in a sense to be defined) then simple limit results are possible. Implicit in the derivation of these limit results is a transformation of our target and proposal: X(d) X (d) /k (d) ( A random walk on target densit and Ỹ(d) Y (d) /k (d). We define a transformed scale parameter k (d) µ d := 1 d 1/2 k (d) 2 k (d) λ d. (12) ) d ( ) ( πd k (d) using proposal densit k (d) ) d ( ) rd k (d) and scale parameter 2µ d is therefore equivalent to a random walk on π d () using proposal r d () and 15

16 a scale parameter l = d 1/2 λ d, a quantit which is familiar from the diffusion-based approach to optimal scaling (see Section 1.1). The following theorem characterises the limiting behaviour for EAR and ESJD for fied values, µ, of the transformed scale parameter; it is proved in Appendi A.2. Theorem 3 Let {X (d) } be a sequence of d-dimensional unimodal sphericall smmetric target random variables and let {Y (d) } be the corresponding sequence of jump proposals. If there { } eist such that X (d) /k (d) { (i) If there eist D R where R has no point mass at 0 then for fied µ: k (d) } such that Y (d) /k (d) D Y then [ ( α d (µ) 2E Φ µy R k (d) )]. (13) (ii) If in fact Y (d) /k (d) m.s. Y with E [ Y 2] < then d 4k (d) [ ( 2 S2 d (µ) 2µ2 E Y 2 Φ µy R )]. (14) The remainder of this paper focusses on an important corollar to Theorem 3, which is obtained b setting Y = 1. Corollar 3 Let { X (d)}, { Y (d)} {, k (d) } { and k (d) non-negative random variable with no point mass at 0. (i) If X (d) /k (d) D R and Y (d) (d) m.s. /k 1 } be as defined in Theorem 3 and let R be an [ ( α d (µ) 2E Φ µ )] R d [ ( 2 S2 d (µ) 2µ2 E Φ µ )] R 4k (d) (15). (16) (ii) If X (d) /k (d) p 1 and Y (d) (d) m.s. /k 1 d 4k (d) α d (µ) 2Φ ( µ) (17) 2 S2 d (µ) 2µ2 Φ ( µ). (18) 16

17 With these asmptotic forms for EAR and ESJD we are finall equipped to eamine the issue of optimal scaling in the limit as d. 3.2 The validit and eistence of an asmptoticall optimal scaling It was shown in Section 2 that there is at least one finite optimal scaling for an sphericall smmetric unimodal finite-dimensional target with finite second moment provided the second moment of the proposal is also finite. We now investigate the validit and eistence of a finite asmptoticall optimal (transformed) scaling for sphericall smmetric targets as d. 1. Validit: we shall obtain an asmptoticall optimal scaling b maimising the limiting efficienc function. Ideall we would instead find the limit of the sequence of scalings which maimise each finite dimensional efficienc function. We investigate the circumstances under which these are equivalent. 2. Eistence: it is not alwas the case that the limiting efficienc function possesses a finite maimum; eamples are provided. An even stronger validit assumption is implicit in works such as Roberts et al. [9], Roberts and Rosenthal [10] and Bedard [2]. In each of these papers a limiting process is found and the efficienc of this limiting process is maimised to give an asmptoticall optimal scaling. For a given sequence of targets and proposals with optimal scalings ˆλ d, we seek the limiting transformed optimal scaling ˆµ := lim d ˆµ d, where ˆµ d is given in terms of ˆλ d b (12). The optimal scaling as d would therefore be ˆλ d (2k (d) ˆµ)/(d 1/2 k (d) ). However the value ˆµ will be obtained b maimising 2µ 2 Θ( µ) lim d Sd 2 (µ), where Θ is defined as in Theorem 2. The following result indicates when the scaling which optimises the limit is equivalent to the limit of the optimal scalings. A proof is provided in Appendi A.3. Lemma 2 Let {Sd 2 (µ)} be a sequence of functions defined on [0, ) with continuous pointwise limit S 2 (µ). Define For each d N select an ˆµ d M d. M := {arg ma µ S2 (µ)} and M d := {arg ma µ S2 d (µ)}. 17

