A Tutorial on Particle Filtering and Smoothing: Fifteen years later

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1 A Tutoral on Partcle Flterng and Smoothng: Ffteen years later Arnaud Doucet The Insttute of Statstcal Mathematcs, Mnam-Azabu, Mnato-ku, Tokyo , Japan Emal: Adam M Johansen Department of Statstcs, Unversty of Warwck, Coventry, CV4 7AL, UK Emal: AMJohansen@warwckacuk Frst Verson 0 Aprl 008 Ths Verson December 008 wth typographcal correctons March 0

2 Abstract Optmal estmaton problems for non-lnear non-gaussan state-space models do not typcally admt analytc solutons Snce ther ntroducton n 993, partcle flterng methods have become a very popular class of algorthms to solve these estmaton problems numercally n an onlne manner, e recursvely as observatons become avalable, and are now routnely used n felds as dverse as computer vson, econometrcs, robotcs and navgaton The objectve of ths tutoral s to provde a complete, up-to-date survey of ths feld as of 008 Basc and advanced partcle methods for flterng as well as smoothng are presented Keywords: Central Lmt Theorem, Flterng, Hdden Markov Models, Markov chan Monte Carlo, Partcle methods, Resamplng, Sequental Monte Carlo, Smoothng, State-Space models

3 Introducton The general state space hdden Markov models, whch are summarsed n secton, provde an extremely flexble framework for modellng tme seres The great descrptve power of these models comes at the expense of ntractablty: t s mpossble to obtan analytc solutons to the nference problems of nterest wth the excepton of a small number of partcularly smple cases The partcle methods descrbed by ths tutoral are a broad and popular class of Monte Carlo algorthms whch have been developed over the past ffteen years to provde approxmate solutons to these ntractable nference problems Prelmnary remarks Snce ther ntroducton n 993 [], partcle flters have become a very popular class of numercal methods for the soluton of optmal estmaton problems n non-lnear non-gaussan scenaros In comparson wth standard approxmaton methods, such as the popular Extended Kalman Flter, the prncpal advantage of partcle methods s that they do not rely on any local lnearsaton technque or any crude functonal approxmaton The prce that must be pad for ths flexblty s computatonal: these methods are computatonally expensve However, thanks to the avalablty of ever-ncreasng computatonal power, these methods are already used n real-tme applcatons appearng n felds as dverse as chemcal engneerng, computer vson, fnancal econometrcs, target trackng and robotcs Moreover, even n scenaros n whch there are no real-tme constrants, these methods can be a powerful alternatve to Markov chan Monte Carlo MCMC algorthms alternatvely, they can be used to desgn very effcent MCMC schemes As a result of the popularty of partcle methods, a few tutorals have already been publshed on the subject [3, 8, 8, 9] The most popular, [3], dates back to 00 and, lke the edted volume [6] from 00, t s now somewhat outdated Ths tutoral dffers from prevously publshed tutorals n two ways Frst, the obvous: t s, as of December 008, the most recent tutoral on the subject and so t has been possble to nclude some very recent materal on advanced partcle methods for flterng and smoothng Second, more mportantly, ths tutoral was not ntended to resemble a cookbook To ths end, all of the algorthms are presented wthn a smple, unfed framework In partcular, we show that essentally all basc and advanced methods for partcle flterng can be renterpreted as some specal nstances of a sngle generc Sequental Monte Carlo SMC algorthm In our opnon, ths framework s not only elegant but allows the development of a better ntutve and theoretcal understandng of partcle methods It also shows that essentally any partcle flter can be mplemented usng a smple computatonal framework such as that provded by [4] Absolute begnners mght beneft from readng [7], whch provdes an elementary ntroducton to the feld, before the present tutoral Organsaton of the tutoral The rest of ths paper s organsed as follows In Secton, we present hdden Markov models and the assocated Bayesan recursons for the flterng and smoothng dstrbutons In Secton 3, we ntroduce a generc SMC algorthm whch provdes weghted samples from any sequence of probablty dstrbutons In Secton 4, we show how all the basc and advanced partcle flterng methods developed n the lterature can be nterpreted as specal nstances of the generc SMC algorthm presented n Secton 3 Secton 5 s devoted to partcle smoothng and we menton some open problems n Secton 6

4 Bayesan Inference n Hdden Markov Models Hdden Markov Models and Inference Ams Consder an valued dscrete-tme Markov process { n } n such that µ x and n n = x n f x n x n where means dstrbuted accordng to, µ x s a probablty densty functon and f x x denotes the probablty densty assocated wth movng from x to x All the denstes are wth respect to a domnatng measure that we wll denote, wth abuse of notaton, dx We are nterested n estmatng { n } n but only have access to the Y valued process {Y n } n We assume that, gven { n } n, the observatons {Y n } n are statstcally ndependent and ther margnal denstes wth respect to a domnatng measure dy n are gven by Y n n = x n g y n x n For the sake of smplcty, we have consdered only the homogeneous case here; that s, the transton and observaton denstes are ndependent of the tme ndex n The extenson to the nhomogeneous case s straghtforward It s assumed throughout that any model parameters are known Models compatble wth - are known as hdden Markov models HMM or general state-space models SSM Ths class ncludes many models of nterest The followng examples provde an llustraton of several smple problems whch can be dealt wth wthn ths framework More complcated scenaros can also be consdered Example - Fnte State-Space HMM In ths case, we have = {,, K} so Pr = k = µ k, Pr n = k n = l = f k l The observatons follow an arbtrary model of the form Ths type of model s extremely general and examples can be found n areas such as genetcs n whch they can descrbe mperfectly observed genetc sequences, sgnal processng, and computer scence n whch they can descrbe, amongst many other thngs, arbtrary fnte-state machnes Example - Lnear Gaussan model Here, = R nx, Y = R ny, N 0, Σ and n = A n + BV n, Y n = C n + DW n d d where V n N 0, I nv, W n N 0, I nw and A, B, C, D are matrces of approprate dmensons Note that N m, Σ denotes a Gaussan dstrbuton of mean m and varance-covarance matrx Σ, whereas N x; m, Σ denotes the Gaussan densty of argument x and smlar statstcs In ths case µ x = N x; 0, Σ, f x x = N x ; Ax, BB T and g y x = N y; Cx, DD T As nference s analytcally tractable for ths model, t has been extremely wdely used for problems such as target trackng and sgnal processng Example 3 - Swtchng Lnear Gaussan model We have = U Z wth U = {,, K} and Z = R nz Here n = U n, Z n where {U n } s a fnte state-space Markov chan such that Pr U = k = µ U k, Pr U n = k U n = l = f U k l and condtonal upon {U n } we have a lnear Gaussan model wth Z U N 0, Σ U and Z n = A Un Z n + B Un V n, Y n = C Un Z n + D Un W n

