{"title": "Greedy Importance Sampling", "book": "Advances in Neural Information Processing Systems", "page_first": 596, "page_last": 602, "abstract": null, "full_text": "Greedy importance sampling \n\nDale Schuurmans \n\nDepartment of Computer Science \n\nUniversity of Waterloo \ndale@cs.uwaterloo.ca \n\nAbstract \n\nI present a simple variation of importance sampling that explicitly search(cid:173)\nes for  important regions in the target distribution.  I prove that the tech(cid:173)\nnique yields unbiased estimates,  and show empirically it can reduce the \nvariance  of standard Monte Carlo estimators.  This is achieved  by con(cid:173)\ncentrating samples in more significant regions of the sample space. \n\n1  Introduction \n\nIt is  well  known  that general  inference  and  learning  with  graphical  models  is  computa(cid:173)\ntionally hard  [1]  and  it is therefore  necessary  to consider restricted architectures  [13],  or \napproximate  algorithms to  perform  these  tasks  [3,  7].  Among  the most  convenient  and \nsuccessful techniques are stochastic methods which are guaranteed to converge to a correct \nsolution in the limit oflarge samples [10,  11,  12,  15].  These methods can be easily applied \nto complex inference problems that overwhelm deterministic approaches. \n\nThe  family  of stochastic  inference methods can  be  grouped into the  independent Monte \nCarlo methods (importance sampling and rejection sampling [4,  10, 14]) and the dependent \nMarkov Chain Monte Carlo (MCMC) methods (Gibbs sampling, Metropolis sampling, and \n\"hybrid\" Monte Carlo) [5,  10,  11,  15].  The goal of all these methods is to simulate drawing \na random sample from a target distribution P (x)  (generally defined by a Bayesian network \nor graphical model) that is difficult to sample from directly. \n\nThis  paper investigates  a  simple modification of importance sampling  that demonstrates \nsome advantages over independent and dependent-Markov-chain methods.  The idea is to \nexplicitly search  for  important regions  in  a  target  distribution P  when  sampling  from  a \nsimpler proposal distribution Q.  Some MCMC methods,  such as Metropolis and \"hybrid\" \nMonte Carlo, attempt to do something like this by biasing a local random search towards \nhigher probability regions,  while preserving the  asymptotic \"fair sampling\" properties of \nthe  exploration  [11,  12].  Here  I  investigate  a  simple direct  approach  where  one draws \npoints from a proposal distribution Q but then explicitly searches in P  to find points from \nsignificant regions.  The main challenge is to maintain  correctness  (i.e.,  unbiased ness)  of \nthe resulting procedure, which we achieve by independently sampling search subsequences \nand  then  weighting  the  sample  points  so  that  their expected  weight  under the  proposal \ndistribution Q matches their true probability under the target P. \n\n\fGreedy Importance Sampling \n\n597 \n\nImportance sampling \n\u2022 Draw Xl , ... , X n  independently from Q. \n\u2022 Weight each point Xi  by W(Xi)  =  ~I::l. \n\u2022 For a random variable,  f, estimate E p (.,) f(x) \n\nby \n\nA \n\nf  =  n L.Ji=I  f(Xi)W(Xi). \n\nI  \",n \n\n\"Indirect\" importance sampling \n\u2022 Draw Xl, ... ,xn independently from Q. \n.Weighteachpointxibyu(xi) = (3~~i/. \n\u2022 For a random variable,  f, estimate Ep(.,)f(x) \nf  =  L.Ji=I  f(Xi)U(Xi)/ L.Ji=I U(Xi). \n\n\",n \n\n\",n \n\nby \n\nA \n\nFigure 1:  Regular and \"indirect\" importance sampling procedures \n\n2  Generalized importance sampling \n\nMany  inference  problems  in  graphical  models  can  be  cast  as  determining  the  expected \nvalue of a random variable of interest, f, given observations drawn  according to  a target \ndistribution P. That is, we are interested in computing the expectation Ep(x) f(x).  Usually \nthe random variable f  is simple,  like the  indicator of some event,  but the distribution P \nis generally not in  a form that  we can  sample from  efficiently.  