18 (i) If M = {ˆµ} and ˆµ d < a < d then ˆµ d ˆµ. (ii) If M = {ˆµ} a sequence µ d ˆµ where each µ d is a local maimum of S2 d ( ). (iii) If M = φ (S 2 has no finite maimum) then ˆµ d i.e. lim d (minm d ) = (iv) If M φ and ˆµ d < a < d then Sd 2(ˆµ d) S 2 (ˆµ) for an ˆµ M. We now highlight certain aspects of Lemma 2 through reference to the miture target (11), and specificall to Target Eamples 6-8 from Section 2.1. Later in this section Lemma 2 is also applied to Target Eamples 1-5. In all that follows consider the sequence of graphs of S 2 d (µ d) against µ d, where µ d is given b (12); consider also the graph of the pointwise limit, S 2 (µ), against µ. For all targets of the form (11) with sufficientl large d (so that the components are sufficientl separated in scale) each graph of S 2 d (µ d) against µ d has two peaks, and a different rescaling k (d) each component of the miture. Choosing to rescale b the higher k (d) applies to would stabilise the right hand peak while the left would approach µ d = 0. However (unless p d 1) this choice of scaling would create a point mass at the origin in the limiting radial distribution function Θ ( ), which is forbidden in the statement of Theorem 3. In order to appl the theorem we must therefore rescale b the lower k (d) which stabilises the left hand peak while the right hand peak drifts off to µ d = and is therefore not present in the pointwise limit. The eistence and consistenc of a limiting optimal scaling then depend on the relative heights of the peaks which in turn depends on the limiting behaviour of p d. First consider an target with p d > 1/d 2 such as Target Eamples 6 and 7. For a given dimension this would produce plots similar to the right hand panels of Figure 1 but with the right hand peak higher than the left hand peak and therefore providing the optimal scaling, ˆµ d. The limit of the scalings which maimise each finite dimensional ESJD is therefore not the same as the scaling which optimises the limiting ESJD. In Lemma 2 Parts (i) and (iv) this situation is prevented through the condition ˆµ d < a <. Suppose in fact that p d p > 0, so that rescaling via the lower k (d) produces a point mass at in the limiting rescaled radial distribution. Consider first the optimal scaling obtained from the limiting form of the ESJD. B Theorem 2, Θ( 1 ) p/2 as 1. Hence the limiting ESJD given in Corollar 3 increases without bound as µ ; this is an eample of Case (iii) in Lemma 18

19 2. The optimal scaling for eploring a real d dimensional target follows the portion of the target with the larger scale parameter, and so in the limit accepted jumps onl arise from this portion of the target. The limit of the optimal EAR is therefore the limiting optimal EAR for the larger component multiplied b a factor p, as suggested b the results for Target Eample 6. If p d p = 0 then onl the left hand peak affects the forms in Corollar 3; the optimal scaling and acceptance rate calculated from this corollar are therefore identical to those for Target Eample 1. However the true optimal scaling follows the right hand peak and so the true limiting optimal acceptance rate is 0, as suggested b the results for Target Eample 7. Alternativel if p d < 1/d 2 then for large enough d the stabilised left hand peak dominates, ˆµ d is bounded and the limit of the maima is the maimum of the limit function. The true limiting optimal acceptance rate is eactl that of the lower component as suggested for Target Eample 8, and this is given correctl b Corollar 3. Provided p d 0 the limiting forms for EAR and ESJD are unaffected b the speed at which this limit is approached. The limiting forms are therefore uninformative about whether or not the second peak is important. This is a fundamental issue with the identifiabilit of a limiting optimal scaling from the limiting ESJD. The above clearl generalises from the specific form (11) so that failure of the boundedness condition on ˆµ d in Lemma 2 intuitivel corresponds to a target sequence which contains a miture of scales which produce local maima in ESJD and whose ratio increases without bound. In general, targets that var on at least two ver different scales are not amenable to the current approach. Indeed the ver eistence of a single optimal scaling is highl debatable. We wish to work with the limit S 2 (µ), accepting its potential limitation. Therefore define ˆµ := min M (or ˆµ := if M = φ), to be the asmptoticall optimal transformed scaling (AOTS), and ˆλ d = (2k (d) ˆµ)/(d 1/2 k (d) ) to be the asmptoticall optimal scaling (AOS). These are equivalent to the limit of the optimal (transformed) scalings provided ˆµ d < a <, d. Similarl the asmptoticall optimal epected acceptance rate (AOA) is the limiting EAR that results from using the AOTS. We now turn to the eistence of an asmptoticall optimal scaling. The practising statistician is free to choose the proposal distribution and we therefore assume throughout the remainder of our 19