5 5 0 Smulated Volatlty Sequence Volatlty Observatons Fgure : A smulaton of the stochastc volatlty model descrbed n example 4 d d where V n N 0, I nv, W n N 0, I nw and {A k, B k, C k, D k ; k =,, K} are matrces of approprate dmensons In ths case we have µ x = µ u, z = µ U u N z; 0, Σ u, f x x = f u, z u, z = f U u u N z ; A u z, B u Bu T and g y x = g y u, z = N y; Cu z, D u Du T Ths type of model provdes a generalsaton of that descrbed n example wth only a slght ncrease n complexty σ Example 4 - Stochastc Volatlty model We have = Y = R, N 0, α and n = α n + σv n, Y n = β exp n / W n where V n d d N 0, and W n N 0, In ths case we have µ x = N x; 0, σ α, f x x = N x ; αx, σ and g y x = N y; 0, β exp x Note that ths choce of ntal dstrbuton ensures that the margnal dstrbuton of n s also µ x for all n Ths type of model, and ts generalsatons, have been very wdely used n varous areas of economcs and mathematcal fnance: nferrng and predctng underlyng volatlty from observed prce or rate data s an mportant problem Fgure shows a short secton of data smulated from such a model wth parameter values α = 09, σ = 0 and β = 05 whch wll be used below to llustrate the behavour of some smple algorthms Equatons - defne a Bayesan model n whch defnes the pror dstrbuton of the process of nterest { n } n and defnes the lkelhood functon; that s: p x :n = µ x n f x k x k 3 k= 3

6 and p y :n x :n = where, for any sequence {z n } n, and any j, z :j := z, z +,, z j n g y k x k, 4 In such a Bayesan context, nference about :n gven a realsaton of the observatons Y :n = y :n reles upon the posteror dstrbuton p x :n y :n = p x :n, y :n, 5 p y :n where k= p x :n, y :n = p x :n p y :n x :n, 6 and p y :n = p x :n, y :n dx :n 7 For the fnte state-space HMM model dscussed n Example, the ntegrals correspond to fnte sums and all these dscrete probablty dstrbutons can be computed exactly For the lnear Gaussan model dscussed n Example, t s easy to check that p x :n y :n s a Gaussan dstrbuton whose mean and covarance can be computed usng Kalman technques; see [], for example However, for most non-lnear non-gaussan models, t s not possble to compute these dstrbutons n closed-form and we need to employ numercal methods Partcle methods are a set of flexble and powerful smulaton-based methods whch provde samples approxmately dstrbuted accordng to posteror dstrbutons of the form p x :n y :n and facltate the approxmate calculaton of p y :n Such methods are a subset of the class of methods known as Sequental Monte Carlo SMC methods In ths tutoral, we wll revew varous partcle methods to address the followng problems: Flterng and Margnal lkelhood computaton: Assume that we are nterested n the sequental approxmaton of the dstrbutons {p x :n y :n } n and margnal lkelhoods {p y :n } n That s, we wsh to approxmate p x y and p y at the frst tme nstance, p x : y : and p y : at the second tme nstance and so on We wll refer to ths problem as the optmal flterng problem Ths s slghtly at varance wth the usage n much of the lterature whch reserves the term for the estmaton of the margnal dstrbutons {p x n y :n } n rather than the jont dstrbutons {p x :n y :n } n We wll descrbe basc and advanced partcle flterng methods to address ths problem ncludng auxlary partcle flterng, partcle flterng wth MCMC moves, block samplng strateges and Rao-Blackwellsed partcle flters Smoothng: Consder attemptng to sample from a jont dstrbuton p x :T y :T and approxmatng the assocated margnals {p x n y :T } where n =,, T Partcle flterng technques can be used to solve ths problem but perform poorly when T s large for reasons detaled n ths tutoral We wll descrbe several partcle smoothng methods to address ths problem Essentally, these methods rely on the partcle mplementaton of the forward flterng-backward smoothng formula or of a generalsed verson of the twoflter smoothng formula Flterng and Margnal Lkelhood The frst area of nterest, and that to whch the vast majorty of the lterature on partcle methods has been dedcated from the outset, s the problem of flterng: charactersng the dstrbuton of the state of the hdden Markov model at the present tme, gven the nformaton provded by all of the observatons receved up to the present tme Ths can be thought of as a trackng problem: keepng track of the current locaton of the system gven nosy observatons and, ndeed, ths s an extremely popular area 4

7 of applcaton for these methods The term s sometmes also used to refer to the practce of estmatng the full trajectory of the state sequence up to the present tme gven the observatons receved up to ths tme We recall that, followng -, the posteror dstrbuton p x :n y :n s defned by 5 the pror s defned n 3 and the lkelhood n 4 The unnormalsed posteror dstrbuton p x :n, y :n gven n 5 satsfes p x :n, y :n = p x :n, y :n f x n x n g y n x n 8 Consequently, the posteror p x :n y :n satsfes the followng recurson p x :n y :n = p x :n y :n f x n x n g y n x n, 9 p y n y :n where p y n y :n = p x n y :n f x n x n g y n x n dx n :n 0 In the lterature, the recurson satsfed by the margnal dstrbuton p x n y :n s often presented It s straghtforward to check by ntegratng out x :n n 9 that we have p x n y :n = g y n x n p x n y :n, p y n y :n where p x n y :n = f x n x n p x n y :n dx n Equaton s known as the predcton step and s known as the updatng step However, most partcle flterng methods rely on a numercal approxmaton of recurson 9 and not of - If we can compute {p x :n y :n } and thus {p x n y :n } sequentally, then the quantty p y :n, whch s known as the margnal lkelhood, can also clearly be evaluated recursvely usng where p y k y :k s of the form 0 p y :n = p y n p y k y :k 3 k= 3 Smoothng One problem, whch s closely related to flterng, but computatonally more challengng for reasons whch wll become apparent later, s known as smoothng Whereas flterng corresponds to estmatng the dstrbuton of the current state of an HMM based upon the observatons receved up untl the current tme, smoothng corresponds to estmatng the dstrbuton of the state at a partcular tme gven all of the observatons up to some later tme The trajectory estmates obtaned by such methods, as a result of the addtonal nformaton avalable, tend to be smoother than those obtaned by flterng It s ntutve that f estmates of the state at tme n are not requred nstantly, then better estmaton performance s lkely to be obtaned by takng advantage of a few later observatons Desgnng effcent sequental algorthms for the soluton of ths problem s not qute as straghtforward as t mght seem, but a number of effectve strateges have been developed and are descrbed below More formally: assume that we have access to the data y :T, and wsh to compute the margnal dstrbutons {p x n y :T } where n =,, T or to sample from p x :T y :T In prncple, the margnals {p x n y :T } could be obtaned drectly by consderng the jont dstrbuton p x :T y :T and ntegratng out the varables x :n, x n+:t Extendng ths reasonng n the context of partcle methods, one can smply use the dentty px n y :T = px :T y :T dx :n dx n+:t and take the same approach whch s used n partcle flterng: 5