Importance  sampling is  a \nuseful technique for estimating Ep(x) f (x)  in these cases.  The idea is to draw independent \npoints xl, .. \"  Xn  from  a  simpler \"proposal\"  distribution Q,  but then  weight these  points \nby  w(x)  = P(x)/Q(x)  to  obtain  a  \"fair\"  representation  of P.  Assuming  that  we  can \nefficiently evaluate P(x) at  each  point, the weighted sample can  be used to  estimate de(cid:173)\nsired expectations (Figure 1).  The correctness (i.e., unbiasedness) of this procedure is easy \nto  establish,  since  the  expected  weighted  value of f  under Q is just Eq(x)f(x)w(x)  = \nEXEX [f(x)w(x)] Q(x) = Ex EX  [f(x)~t:n Q(x) = EXEX f(x)P(x) = Ep(x)f(x), \nThis technique can be implemented using \"indirect\" weights u( x)  = f3P (x) / Q ( x)  and an \nalternative estimator (Figure 1) that only requires us to compute a fixed multiple of P (x). \nThis preserves asymptotic correctness because  ~ E7=1  f(xdu(xd and ~ E?=l U(Xi)  con(cid:173)\nverge to f3Ep(x)f(x)  and f3  respectively,  which yields j  -t Ep(x)f(x)  (generally [4]).  It \nwill always be possible to apply this extended approach below, but we drop it for now. \n\nImportance  sampling is  an  effective  estimation  technique when  Q  approximates P  over \nmost of the domain, but it fails when Q misses high probability regions of P  and system(cid:173)\natically yields samples with small  weights.  In this case,  the reSUlting estimator will have \nhigh variance because the sample will almost always contain unrepresentative points but is \nsometimes dominated by a few  high weight points.  To  overcome  this problem it is  criti(cid:173)\ncal to obtain data points from the important regions of P.  Our goal is to avoid generating \nsystematically under-weight samples  by  explicitly searching for significant regions in the \ntarget distribution P. To do this, and maintain the unbiased ness of the resulting procedure, \nwe develop a series of extensions to importance sampling that are each provably correct. \n\nThe first extension is to consider sampling blocks of points instead of just individual points. \nLet  B  be  a  partition of X  into  finite  blocks  B,  where  UBE8 B  =  X,  B  n B'  =  0, \nand  each  B  is  finite.  (Note  that  B  can  be  infinite.)  The  \"block\"  sampling  procedure \n(Figure  2)  draws  independent  blocks  of points  to  construct  the  final  sample,  but  then \nweights  points  by  their  target  probability  P(x)  divided  by  the  total  block  probability \nQ (B (x)),  For discrete spaces it is easy  to verify that this procedure yields unbiased esti-\nmates,  since Eq(x)  [EXjEB(X)  f(xj)w(Xj)]  = EXEX  [L:XjEB(X)  f(xj)w(Xj)]  Q(x) = \nEBE8 EXiEB  [EXjEB f(xj)w(Xj)]  Q(Xi)  =  EBE8  [EXjEB f(xj)w(Xj)]  Q(B)  = \nEBE8 [EXjEB f(xj) ~\u00ab~n Q(B) = EBE8 [EXjEB f(xj )P(Xj)]  =  L:xEx f(x)P(x). \n\n\f598 \n\nD.  Schuurmans \n\n\"Block\" importance sampling \n\u2022 Draw Xl , ... , Xn  independently from  Q. \n\u2022 For Xi, recover block Bi  =  {Xi,l, ... ,Xi,bJ. \n\u2022 Create a large sample out of the blocks \nXl ,1, ... , Xl ,bl , X2 ,1 , ... , X2,b2'  \u2022\u2022\u2022 , Xn,l, ... , Xn ,b\" . \n\u2022 Weighteachx '  ' byw(x.') = \n\nP(zi,j) \n\nI ,} \n\nI,} \n\n\",oi  Q(z,,)' \nL.Jj=l \n\n' ,J \n\n\u2022 For a random variable,  f, estimate Ep(z) f(x) \nj  =  ~ 2:~=1 2:~~1 f(Xi ,j)W(Xi,j). \n\nby \n\n\"Sliding window\" importance sampling \n\u2022  Draw Xl, ... , xn independently from Q. \n\u2022  For Xi , recover block Bi , and let Xi ,l  =  Xi : \n- Get Xi,l 'S successors Xi,l, Xi,2, ... , Xi ,m \nby climbing up m - 1 steps from  Xi ,l . \n\n- Get predecessorsxi,_m+l\"  ... ,Xi,-l , Xi,O \n\nby climbing down m - 1 steps from  Xi,l . \n\n- Weight W(Xi ,j)= P(Xi ,i)/2:!