20 discussion of sphericall smmetric targets that there is a sequence k (d) such that the transformed proposal satisfies Y (d) /k (d) m.s. 1. First consider the special case where there is a sequence k (d) satisfies X (d) /k (d) p such that the transformed target 1. Differentiating (18) we see that the optimal scaling must satisf: 2Φ( ˆµ p ) = ˆµ p φ( ˆµ p ), which gives ˆµ p : Substituting into (17) provides the EAR at this optimal scaling: ˆα p : 0.234, as suggested b the finite dimensional results for Target Eamples 1 and 2. More generall X (d) /k (d) D R. Following our discussions on validit we now assume that R contains no point mass at 0 or. In general we seek a finite scaling ˆµ which maimises the pointwise limit of S 2 d as given in (16); we then compute the EAR using (15). We illustrate this process with reference to three of our non-standard eamples from Section 2.1. For Target Eamples 3 and 4 the marginal radial distribution is respectivel positive Gaussian (θ d (r) e 1 2 r2 ) and eponential (θ d (r) = e r ). S 2 (µ) is maimised respectivel at ˆµ = 1.67 and ˆµ = 2.86, which correspond to EARs of respectivel and 0.055, consistent with the findings in Section 2.1. In Target Eamples 1-4, the ESJD of each element in the sequence has a single maimum, so b Lemma 2(ii) the limit of these maima is the maimum of the limit function, subject to the scaling k (d). For Target Eample 5 the limiting transformed marginal radial densit is θ (r) e (log r)2 and numerical evaluation shows S 2 (µ) to be bounded above but to increase monotonicall with µ; this corresponds to Part (iii) of Lemma 2. As with Target Eample 6 this provides a situation where S 2 (µ) is increasing as µ. However unlike Target Eample 6, here there is no radial point mass at, S 2 (µ) is bounded and the limiting optimal EAR is Asmptoticall optimal scaling and EAR If Y (d) /k (d) m.s. 1 then for µ to be optimal we require 2Θ( µ) = µθ ( µ). (19) There ma not alwas be a solution for µ (see Section 3.2) but when there is, denote this value as ˆµ. Asmptoticall optimal scaling is therefore achieved b setting µ = ˆµ, so that re-arranging (12) 20

21 we obtain the following corollar to Theorem 3: Corollar 4 Let {X (d) } be a sequence of d-dimensional sphericall smmetric unimodal target distributions and let {Y (d) } be a sequence of jump proposal distributions. If there eist sequences {k (d) } and {k (d) } such that the marginal radial distribution function of X (d) satisfies X (d) /kd where R has no point mass at 0, Y (d) /k (d) the asmptoticall optimal scaling (AOS) satisfies D m.s. 1, and provided there is a solution ˆµ to (19) then ˆλ d = 2ˆµ k (d) d 1/2 k (d). R Target Eamples 1 and 2 satisf X (d) /k (d) p 1, with k (d) = d 1/2 and k (d) we therefore epect an AOTS of ˆµ p For Target Eamples 3 and 4, k (d) = d respectivel; = 1, and the AOTSs are respectivel ˆµ 1.67 and ˆµ Figure 4 shows plot of the true optimal scale parameter against dimension evaluated numericall using (9). The asmptotic approimation given in Corollar 4 appears as a dotted line. Both aes are log transformed and in all cases the finite dimensional optimal scalings are seen to approach their asmptotic values as d increases. For the Gaussian target ver close agreement is attained even in one-dimension since the asmptotic Gaussian approimation to the marginal radial distribution function is eact for all finite d. Target 1 Target 2 log(ˆλ) log(ˆλ) log(ˆλ) log(d) log(d) Target 3 Target log(ˆλ) log(d) log(d) Figure 4: Plots of log ˆλ against log d for Target Eamples 1-4. Optimal values predicted using Corollar 4 appear as dotted lines. 21