8 use Monte Carlo algorthms to obtan an approxmate charactersaton of the jont dstrbuton and then use the assocated margnal dstrbuton to approxmate the dstrbutons of nterest Unfortunately, as s detaled below, when n T ths strategy s doomed to falure: the margnal dstrbuton px n y :n occupes a prvleged role wthn the partcle flter framework as t s, n some sense, better charactersed than any of the other margnal dstrbutons For ths reason, t s necessary to develop more sophstcated strateges n order to obtan good smoothng algorthms There has been much progress n ths drecton over the past decade Below, we present two alternatve recursons that wll prove useful when numercal approxmatons are requred The key to the success of these recursons s that they rely upon only the margnal flterng dstrbutons {p x n y :n } 3 Forward-Backward Recursons The followng decomposton of the jont dstrbuton p x :T y :T T p x :T y :T = p x T y :T p x n x n+, y :T n= T = p x T y :T p x n x n+, y :n, 4 shows that, condtonal on y :T, { n } s an nhomogeneous Markov process Eq 4 suggests the followng algorthm to sample from p x :T y :T Frst compute and store the margnal dstrbutons {p x n y :n } for n =,, T Then sample T p x T y :T and for n = T, T,,, sample n p x n n+, y :n where n= p x n x n+, y :n = f x n+ x n p x n y :n p x n+ y :n It also follows, by ntegratng out x :n, x n+:t n Eq 4, that f xn+ x n p x n y :T = p x n y :n p x n+ y :n p x n+ y :T dx n+ 5 So to compute {p x n y :T }, we smply modfy the backward pass and, nstead of samplng from p x n x n+, y :n, we compute p x n y :T usng 5 3 Generalsed Two-Flter Formula The two-flter formula s a well-establshed alternatve to the forward-flterng backward-smoothng technque to compute the margnal dstrbutons {p x n y :T } [4] It reles on the followng dentty p x n y :T = p x n y :n p y n:t x n, p y n:t y :n where the so-called backward nformaton flter s ntalsed at tme n = T by p y T x T = g y T x T and satsfes T T p y n:t x n = f x k x k g y k x k dx n+:t 6 k=n+ k=n = g y n x n f x n+ x n p y n+:t x n+ dx n+ 6

9 The backward nformaton flter s not a probablty densty n argument x n and t s even possble that p yn:t x n dx n = Although ths s not an ssue when p y n:t x n can be computed exactly, t does preclude the drect use of SMC methods to estmate ths ntegral To address ths problem, a generalsed verson of the two-flter formula was proposed n [5] It reles on the ntroducton of a set of artfcal probablty dstrbutons { p n x n } and the jont dstrbutons p n x n:t y n:t p n x n T k=n+ f x k x k T g y k x k, 7 whch are constructed such that ther margnal dstrbutons, p n x n y n:t p n x n p y n:t x n, are smply ntegrable versons of the backward nformaton flter It s easy to establsh the generalsed two-flter formula px y :T µ x px y :T, px n y :T px n y :n px n y n:t 8 p x p n x n whch s vald whenever the support of p n x n ncludes the support of the pror p n x n ; that s k=n n p n x n = µ x f x k x k dx :n > 0 p n x n > 0 k= The generalsed two-flter smoother for {px n y n:t } proceeds as follows Usng the standard forward recurson, we can compute and store the margnal dstrbutons {p x n y :n } Usng a backward recurson, we compute and store { px n y n:t } Then for any n =,, T we can combne p x n y :n and px n y n:t to obtan px n y :T In [4], ths dentty s dscussed n the partcular case where p n x n = p n x n However, when computng { px n y n:t } usng SMC, t s necessary to be able to compute p n x n exactly hence ths rules out the choce p n x n = p n x n for most non-lnear non-gaussan models In practce, we should select a heavy-taled approxmaton of p n x n for p n x n n such settngs It s also possble to use the generalsed-two flter formula to sample from px :T y :T ; see [5] for detals 4 Summary Bayesan nference n non-lnear non-gaussan dynamc models reles on the sequence of posteror dstrbutons {p x :n y :n } and ts margnals Except n smple problems such as Examples and, t s not possble to compute these dstrbutons n closed-form In some scenaros, t mght be possble to obtan reasonable performance by employng functonal approxmatons of these dstrbutons Here, we wll dscuss only Monte Carlo approxmatons of these dstrbutons; that s numercal schemes n whch the dstrbutons of nterest are approxmated by a large collecton of N random samples termed partcles The man advantage of such methods s that under weak assumptons they provde asymptotcally e as N consstent estmates of the target dstrbutons of nterest It s also noteworthy that these technques can be appled to problems of moderately-hgh dmenson n whch tradtonal numercal ntegraton mght be expected to perform poorly 3 Sequental Monte Carlo Methods Over the past ffteen years, partcle methods for flterng and smoothng have been the most common examples of SMC algorthms Indeed, t has become tradtonal to present partcle flterng and SMC as beng the same thng n much of the lterature Here, we wsh to emphasse that SMC actually encompasses a broader range of algorthms and by dong so we are able to show that many more advanced technques for 7