=i_m+lQ (x; ,k) \n\u2022  Create final sample from successor points \nXI ,I, ... , Xl ,m, X2,1 ,  \u2022\u2022. , X2 ,m, \u2022.. , Xn ,l, ... , Xn ,m. \n\u2022  For a random variable,  f, estimate Ep(z) f(x) \nf  =  n 6i=1 6j=1 f(Xi,j)W(Xi,j) . \n\n1  \",n  \",m \n\nby \n\nA \n\nFigure 2: \"Block\" and \"sliding window\" importance sampling procedures \n\nCrucially, this argument does not depend on how the partition of X  is chosen.  In fact,  we \ncould fix  any partition, even one that depended on the target distribution P, and still obtain \nan unbiased procedure (so long as  the partition remains fixed) .  Intuitively, this works be(cid:173)\ncause blocks are  drawn  independently from  Q  and the weighting scheme still produces a \n\"fair\" representation of P . (Note that the results presented in this paper can all be extended \nto continuous spaces under mild technical restrictions. However, for the purposes of clarity \nwe will restrict the technical presentation in this paper to the discrete case.) \n\nThe  second  extension  is  to allow countably infinite blocks that  each  have  a  discrete  to(cid:173)\ntal  order  . . .  <  Xi -1  <  Xi  <  Xi +1  <  .. . defined  on their elements.  This  order could \nreflect  the  relative  probability  of  Xi  and  X j  under  P,  but  for  now  we  just consider  it \nto  be  an  arbitrary  discrete  order.  To  cope  with  blocks  of unbounded  length,  we  em(cid:173)\nploy a  \"sliding  window\"  sampling  procedure that  selects  a contiguous sub-block of size \nm  from  within a  larger  selected  block (Figure  2).  This  procedure  builds  each  indepen(cid:173)\ndent  subsample  by  choosing  a  random  point  Xl  from  the  proposal  distribution  Q,  de(cid:173)\ntermining  its  containing block B(xt),  and  then  climbing  up m  - 1 steps  to  obtain  the \nsuccessors  Xl,  X2,  \u2022\u2022. ,  Xm ,  and  climbing  down  m  - 1  steps  to  obtain  the  predecessors \nX- m +1 , ... ,  X-I, Xo .  The successor points  (including  Xl)  appear  in the final  sample,  but \nthe  predecessors  are  only used  to determine  the  weights  of the  sample  points.  Weights \nare  determined  by  the  target  probability P (x)  divided by  the  probability  that  the  point \nX  appears  in  a  random  reconstruction  under  Q.  This  too  yields  an  unbiased  estimator \n\nsinceEQ(x) [2:~lf(xj)w(Xj)] =  2: XtEX  [2:~;~-1 f(xj)2:~=j~~:: Q(Xk)]  Q(Xl )  = \n\n'\" \n6BEB \n\n2: \n\nx t EB \n\n2:l +m - 1  f(xj)P(Xj)Q(xd  -\n\nj=l  \"'J. \n\n6k=J-m+l \n\n'\" \n\n2: \n\nQ(Xk)  - 6BEB  Xj EB \n\n2:j \n\nl=j-m+1 \"'J . \n\nf(xj)P(Xj)Q(xt}  -\nQ(Xk)  -\n6k=J-m+l \n\n2:BEB2:x j EBf(xj )P(Xj )Ei::=::: ~;::~ = 2:BEB2:xjEBf(xj )P(Xj)= 2: xEx f(x)P(x). \n\n(The middle line breaks the sum into disjoint blocks and then reorders the sum so that in(cid:173)\nstead of first  choosing the start point Xl  and then  XL'S  successors  Xl, .. \u2022 , Xl+m-l.  we first \nchoose the successor point Xj  and then the start points Xj-m+1 , ... ,  Xj  that could have led \nto Xj).  Note that this derivation does not depend on the particular block partition nor on the \nparticular discrete orderings, so long as  they remain fixed.  This means that,  again,  we can \nuse partitions and orderings that explicitly depend on P and still obtain a correct procedure. \n\n\fGreedy Importance Sampling \n\n599 \n\n\"Greedy\" importance sampling (I-D) \ne Draw Xl , ... , Xn  independently from Q. \neForeachxi , letxi,l  =Xi : \n- Compute successors Xi,l, Xi,2, ... ,Xi,m  by taking \n\nm  - 1 size \u20ac  steps in the direction of increase. \n\nIf(x)P(x)1 \n\ncollision \n\n- If an improper ascent or descent occurs, \n\n- Compute predecessors Xi,-m+l, ... ,Xi ,-l ,Xi ,O by \ntaking m -1 size \u20ac  steps in the direction of decrease. \n\n- Weightw(xi,j) =  P(Xi,j)/L:~=j_m+l Q(Xi ,k) .  