22 Let us now eplore the AOA, if it eists, and define α (µ) := lim d α d (µ). From Theorem 2 and Corollar 3(i) α (µ) = 2Θ ( µ) = 2E R [Φ ( )] µ, R where R is the marginal radius of the limit of the sequence of scaled targets. We build upon Theorem 3, which eplicitl requires that the rescaled marginal radial distribution have no point mass at 0. Following the discussion in Section 3.2 the condition that there be an optimal µ implies that the limiting marginal radius has no point mass at infinit. The following is proved in Appendi A.4. Theorem 4 Let X (d) be a sequence of d-dimensional sphericall smmetric unimodal target distributions and let Y (d) be a sequence of jump proposal distributions. Let there eist k (d) such that Y (d) /k (d) m.s. 1 and X (d) /k (d) is a limiting (non-zero) AOA it is and k (d) D R for some R with no point mass at 0. If there α (µ) ˆα p Equalit is achieved if and onl if there eist k (d) such that X (d) /k (d) p 1. For standard proposals, the often used optimal EAR of therefore provides an upper bound on the possible optimal EARs for sphericall smmetric targets, and it is achieved if and onl if the mass of the target converges (after rescaling) to an infinitesimall thin shell. 4 Ellipticall smmetric distributions As discussed in Section 1.2 a unimodal ellipticall smmetric target X ma be defined in terms of an associated orthogonal linear map T such that X := T(X) is sphericall smmetric with unit scale parameter. Since T is linear, the jump proposal in the transformed space is Y := T(Y). The ESJD (4) is preserved under the transformation, since Yi 2/β2 i = Y i 2, and thus we simpl appl Theorem 1 in the transformed space. Write F1 d ( ) for the one-dimensional marginal densit of 22

23 sphericall smmetric X, We wish to optimise the ESJD [ ( Sd 2 (λ) := 2λ2 E Y 2 F1 d 1 )] 2 λ Y. (20) Here epectation is with respect to Lebesgue measure r ( ) of Y. Acceptance in the original space is equivalent to acceptance in the transformed space and the EAR is therefore given b [ ( α d (λ) = 2E F1 d 1 )] 2 λ Y. (21) Corollaries 1 and 2 are now seen to hold for all unimodal ellipticall smmetric targets. For Corollar 3(i) to be applicable in the transformed space we require there to eist k (d) and k (d) such that T (d) ( X (d)) k (d) D R and T (d) ( Y (d)) k (d) m.s. 1. (22) Since X (d) is sphericall smmetric, it is natural to request convergence of X (d) /k (d) eplicitl in the statement of the theorem. Bedard [2] considers a situation analogous to this, but with R = 1. The working statistician is free to chose a jump proposal such that Y (d) /k (d) m.s. 1. If Y (d) is in fact sphericall smmetric this convergence carries through to the transformed space provided the eccentricit of the original target is not too severe. A proof of the following appears in Appendi A.5. Theorem 5 Let {X (d) } be a sequence of ellipticall smmetric unimodal targets and {T (d) } a := T (d) ( X (d)) be sphericall with unit scale parameter. Let { } { } {Y (d) } be a sequence of sphericall smmetric proposals and let there eist and such sequence of linear maps such that X (d) that X (d) k (d) D R and Y (d) k (d) m.s. 1. ( ) Denote b ν i the eigenvalues of T (d), and define k (d) 1/2 = ν 2 (d) k, where ν 2 := d 1 d i=1 ν2 i. If k (d) k (d) ν ma (d) 2 d i=1 ν i(d) 2 0 (23) 23