10 approxmate flterng and smoothng can be descrbed usng precsely the same framework and termnology as the basc algorthm SMC methods are a general class of Monte Carlo methods that sample sequentally from a sequence of target probablty denstes {π n x :n } of ncreasng dmenson where each dstrbuton π n x :n s defned on the product space n Wrtng π n x :n = γ n x :n Z n 9 we requre only that γ n : n R + s known pontwse; the normalsng constant Z n = γ n x :n dx :n 0 mght be unknown SMC provde an approxmaton of π x and an estmate of Z at tme then an approxmaton of π x : and an estmate of Z at tme and so on For example, n the context of flterng, we could have γ n x :n = p x :n, y :n, Z n = p y :n so π n x :n = p x :n y :n However, we emphasse that ths s just one partcular choce of target dstrbutons Not only can SMC methods be used outsde the flterng context but, more mportantly for ths tutoral, some advanced partcle flterng and smoothng methods dscussed below do not rely on ths sequence of target dstrbutons Consequently, we beleve that understandng the man prncples behnd generc SMC methods s essental to the development of a proper understandng of partcle flterng and smoothng methods We start ths secton wth a very basc revew of Monte Carlo methods and Importance Samplng IS We then present the Sequental Importance Samplng SIS method, pont out the lmtatons of ths method and show how resamplng technques can be used to partally mtgate them Havng ntroduced the basc partcle flter as an SMC method, we show how varous advanced technques whch have been developed over the past ffteen years can themselves be nterpreted wthn the same formalsm as SMC algorthms assocated wth sequences of dstrbutons whch may not concde wth the flterng dstrbutons These alternatve sequences of target dstrbutons are ether constructed such that they admt the dstrbutons {p x :n y :n } as margnal dstrbutons, or an mportance samplng correcton s necessary to ensure the consstency of estmates 3 Bascs of Monte Carlo Methods Intally, consder approxmatng a generc probablty densty π n x :n for some fxed n If we sample N ndependent random varables, :n π n x :n for =,, N, then the Monte Carlo method approxmates π n x :n by the emprcal measure π n x :n = N δ :n x :n, = where δ x0 x denotes the Drac delta mass located at x 0 Based on ths approxmaton, t s possble to approxmate any margnal, say π n x k, easly usng π n x k = N δ k x k, and the expectaton of any test functon ϕ n : n R gven by I n ϕ n := ϕ n x :n π n x :n dx :n, We persst wth the abusve use of densty notaton n the nterests of smplcty and accessblty; the alternatons requred to obtan a rgorous formulaton are obvous = 8

11 s estmated by In MC ϕ n := ϕ n x :n π n x :n dx :n = N ϕ n :n It s easy to check that ths estmate s unbased and that ts varance s gven by V [ In MC ϕ n ] = ϕ n x :n π n x :n dx :n In ϕ n N The man advantage of Monte Carlo methods over standard approxmaton technques s that the varance of the approxmaton error decreases at a rate of O/N regardless of the dmenson of the space n However, there are at least two man problems wth ths basc Monte Carlo approach: Problem : If π n x :n s a complex hgh-dmensonal probablty dstrbuton, then we cannot sample from t Problem : Even f we knew how to sample exactly from π n x :n, the computatonal complexty of such a samplng scheme s typcally at least lnear n the number of varables n So an algorthm samplng exactly from π n x :n, sequentally for each value of n, would have a computatonal complexty ncreasng at least lnearly wth n = 3 Importance Samplng We are gong to address Problem usng the IS method Ths s a fundamental Monte Carlo method and the bass of all the algorthms developed later on IS reles on the ntroducton of an mportance densty q n x :n such that π n x :n > 0 q n x :n > 0 In ths case, we have from 9-0 the followng IS denttes where w n x :n s the unnormalsed weght functon π n x :n = w n x :n q n x :n, Z n Z n = w n x :n q n x :n dx :n w n x :n = γ n x :n q n x :n In partcular, we can select an mportance densty q n x :n from whch t s easy to draw samples; eg a multvarate Gaussan Assume we draw N ndependent samples :n q n x :n then by nsertng the Monte Carlo approxmaton of q n x :n that s the emprcal measure of the samples :n nto we obtan π n x :n = Wnδ :n x :n, 3 where = Ẑ n = N W n = w n :n Some authors use the terms proposal densty or nstrumental densty nterchangeably = 4 w n :n 5 N j= w n j :n 9

12 Compared to standard Monte Carlo, IS provdes an unbased estmate of the normalsng constant wth relatve varance ] V IS [Ẑn = π n x :n N q n x :n dx :n 6 If we are nterested n computng I n ϕ n, we can also use the estmate In IS ϕ n = Z n ϕ n x :n π n x :n dx :n = Wnϕ n :n Unlke In MC ϕ n, ths estmate s based for fnte N However, t s consstent and t s easy to check that ts asymptotc bas s gven by lm N In IS ϕ n I n ϕ n π = n x :n N q n x :n ϕ n x :n I n ϕ n dx :n When the normalsng constant s known analytcally, we can calculate an unbased mportance samplng estmate however, ths generally has hgher varance and ths s not typcally the case n the stuatons n whch we are nterested Furthermore, In IS ϕ satsfes a Central Lmt Theorem CLT wth asymptotc varance N = π n x :n q n x :n ϕ n x :n I n ϕ n dx :n 7 The bas beng O/N and the varance O/N, the mean-squared error gven by the squared bas plus the varance s asymptotcally domnated by the varance term For a gven test functon, ϕ n x :n, t s easy to establsh the mportance dstrbuton mnmsng the asymptotc varance of In IS ϕ n However, such a result s of mnmal nterest n a flterng context as ths dstrbuton depends on ϕ n x :n and we are typcally nterested n the expectatons of several test functons Moreover, even f we were nterested n a sngle test functon, say ϕ n x :n = x n, then selectng the optmal mportance dstrbuton at tme n would have detrmental effects when we wll try to obtan a sequental verson of the algorthms the optmal dstrbuton for estmatng ϕ n x :n wll almost certanly not be even smlar to the margnal dstrbuton of x :n n the optmal dstrbuton for estmatng ϕ n x :n and ths wll prove to be problematc A more approprate approach n ths context s to attempt to select the q n x :n whch mnmses the varance of the mportance weghts or, equvalently, the varance of Ẑn Clearly, ths varance s mnmsed for q n x :n = π n x :n We cannot select q n x :n = π n x :n as ths s the reason we used IS n the frst place However, ths smple result ndcates that we should am at selectng an IS dstrbuton whch s close as possble to the target Also, although t s possble to construct samplers for whch the varance s fnte wthout satsfyng ths condton, t s advsable to select q n x :n so that w n x :n < C n < 33 Sequental Importance Samplng We are now gong to present an algorthm that admts a fxed computatonal complexty at each tme step n mportant scenaros and thus addresses Problem Ths soluton nvolves selectng an mportance dstrbuton whch has the followng structure q n x :n = q n x :n q n x n x :n n = q x q k x k x :k 8 k= 0