k \n\ne Create the final  sample from  successor points \n\ntruncate paths as shown on the upper right. \n\nXI ,l, \u2022\u2022\u2022 ,  XI ,m ,  X'2 , I ,  \u2022\u2022\u2022 , X2 ,m ,  \u2022\u2022\u2022 ,Xn ,l, \u2022\u2022\u2022 , ::tn ,m-\n\ne For a random variable,  f, estimate Ep(z) f(x) \n\nby \n\nA \n\nf  =  -;;  L..ti=l  L.Jj=l  f(Xi ,j)W(Xi ,j) . \n\n1  \",n  \",m \n\nx\u00b7  x\" \n\n~ \nif 6 \"\\, \n\nmerge \n\nFigure 3:  \"Greedy\" importance sampling procedure; \"colliding\" and \"merging\" paths. \n\n3  Greedy importance sampling:  I-dimensional case \n\nFinally,  we  apply  the sliding window procedure  to conduct an explicit search  for impor(cid:173)\ntant regions in  X.  It is well  known that the optimal proposal distribution for importance \nsampling isjust Q* (x)  =  If(x )P(x)11 EXEX If(x )P(x) 1 (which minimizes variance [2]). \nHere we apply the sliding window procedure using an order structure that is determined by \nthe objective If(x )P(x )1 . The hope is to obtain reduced variance by sampling independent \nblocks of points where each block (by virtue of being constructed via an explicit search) is \nlikely to contain  at least  one or two high weight points.  That is, by capturing a moderate \nsize sample of independent high weight points we intuitively expect to outperform standard \nmethods that are unlikely to observe such points by chance. Our experiments below verify \nthis intuition (Figure 4). \n\nThe main technical issue is maintaining unbiasedness,  which is easy to establish in the 1-\ndimensional case.  In the simple  I-d setting, the \"greedy\"  importance sampling procedure \n(Figure 3) first draws an  initial point Xl from Q and then follows the direction of increas(cid:173)\ning  If(x)P(x)l, taking fixed  size \u20ac  steps,  until either m  - 1 steps have been taken  or we \nencounter a critical point. A single \"block\" in our final  sample is comprised of a complete \nsequence captured in one ascending search.  To weight the sample points we account for all \npossible ways each  point could appear in  a subsample, which, as  before, entails climbing \ndown m-l steps in the descent direction (to calculate the denominators). The unbiasedness \nof the procedure then follows directly from  the previous section, since greedy importance \nsampling is equivalent to sliding window importance sampling in this setting. \n\nThe only nontrivial issue is to maintain disjoint search paths.  Note that a search path must \nterminate whenever it steps from a point x\u00b7  to a point x**  with lower value; this indicates \nthat a collision has occurred because some other path must reach  x\u00b7 from the \"other side\" \nof the critical point (Figure 3).  At a collision, the largest ascent point x\u00b7 must be allocated \nto a single path. A reasonable policy is to allocate x\u00b7 to the path that has the lowest weight \npenultimate point (but the only critical  issue  is  ensuring that  it gets  assigned  to a single \nblock). By ensuring that the critical point is included in only one of the two distinct search \npaths, a practical estimator can be obtained that exhibits no bias (Figure 4). \n\nTo  test the effectiveness of the greedy approach I conducted several  I-dimensional experi(cid:173)\nments which varied the relationship between P, Q and the random variable f  (Figure 4). In \n\n\f600 \n\nD.  Schuurmans \n\nthese experiments  greedy  importance sampling strongly outperformed standard methods, \nincluding regular importance sampling and directly sampling from the target distribution P \n(rejection sampling and Metropolis sampling were  not competitive).  The results not only \nverify the unbiasedness of the greedy procedure, but also show that it obtains significantly \nsmaller variances across a wide range of conditions.  Note that the greedy procedure actu(cid:173)\nally uses m  out of 2m - 1 points sampled for each block and therefore effectively uses  a \ndouble sample.  However,  Figure 4 shows that the greedy approach often obtains variance \nreductions that are far greater than 2 (which corresponds to a standard deviation reduction \nof V2). \n\n4  Multi-dimensional case \n\nOf course, this technique is worthwhile only if it can be applied to multi-dimensional prob(cid:173)\nlems.  