24 then for fied µ := 1 2 d 1/2 k (d) k (d) λ d (24) the EAR and the ESJD satisf d 4k (d) α d (µ) 2 S2 d (µ) 2µ2 E [ ( 2E Φ µ )] (25) R [ ( Φ µ )], (26) R where Φ() is the cumulative distribution function of a standard Gaussian. If in fact R = 1 then d 4k (d) α d (µ) 2 Φ ( µ) (27) 2 S2 d (µ) 2µ2 Φ ( µ). (28) Naturall (28) leads to the same optimal ˆµ p as for a sphericall smmetric target, so the AOA is still approimatel and the AOS satisfies ˆλ d = 2ˆµ p k (d) d 1/2 k (d) 1 (ν 2 ) 1/2. Similarl (25) and (26) lead again to α(ˆµ) α(ˆµ p ) Discussion We have investigated optimal scaling of the random walk Metropolis algorithm on unimodal ellipticall smmetric targets. An approach through finite dimensions using epected square jumping distance (ESJD) as a measure of efficienc both agrees with and etends the eisting literature, which is based upon diffusion limits. We obtained eact analtical epressions for the epected acceptance rate (EAR) and the ESJD in finite dimension d. For an RWM algorithm on a sphericall smmetric unimodal target it was shown that EAR decreases monotonicall from 1 to 0 as the proposal scaling parameter increases from 0 to. This bijective mapping justifies to an etent the use of acceptance rate as a pro 24

25 for the scale parameter. The theor for finite-dimensional targets was then shown to etend to ellipticall smmetric targets. An asmptotic theor was developed for the behaviour of the RWM algorithm as dimension d. It was shown that the asmptoticall optimal EAR of etends to the class of sphericall smmetric unimodal targets if and onl if the mass of the targets converges to a spherical shell which becomes infinitel thin relative to its radius, with a similar but slightl stronger condition on the proposal. The optimal acceptance rate was then eplored for target sequences for which the shell condition fails. In such cases the asmptoticall optimal EAR (if it eists) was shown to be strictl less than An asmptotic form for the optimal scale parameter showed that the dimension dependent rescalings which stabilise the radial mass for both the proposal and target must be taken into account. Much of the eisting literature e.g. Roberts et al. [9] uses independent and identicall distributed (i.i.d.) target components and i.i.d. proposal components so that these two etra effects cancel and ˆλ d d 1/2. The class for which the limit results are valid was then etended to include all algorithms on ellipticall smmetric targets such that the same shell conditions are satisfied once the target has been transformed to spherical smmetr b an orthogonal linear map. If the original target is eplored b a sphericall smmetric proposal then an additional constraint applies to the eigenvalues of the linear map which forbids the scale parameter of the smallest principle component from being too much smaller than all the other scale parameters and is equivalent to the condition of Bedard [2], derived for targets with independent components which are identical up to a scaling. The optimalit limit results are not alwas valid for targets with at least two ver different (but important) scales of variation; however the suitabilit of the RWM to such targets is itself questionable. Eplicit forms for EAR and ESJD in terms of marginal radial densities were also used to eplore specific combinations of target and proposal in finite dimensions. Numerical and analtical results agreed with our limit theor and with a simulation stud in Roberts and Rosenthal [10]. 25

26 A Appendi A.1 Proof of Theorem 1 The proof of Theorem 1 relies on a partitioning of the space of possible values for (and so for ) given into four disjoint regions: the identit region: R id () := {}, the equalit region: R eq () := { } R d : R id (), π( )q( ) π()q( ) = 1 the acceptance region: R a () := { R d : α(, ) = 1, {R eq () R id ()}}, the rejection region: R r () := { R d : α(, ) < 1}. Here π( ) is the target (Lebesgue) densit. For vectors (, ) in R d R d we emplo the shorthand R ID := {(, ) : R d, R id ()}, with regions R EQ, R A, and R R defined analogousl. The following lemma holds for almost an Metropolis-Hastings algorithm and allows us to simplif the calculations of ESJD and EAR. It is convenient to be able to refer to the proposed jump, Y := X X. Lemma 3 Consider an Metropoplis-Hastings Markov chain with stationar Lebesgue densit π( ). At stationarit let X denote the current element, X the proposed net element, and X the realised net element. Let proposals be drawn from Lebesgue densit q ( ) and assume that d q( ) = 0. (29) R eq() Also denote the probabilit of accepting proposal b α(, ) and the joint laws of (X,X ) and (X,X ) respectivel b A (d,d ) := π()d q ( ) d (30) 26