13 Practcally, ths means that to obtan partcles :n q n x :n at tme n, we sample q x at tme then k q k xk :k at tme k for k =,, n The assocated unnormalsed weghts can be computed recursvely usng the decomposton w n x :n = γ n x :n q n x :n = γ n x :n q n x :n γ n x :n γ n x :n q n x n x :n 9 whch can be wrtten n the form w n x :n = w n x :n α n x :n n = w x α k x :k where the ncremental mportance weght functon α n x :n s gven by α n x :n = k= γ n x :n γ n x :n q n x n x :n 30 The SIS algorthm proceeds as follows, wth each step carred out for =,, N: Sequental Importance Samplng At tme n = Sample q x Compute the weghts w and W w At tme n Sample n q n x n :n Compute the weghts w n :n = wn :n αn :n, Wn w n :n At any tme, n, we obtan the estmates π n x :n Eq 3 and Ẑn Eq 4 of π n x :n and Z n, respectvely It s straghtforward to check that a consstent estmate of Z n /Z n s also provded by the same set of samples: Ẑ n = W Z n α n :n n = Ths estmator s motvated by the fact that α n x :n π n x :n q n x n x :n dx :n = γn x :n π n x :n q n x n x :n dx :n = γ n x :n q n x n x :n Z n Z n

14 In ths sequental framework, t would seem that the only freedom the user has at tme n s the choce of q n x n x :n 3 A sensble strategy conssts of selectng t so as to mnmse the varance of w n x :n It s straghtforward to check that ths s acheved by selectng q opt n x n x :n = π n x n x :n as n ths case the varance of w n x :n condtonal upon x :n s zero and the assocated ncremental weght s gven by αn opt x :n = γ n x :n γ n x :n = γn x :n dx n γ n x :n Note that t s not always possble to sample from π n x n x :n Nor s t always possble to compute αn opt x :n In these cases, one should employ an approxmaton of qn opt x n x :n for q n x n x :n In those scenaros n whch the tme requred to sample from q n x n x :n and to compute α n x :n s ndependent of n and ths s, ndeed, the case f q n s chosen sensbly and one s concerned wth a problem such as flterng, t appears that we have provded a soluton for Problem However, t s mportant to be aware that the methodology presented here suffers from severe drawbacks Even for standard IS, the varance of the resultng estmates ncreases exponentally wth n as s llustrated below; see also [8] As SIS s nothng but a specal verson of IS n whch we restrct ourselves to an mportance dstrbuton of the form 8 t suffers from the same problem We demonstrate ths usng a very smple toy example Example Consder the case where = R and We select an mportance dstrbuton π n x :n = γ n x :n = q n x :n = n π n x k = k= n k= Z n = π n/ n N x k ; 0,, 3 k= exp x k, n q k x k = k= In ths case, we have V IS [Ẑn ] < only for σ > and V IS [Ẑn ] Z n = N [ σ 4 n N x k ; 0, σ k= σ n/ ] It can easly be checked that σ 4 σ > for any < σ : the varance ncreases exponentally wth n even n ths smple case For example, f we select σ = then we have a reasonably good mportance dstrbuton as q k x k π n x k but N VIS[Ẑn] Z n 03 n/ whch s approxmately equal to 9 0 for n = 000! We would need to use N 0 3 partcles to obtan a relatve varance VIS[Ẑn] Zn clearly mpractcable = 00 Ths s 3 However, as we wll see later, the key to many advanced SMC methods s the ntroducton of a sequence of target dstrbutons whch dffer from the orgnal target dstrbutons

15 34 Resamplng We have seen that IS and thus SIS provdes estmates whose varance ncreases, typcally exponentally, wth n Resamplng technques are a key ngredent of SMC methods whch partally solve ths problem n some mportant scenaros Resamplng s a very ntutve dea whch has major practcal and theoretcal benefts Consder frst an IS approxmaton π n x :n of the target dstrbuton π n x :n Ths approxmaton s based on weghted samples from q n x :n and does not provde samples approxmately dstrbuted accordng to π n x :n To obtan approxmate samples from π n x :n, we can smply sample from ts IS approxmaton π n x :n ; that s we select :n wth probablty Wn Ths operaton s called resamplng as t corresponds to samplng from an approxmaton π n x :n whch was tself obtaned by samplng If we are nterested n obtanng N samples from π n x :n, then we can smply resample N tmes from π n x :n Ths s equvalent to assocatng a number of offsprng Nn wth each partcle :n n such a way that Nn :N = Nn,, Nn N follow a multnomal dstrbuton wth parameter vector N, Wn :N and assocatng a weght of /N wth each offsprng We approxmate π n x :n by the resampled emprcal measure π n x :n = = N n N δ :n x :n 3 where E [ ] Nn W :N n = NW n Hence π n x :n s an unbased approxmaton of π n x :n Improved unbased resamplng schemes have been proposed n the lterature These are methods of selectng Nn such that the unbasedness property s preserved, and such that V [ ] Nn W :N n s smaller than that obtaned va the multnomal resamplng scheme descrbed above To summarze, the three most popular algorthms found n the lterature are, n descendng order of popularty/effcency: Systematc Resamplng Sample U U [ ] { 0, N and defne U = U + N for =,, N, then set Nn = U j : k= W n k U j } k= W n k 0 wth the conventon k= := 0 It s straghtforward to establsh that ths approach s unbased Resdual Resamplng Set Ñ n = NWn :N, sample N n from a multnomal of parameters N, W :N n where W n Wn N Ñn then set Nn = Ñ n+ N n Ths s very closely related to breakng the emprcal CDF up nto N components and then samplng once from each of those components: the stratfed resamplng approach of [7] Multnomal Resamplng Sample Nn :N from a multnomal of parameters N, Wn :N Note that t s possble to sample effcently from a multnomal dstrbuton n O N operatons However, the systematc resamplng algorthm ntroduced n [5] s the most wdely-used algorthm n the lterature as t s extremely easy to mplement and outperforms other resamplng schemes n most scenaros although ths s not guaranteed n general [3] Resamplng allows us to obtan samples dstrbuted approxmately accordng to π n x :n, but t should be clear that f we are nterested n estmatng I n ϕ n then we wll obtan an estmate wth lower varance usng π n x :n than that whch we would have obtaned by usng π n x :n By resamplng we ndeed add some extra nose as shown by [9] However, an mportant advantage of resamplng s that t allows us to remove of partcles wth low weghts wth a hgh probablty In the sequental framework n whch we are nterested, ths s extremely useful as we do not want to carry forward partcles wth low weghts and we want to focus our computatonal efforts on regons of hgh probablty mass Clearly, there s always the possblty than a partcle havng a low weght at tme n could have an hgh weght at tme n +, n whch case resamplng could be wasteful It s straghtforward to consder artfcal problems for whch ths 3