In  principle,  it  is  straightforward  to  apply  the  greedy  procedure  of Section  3  to \nmulti-dimensional sample spaces. The only new issue is that discrete search paths can now \npossibly \"merge\"  as  well  as  \"collide\";  see  Figure  3.  (Recall  that  paths  could  not merge \nin the  previous case.)  Therefore,  instead of decomposing the domain into a collection of \ndisjoint search  paths,  the objective  If(x)P(x)1  now decomposes  the domain into a forest \nof disjoint search trees.  However,  the same principle could be used to devise an  unbiased \nestimator in  this case:  one could assign a weight to a sample point x  that is just its target \nprobability P (x)  divided by the total  Q-probability of the subtree of points that lead to x \nin fewer than m  steps. This weighting scheme can be shown to yield an unbiased estimator \nas  before.  However,  the resulting procedure is  impractical because  in  an  N-dimensional \nsample space a search tree will typically have a branching factor of n(N); yielding expo(cid:173)\nnentially large trees.  Avoiding the need  to exhaustively examine  such trees is the critical \nissue in applying the greedy approach to multi-dimensional spaces. \n\nThe  simplest conceivable strategy is just to  ignore merge  events.  Surprisingly, this turns \nout to work reasonably well in many  circumstances.  Note that merges  will be a measure \nzero event in many  continuous domains.  In such cases  one could hope to  ignore merges \nand  trust  that  the  probability of \"double counting\"  such  points  would remain  near  zero. \nI conducted simple experiments with a version of greedy importance sampling procedure \nthat ignored merges.  This procedure searched in the gradient ascent direction of the objec(cid:173)\ntive  If{x)p{x)1  and heuristically inverted search steps by climbing in the gradient descent \ndirection.  Figures 5 and 6 show that, despite the heuristic nature of this procedure, it nev(cid:173)\nertheless demonstrates credible performance on simple tasks. \n\nThe first  experiment is a simple demonstration from  [12,  10]  where the task is to sample \nfrom a bivariate Gaussian distribution P  of two highly correlated random variables using a \n\"weak\" proposal distribution Q that is standard normal (depicted by the elliptical and circu(cid:173)\nlar one standard deviation contours in Figure 5 respectively).  Greedy importance sampling \nonce again performs very well (Figure 5); achieving unbiased estimates with lower variance \nthan standard Monte Carlo estimators, including common MCMC methods. \n\nTo conduct a more significant study,  I applied the heuristic greedy method to an  inference \nproblem in graphical models:  recovering the hidden state sequence from a dynamic proba(cid:173)\nbilistic model, given a sequence of observations.  Here I considered a simple Kalman  filter \nmodel  which had one state variable and one observation variable per time-step,  and  used \ntheconditionaldistributionsXtlXt_ l  \"-'  N(Xt_l,O';),  ZtlXt \"\"  N(xt,O'~) and initial dis(cid:173)\ntribution Xl  \"-'  N(O,O';) . The problem was to infer the value of the final  state variable Xt \ngiven the observations Zl, Z2,  \"', Zt.  Figure 6 again demonstrates that the greedy approach \n\n\fGreedy Importance Sampling \n\n601 \n\nhas  a strong advantage over standard  importance sampling.  (In fact,  the greedy approach \ncan  be applied  to  \"condensation\"  [6,  8]  to  obtain further  improvements  on  this  task,  but \nspace bounds preclude a detailed discussion.) \n\nOverall, these preliminary results show that despite the heuristic choices made in this sec(cid:173)\ntion,  the  greedy  strategy  still performs well  relative  to  common  Monte Carlo estimators, \nboth  in  terms  of bias  and  variance  (at  least on  some  low  and  moderate  dimension  prob(cid:173)\nlems).  