27 and A(d,d ) := A (d,d )α(, ) 1 { } + π()d d q ( ) (1 α(, )) 1 { =}. (31) Finall let h(, ) be an function satisfing the following two conditions: h(, ) = c h(,), (with c = ±1), (32) h(,) = 0. (33) Subject to the above conditions E [ h(x,x ) ] = (1 + c) d d π() q ( ) h(, ), (, ) R A E [α(x,x )] = 2 d d π()q ( ). (, ) R A Proof: First note that an echangeabilit between the regions R a ( ) and R r ( ) follows directl from their definitions R a () R r ( ). Consecutivel appling this echangeabilit, reversibilit, and the smmetr of h(, ) we find: A(d,d )h(, ) = A(d,d )h(, ) (, ) R A (,) R R = A(d,d)h(, ) = c A(d,d)h(,). (,) R R (,) R R The set R ID corresponds to the second term in A(d,d ); this is in general not null with respect to A(, ); however (33) implies that h(, ) = 0 in R ID. Further, α(, ) = 1 (, ) R EQ (, ), 27

28 and (29) holds. Therefore A(d,d )h(, ) = 0. (, ) R ID R EQ Thus R ID and R EQ contribute nothing to the overall epectation of h(x,x ). Since α(, ) = 1 (, ) R A (, ) the first result then follows. The proof of the second result is similar to that of the first and is omitted. Sherlock [11] shows that for a smmetric proposal (such as the RWM) Lemma 3 ma be etended to deal with cases where the densit contains a series of plateau and hence R EQ is not null. R EQ is then partitioned into a null set and pseudo-acceptance and rejection regions which are eactl as would be found if each plateau in fact had a small downward slope awa from the origin. In the region R A, where acceptance is guaranteed, we have = and = so that for integrals over R A we need not distinguish between proposed and accepted values. Appling Lemma 3 with h(, ) = 2 β = 2 we have α d (λ) = 2 λ d R A d d π() r (/λ) (34) S 2 d (λ) = 2 λ d R A d d 2 π() r (/λ). (35) First consider target densities which decrease with strict monotonicit from the mode. In this case R A corresponds to the region where π( + ) > π() + 2 < 2.ŷ < 1, (36) 2 where ŷ is the unit vector in the direction of. So (, + ) R A R d and ŷ <

29 Thus (34) and (35) become α d (λ) = 2 λ d R d d r (/λ) F 1 d S 2 d (λ) = 2 λ d R d d 2 r (/λ) F 1 d ( 12 ) ( 12 ), and the required results follow directl. Without strict monotonicit R A must simpl be etended to include the pseudo acceptance regions defined in Sherlock [11]. A.2 Proof of Theorem 3 Theorem 3 follows almost directl from the following simple lemma, the proof of which follows from a standard measure theor argument which we omit. Lemma 4 Let U d be a sequence of random variables and let G d ( ) G( ) be a sequence of monotonic functions with 0 G d (u) 1 and G( ) continuous. Then U d p U E [G d (U d )] E [G(U)], and U d m.s. U E [ U 2 d G d(u d ) ] E [ U 2 G(U) ]. Now note that 1 2 λ d Y (d) = µ Y (d) (d) /k k (d) /d 1/2 whence (5) and (6) become ( α d (µ) = 2E [F Y (d) 1 d µ S 2 d (µ) = k (d) k (d) d 1/2 )] and 8µ2 k (d) 2 ( ) Y (d) 2 ( Y (d) E F d 1 d µ k (d) k (d) k (d) d 1/2 ). Again denote the limiting 1-dimensional distribution function corresponding to R b Θ(). In Lemma 4 substitute U d = ( ) Y (d) (d) /k, U = Y, G d (u) = F 1 d µ k(d) and G(u) = Θ( µ u). d 1/2 u Note also that since G(.) and G d (.) are bounded the convergence in the first part of the lemma holds if U d D U. The theorem then follows directl. 29