16 s the case However, we wll show that n the estmaton problems we are lookng at the resamplng step s provably benefcal Intutvely, resamplng can be seen to provde stablty n the future at the cost of an ncrease n the mmedate Monte Carlo varance Ths concept wll be made more precse n secton A Generc Sequental Monte Carlo Algorthm SMC methods are a combnaton of SIS and resamplng At tme, we compute the IS approxmaton π x of π x whch s a weghted collecton of partcles { W, } Then we use a resamplng step to elmnate wth hgh probablty those partcles wth low weghts and multply those wth hgh weghts We denote by { N, } the collecton of equally-weghted resampled partcles Remember that each orgnal partcle has N offsprng so there exst N dstnct ndexes j j j N such that j = j = = jn = After the resamplng step, we follow the SIS strategy and sample q x Thus, s approxmately dstrbuted accordng to π x q x x Hence the correspondng mportance weghts n ths case are smply equal to the ncremental weghts α x : We then resample the partcles wth respect to these normalsed weghts and so on To summarse, the algorthm proceeds as follows ths algorthm s sometmes referred to as Sequental Importance Resamplng SIR or Sequental Importance Samplng and Resamplng SIS/R Sequental Monte Carlo At tme n = Sample q x Compute the weghts w and W w Resample { } { W, to obtan N equally-weghted partcles At tme n Sample n q n x n :n and set :n Compute the weghts α n :n and W n α n :n :n, n N, Resample { } { Wn, :n to obtan N new equally-weghted partcles } N, :n } At any tme n, ths algorthm provdes two approxmatons of π n x :n We obtan π n x :n = Wnδ :n x :n 33 = after the samplng step and π n x :n = N = δ x :n 34 :n 4

17 after the resamplng step The approxmaton 33 s to be preferred to 34 We also obtan an approxmaton of Z n /Z n through Ẑ n = α n Z n N :n = As we have already mentoned, resamplng has the effect of removng partcles wth low weghts and multplyng partcles wth hgh weghts However, ths s at the cost of mmedately ntroducng some addtonal varance If partcles have unnormalsed weghts wth a small varance then the resamplng step mght be unnecessary Consequently, n practce, t s more sensble to resample only when the varance of the unnormalsed weghts s superor to a pre-specfed threshold Ths s often assessed by lookng at the varablty of the weghts usng the so-called Effectve Sample Sze ESS crteron [30, pp 35-36], whch s gven at tme n by N ESS = W n = Its nterpretaton s that n a smple IS settng, nference based on the N weghted samples s approxmately equvalent n terms of estmator varance to nference based on ESS perfect samples from the target dstrbuton The ESS takes values between and N and we resample only when t s below a threshold N T ; typcally N T = N/ Alternatve crtera can be used such as the entropy of the weghts { Wn} whch acheves ts maxmum value when Wn = N In ths case, we resample when the entropy s below a gven threshold Sequental Monte Carlo wth Adaptve Resamplng At tme n = Sample q x Compute the weghts w and W w If resamplng crteron satsfed then resample { } { W, to obtan N equally weghted partcles { } { } { } set W,, otherwse set W, { W, } At tme n N, Sample n q n x n :n and set :n :n, n N, } and Compute the weghts α n :n and W n W n α n :n If resamplng crteron satsfed, then resample { } { Wn, :n to obtan N equally weghted partcles { } { } { } and set W n, n, otherwse set W n, n { Wn, n} N, n N, :n } 5

18 In ths context too we have two approxmatons of π n x :n π n x :n = π n x :n = Wnδ :n x :n, 35 = = W nδ x :n :n whch are equal f no resamplng step s used at tme n We may also estmate Z n /Z n through Ẑ n Z n = W n α n :n 36 = SMC methods nvolve systems of partcles whch nteract va the resamplng mechansm and, consequently, obtanng convergence results s a much more dffcult task than t s for SIS where standard results d asymptotcs apply However, there are numerous sharp convergence results avalable for SMC; see [0] for an ntroducton to the subject and the monograph of Del Moral [] for a complete treatment of the subject An explct treatment of the case n whch resamplng s performed adaptvely s provded by [] The presence or absence of degeneracy s the factor whch most often determnes whether an SMC algorthm works n practce However strong the convergence results avalable for lmtngly large samples may be, we cannot expect good performance f the fnte sample whch s actually used s degenerate Indeed, some degree of degeneracy s nevtable n all but trval cases: f SMC algorthms are used for suffcently many tme steps every resamplng step reduces the number of unque values representng, for example For ths reason, any SMC algorthm whch reles upon the dstrbuton of full paths x :n wll fal for large enough n for any fnte sample sze, N, n spte of the asymptotc justfcaton It s ntutve that one should endeavour to employ algorthms whch do not depend upon the full path of the samples, but only upon the dstrbuton of some fnte component x n L:n for some fxed L whch s ndependent of n Furthermore, ergodcty a tendency for the future to be essentally ndependent of the dstant past of the underlyng system wll prevent the accumulaton of errors over tme These concepts are precsely charactersed by exstng convergence results, some of the most mportant of whch are summarsed and nterpreted n secton 36 Although sample degeneracy emerges as a consequence of resamplng, t s really a manfestaton of a deeper problem one whch resamplng actually mtgates It s nherently mpossble to accurately represent a dstrbuton on a space of arbtrarly hgh dmenson wth a sample of fxed, fnte sze Sample mpovershment s a term whch s often used to descrbe the stuaton n whch very few dfferent partcles have sgnfcant weght Ths problem has much the same effect as sample degeneracy and occurs, n the absence of resamplng, as the nevtable consequence of multplyng together ncremental mportance weghts from a large number of tme steps It s, of course, not possble to crcumvent ether problem by ncreasng the number of samples at every teraton to mantan a constant effectve sample sze as ths would lead to an exponental growth n the number of samples requred Ths sheds some lght on the resamplng mechansm: t resets the system n such a way that ts representaton of fnal tme margnals remans well behaved at the expense of further dmnshng the qualty of the path-samples By focusng on the fxed-dmensonal fnal tme margnals n ths way, t allows us to crcumvent the problem of ncreasng dmensonalty 36 Convergence Results for Sequental Monte Carlo Methods Here, we brefly dscuss selected convergence results for SMC We focus on the CLT as t allows us to clearly understand the benefts of the resamplng step and why t works If multnomal resamplng s used at every teraton 4, then the assocated SMC estmates of Ẑn/Z n and I n ϕ n satsfy a CLT and ther respectve 4 Smlar expressons can be establshed when a lower varance resamplng strategy such as resdual resamplng s used and when resamplng s performed adaptvely [] The results presented here are suffcent to gude the desgn of partcular 6