However,  the heuristic nature of this procedure makes it extremely unsatisfying. In \nfact,  merge points can  easily make up a significant fraction  of finite domains.  It turns out \nthat a rigorously unbiased and  feasible  procedure can  be  obtained as  follows.  First, take \ngreedy fixed size steps in axis parallel directions (which ensures the steps can be inverted). \nThen, rather than exhaustively explore an entire predecessor tree to calculate the weights of \na sample point, use the well known technique of Knuth [9] to sample a single path from the \nroot and obtain an  unbiased estimate of the total Q-probability of the tree.  This procedure \nallows one to formulate  an  asymptotically unbiased estimator that is nevertheless feasible \nto implement.  It remains important future work to  investigate this approach and compare \nit to other Monte Carlo estimation methods on large dimensional problems-in particular \nhybrid Monte Carlo [11,  12].  The  current results already  suggest  that the  method  could \nhave benefits. \n\nReferences \n\n[1]  P.  Dagum and M.  Luby.  Approximating probabilistic inference in Bayesian belief networks is \n\nNP-hard.  Artif Intell,  60: 141-153, 1993. \n\n[2]  M. Evans. Chaining via annealing. 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Comput., 29(129): 121-\n\n136,1975. \n\n[10]  D.  MacKay.  Intro to Monte Carlo methods.  In Learning in  Graphical Models. Kluwer,  1998. \n\n[11]  R.  Neal.  Probabilistic inference using Markov chain Monte Carlo methods.  1993. \n\n[12]  R.  Neal.  Bayesian Learning for Neural Networks . Springer, New York, 1996. \n\n[13]  J. Pearl.  Probabilistic Reasoning in  Intelligence Systems.  Morgan Kaufmann,  1988. \n\n[14]  R.  Shacter and  M.  Peot.  Simulation  approaches to  general  probabilistic  inference  in  belief \n\nnetworks.  In  Uncertainty in Artificial Intelligence 5. Elsevier, 1990. \n\n[15]  M.  Tanner.  Tools for statistical inference:  Methods for exploration of posterior distributions \n\nand likelihoodfunctions.  Springer, New York,  1993. \n\n\f602 \n\nD.  Schuurmans \n\n. \n\nf ' \n\n._.11 \n\nmean \nbias \nstdev \n\nDirec \n0.779 \n0.001 \n0.071 \n\nGreed \n0.781 \n0.001 \n0.Q38 \n\nImprt  Direc \n1.038 \n0.777 \n0.002 \n0.003 \n0.065 \n0.088 \n\nGreed \n1.044 \n0.003 \n0.049 \n\n1m  rt \n1.032 \n0.008 \n0.475 \n\nDirec \n0.258 \n0.049 \n0.838 \n\nGreed \n0.208 \n0.000 \n0.010 \n\nImrt  Direc \n6.024 \n0.209 \n0.001 \n0.001 \n0.095 \n0.069 \n\nGreed \n6.028 \n0.004 \n0.037 \n\nImprt \n6.033 \n0.009 \n0.094 \n\nFigure 4:  I-dimensional experiments:  1000 repetitions on estimation samples of size 100. \nProblems with varying relationships between P, Q, I  and II PI. \n\n{/... . \n\n\u2022  . . \n\n/ ..  :~:. \n\n. . \n\n'I:'::'~\" \n\n,1 :.~ \n\n/ . . . .  \n\n.. \n\n..  . \n\n. , \n\nmean \nbias \nstdev \n\nDirect \n0.1884 \n0.0022 \n0.07 \n\nGreedy \n0.1937 \n0.0075 \n0.1374 \n\nImportance \n\n0.1810 \n0.0052 \n0.1762 \n\nRejection \n0.1506 \n0.0356 \n0.2868 \n\nGibbs \n0.3609 \n0.1747 \n0.5464 \n\nMetropolis \n\n8.3609 \n8.1747 \n22.1212 \n\nFigure 5:  2-dimensional experiments:  500 repetitions on estimation samples  of size 200. \nPictures depict: direct, greedy importance, regular importance, and Gibbs sampling, show(cid:173)\ning 1 standard deviation countours (dots are sample points, vertical lines are weights). \n\nmean \nbias \nstdev \n\nImportance \n\n5.2269 \n2.7731 \n1.2107 \n\nGreedy \n6.9236 \n1.0764 \n0.1079 \n\nFigure 6:  A 6-dimensional experiment:  500 repetitions on estimation samples of size 200. \nEstimating the value of Xt  given the observations Zl, \"\"  Zt.  Pictures depict paths sampled \nby regular versus greedy importance sampling. \n\n\f", "award": [], "sourceid": 1691, "authors": [{"given_name": "Dale", "family_name": "Schuurmans", "institution": null}]}