19 asymptotc varances are gven by and N π n x q x dx + n k= π n x q ϕn x x :n π n x :n x dx :n I n ϕ n dx + n k= πn x :k π k x :k q k x k x :k dx k :k π n x :k π k x :k q k x k x :k ϕn x :n π n x k+:n x :k dx k+:n I n ϕ n dx:k + π n x:n π n x :n q n x n x :n ϕ n x :n I n ϕ n dx :n A short and elegant proof of ths result s gven n [, Chapter 9]; see also [9] These expresson are very nformatve They show that the resamplng step has the effect of resettng the partcle system whenever t s appled Comparng 6 to 37, we see that the SMC varance expresson has replaced the mportance dstrbuton q n x :n n the SIS varance wth the mportance dstrbutons π k x :k q k x k x :k obtaned after the resamplng step at tme k Moreover, we wll show that n mportant scenaros the varances of SMC estmates are orders of magntude smaller than the varances of SIS estmates Let us frst revst the toy example dscussed n secton 33 Example contnued In ths case, t follows from 37 that the asymptotc varance s fnte only when σ > and ] [ V SMC [Ẑn Zn π n x n ] π N q x dx + n x k q k x k dx k k= [ σ 4 / ] = n N σ compared to V IS [Ẑn ] Z n = N [ σ 4 σ n/ ] The asymptotc varance of the SMC estmate ncreases lnearly wth n n contrast to the exponental growth of the IS varance For example, f we select σ = then we have a reasonably good mportance dstrbuton as q k x k π n x k In ths case, we saw that t s necessary to employ N 0 3 partcles n order to obtan VIS[Ẑn] Zn requres the use of just N 0 4 partcles: an mprovement by 9 orders of magntude = 0 for n = 000 Whereas to obtan the same performance, VSMC[Ẑn] Z n = 0, SMC Ths scenaro s overly favourable to SMC as the target 3 factorses However, generally speakng, the major advantage of SMC over IS s that t allows us to explot the forgettng propertes of the model under study as llustrated by the followng example Example Consder the followng more realstc scenaro where n n γ n x :n = µ x M k x k x k G k x k k= wth µ a probablty dstrbuton, M k a Markov transton kernel and G k a postve potental functon Essentally, flterng corresponds to ths model wth M k x k x k = f x k x k and the tme nhomogeneous potental functon G k x k = g y k x k In ths case, π k x k x :k = π k x k x k and we would algorthms and the addtonal complexty nvolved n consderng more general scenaros serves largely to produce substantally more complex expressons whch obscure the mportant ponts k= 7

20 typcally select an mportance dstrbuton q k x k x :k wth the same Markov property q k x k x :k = q k x k x k It follows that 37 s equal to π x n N q x dx πn x k :k + π k x k q k x k x k dx k :k and 38, for ϕ n x :n = ϕ x n, equals: k= π x q x ϕ xn π n x n x dx :n I n ϕ dx + n k= π n x k :k π k x k q k x k x k ϕ xn π n x n x k dx k I n ϕ dxk :k + π n xn :n π n x n q n x n x n ϕ x n I n ϕ dx n :n, where we use the notaton I n ϕ for I n ϕ n In many realstc scenaros, the model assocated wth π n x :n has some sort of ergodc propertes; e x k, x k π n x n x k π n x n x k for large enough n k In layman s terms, at tme n what happened at tme k s rrelevant f n k s large enough Moreover, ths often happens exponentally fast; that s for any x k, x k π n x n x k π n x n x k dx n β n k for some β < Ths property can be used to establsh that for bounded functons ϕ ϕ ϕ x n π n x n x k dx n I ϕ βn k ϕ and under weak addtonal assumptons we have for a fnte constant A Hence t follows that π n x k :k π k x k q k x k x k A V SMC [Ẑn ] Z n C n N, ] V SMC [În ϕ D N for some fnte constants C, D that are ndependent of n These constants typcally ncrease polynomally/exponentally wth the dmenson of the state-space and decrease as β 0 37 Summary We have presented a generc SMC algorthm whch approxmates {π n x :n } and {Z n } sequentally n tme Wherever t s possble to sample from q n x n x :n and evaluate α n x :n n a tme ndependent of n, ths leads to an algorthm whose computatonal complexty does not ncrease wth n For any k, there exsts n > k such that the SMC approxmaton of π n x :k conssts of a sngle partcle because of the successve resamplng steps It s thus mpossble to get a good SMC approxmaton of the jont dstrbutons {π n x :n } when n s too large Ths can easly be seen n practce, by montorng the number of dstnct partcles approxmatng π n x 8

21 However, under mxng condtons, ths SMC algorthm s able to provde estmates of margnal dstrbutons of the form π n x n L+:n and estmates of Z n /Z n whose asymptotc varance s unformly bounded wth n Ths property s crucal and explans why SMC methods work n many realstc scenaros Practcally, one should keep n mnd that the varance of SMC estmates can only expected to be reasonable f the varance of the ncremental weghts s small In partcular, ths requres that we can only expect to obtan good performance f π n x :n π n x :n and q n x n x :n π n x n x :n ; that s f the successve dstrbutons we want to approxmate do not dffer much one from each other and the mportance dstrbuton s a reasonable approxmaton of the optmal mportance dstrbuton However, f successve dstrbutons dffer sgnfcantly, t s often possble to desgn an artfcal sequence of dstrbutons to brdge ths transton [, 3] 4 Partcle Flterng Remember that n the flterng context, we want to be able to compute a numercal approxmaton of the dstrbuton {p x :n y :n } n sequentally n tme A drect applcaton of the SMC methods descrbed earler to the sequence of target dstrbutons π n x :n = p x :n y :n yelds a popular class of partcle flters More elaborate sequences of target and proposal dstrbutons yeld varous more advanced algorthms For ease of presentaton, we present algorthms n whch we resample at each tme step However, n practce we recommend only resamplng when the ESS s below a threshold and employng the systematc resamplng scheme 9

22 4 SMC for Flterng Frst, consder the smplest case n whch γ n x :n = p x :n, y :n s chosen, yeldng π n x :n = p x :n y :n and Z n = p y :n Practcally, t s only necessary to select the mportance dstrbuton q n x n x :n We have seen that n order to mnmse the varance of the mportance weghts at tme n, we should select q opt n x n x :n = π n x n x :n where π n x n x :n = p x n y n, x n = g y n x n f x n x n, 39 p y n x n and the assocated ncremental mportance weght s α n x :n = p y n x n In many scenaros, t s not possble to sample from ths dstrbuton but we should am to approxmate t In any case, t shows that we should use an mportance dstrbuton of the form q n x n x :n = q x n y n, x n 40 and that there s nothng to be ganed from buldng mportance dstrbutons dependng also upon y :n, x :n although, at least n prncple, n some settngs there may be advantages to usng nformaton from subsequent observatons f they are avalable Combnng 9, 30 and 40, the ncremental weght s gven by α n x :n = α n x n :n = g y n x n f x n x n q x n y n, x n The algorthm can thus be summarsed as follows SIR/SMC for Flterng At tme n = Sample qx y Compute the weghts w = µ g y q y and W w Resample { } { W, to obtan N equally-weghted partcles At tme n Sample n qx n y n, n and set :n :n, n Compute the weghts α n n :n = g y n nf n n q n yn, n N, Resample { } { Wn, :n to obtan N new equally-weghted partcles } and W n α n n :n N, :n } We obtan at tme n p x :n y :n = Wnδ :n x :n, = 0

23 6 SV Model: SIS Flterng Estmates 4 True Volatlty Flter Mean +/ SD Fgure : Flterng estmates obtaned for the stochastc volatlty model usng SIS At each tme the mean and standard devaton of x n condtonal upon y :n s estmated usng the partcle set It ntally performs reasonably, but the approxmaton eventually collapses: the estmated mean begns to dverge from the truth and the estmate of the standard devaton s naccurately low p y n y :n = Wn α n n :n = However, f we are nterested only n approxmatng the margnal dstrbutons {p x n y :n } and {p y :n } then we need to store only the termnal-value partcles { n :n} to be able to compute the weghts: the algorthm s storage requrements do not ncrease over tme Many technques have been proposed to desgn effcent mportance dstrbutons q x n y n, x n whch approxmate p x n y n, x n In partcular the use of standard suboptmal flterng technques such as the Extended Kalman Flter or the Unscented Kalman Flter to obtan mportance dstrbutons s very popular n the lterature [4, 37] The use of local optmsaton technques to desgn q x n y n, x n centred around the mode of p x n y n, x n has also been advocated [33, 34] 4 Example: Stochastc Volatlty Returnng to example 4 and the smulated data shown n fgure, we are able to llustrate the performance of SMC algorthms wth and wthout resamplng steps n a flterng context An SIS algorthm correspondng to the above SMC algorthm wthout a resamplng step wth N = 000 partcles, n whch the condtonal pror s employed as a proposal dstrbuton leadng to an algorthm n whch the partcle weghts are proportonal to the lkelhood functon produces the output shown n fgure Specfcally, at each teraton, n, of the algorthm the condtonal expectaton and standard devaton of x n, gven y :n s obtaned It can be seen that the performance s ntally good, but after a few teratons the estmate of the mean becomes naccurate, and the estmated standard devaton shrnks to a very small

24 500 n= 000 n=0 000 n= Partcle Count Normalsed Weghts x Normalsed Weghts Normalsed Weghts Fgure 3: Emprcal dstrbutons of the partcle weghts obtaned wth the SIS algorthm for the stochastc volatlty model at teratons, 0 and 50 Although the algorthm s reasonably ntalsed, by teraton 0 only a few tens of partcles have sgnfcant weght and by teraton 50 a sngle partcle s domnant value Ths standard devaton s an estmate of the standard devaton of the condtonal posteror obtaned va the partcle flter: t s not a measure of the standard devaton of the estmator Such an estmate can be obtaned by consderng several ndependent partcle flters run on the same data and would llustrate the hgh varablty of estmates obtaned by a poorly-desgned algorthm such as ths one In practce, approxmatons such as the effectve sample sze are often used as surrogates to characterse the uncertanty of the flter estmates but these perform well only f the flter s provdng a reasonable approxmaton of the condtonal dstrbutons of nterest Fgure 3 supports the theory that the falure of the algorthm after a few teratons s due to weght degeneracy, showng that the number of partcles wth sgnfcant weght falls rapdly The SIR algorthm descrbed above was also appled to ths problem wth the same proposal dstrbuton and number of partcles as were employed n the SIS case For smplcty, multnomal resamplng was appled at every teraton Qualtatvely, the same features would be observed f a more sophstcated algorthm were employed, or adaptve resamplng were used although these approaches would lessen the severty of the path-degeneracy problem Fgure 4 shows the dstrbuton of partcle weghts for ths algorthm Notce that unlke the SIS algorthm shown prevously, there are many partcles wth sgnfcant weght at all three tme ponts It s mportant to note that whle ths s encouragng t s not evdence that the algorthm s performng well: t provdes no nformaton about the path-space dstrbuton and, n fact, t s easy to construct poorly-performng algorthms whch appear to have a good dstrbuton of partcle weghts for nstance, consder a scenaro n whch the target s relatvely flat n ts tals but sharply concentrated about a mode; f the proposal has very lttle mass n the vcnty of the mode then t s lkely that a collecton of very smlar mportance weghts wll be obtaned but the sample thus obtaned does not characterse the dstrbuton of nterest well Fgure 5 shows that the algorthm does ndeed produce a reasonable estmate and plausble credble nterval And, as we expect, a problem does arse when we consder the smoothng dstrbutons px n y :500 as shown n fgure 6: the estmate and credble nterval s unrelable for n 500 Ths s due to the degeneracy caused at the begnnng of the path by repeated resamplng In contrast the