{"title": "Fisher Scoring and a Mixture of Modes Approach for Approximate Inference and Learning in Nonlinear State Space Models", "book": "Advances in Neural Information Processing Systems", "page_first": 403, "page_last": 409, "abstract": null, "full_text": "Fisher Scoring and a Mixture of Modes \nApproach for Approximate Inference and \nLearning in Nonlinear State Space Models \n\nThomas Briegel and Volker Tresp \nSiemens AG, Corporate Technology \n\nDept.  Information and Communications \n\nOtto-Hahn-Ring 6,81730 Munich, Germany \n\n{Thomas.Briegel, Volker.Tresp} @mchp.siemens.de \n\nAbstract \n\nWe  present Monte-Carlo generalized EM  equations for learning in  non(cid:173)\nlinear state space models.  The difficulties lie in the Monte-Carlo E-step \nwhich consists of sampling from  the posterior distribution of the hidden \nvariables given the observations. The new idea presented in this paper is \nto generate samples from a Gaussian approximation to the true posterior \nfrom  which it is easy to obtain independent samples.  The parameters  of \nthe Gaussian approximation are either derived from the extended Kalman \nfilter or the Fisher scoring algorithm. In case the posterior density is mul(cid:173)\ntimodal  we propose to  approximate the posterior by  a sum of Gaussians \n(mixture of modes approach).  We show that sampling from the approxi(cid:173)\nmate posterior densities obtained by the above algorithms leads to better \nmodels  than  using  point estimates  for  the  hidden  states.  In  our exper(cid:173)\niment,  the Fisher scoring algorithm obtained  a better approximation of \nthe posterior mode than the EKF. For a multimodal distribution, the mix(cid:173)\nture of modes approach gave superior results. \n\n1  INTRODUCTION \nNonlinear state space models (NSSM) are a general framework  for representing nonlinear \ntime series.  In particular, any NARMAX model (nonlinear auto-regressive moving average \nmodel  with external inputs) can be translated into an  equivalent NSSM. Mathematically, a \nNSSM is described by the system equation \n\n(1) \n\nwhere Xt  denotes a hidden state variable, (t denotes zero-mean uncorrelated Gaussian noise \nwith  covariance  Qt  and  Ut  is  an  exogenous  (deterministic) input vector.  The time-series \nmeasurements  Yt  are  related  to  the  unobserved  hidden  states  Xt  through the observation \nequation \n\n(2) \nwhere  Vt  is  uncorrelated Gaussian  noise with covariance  lit.  In  the following we  assume \nthat  the  nonlinear mappings  fw{.)  and  gv{.)  are  neural  networks  with  weight  vectors  w \nand  v,  respectively.  The initial state  Xo  is  assumed  to be Gaussian  distributed with mean \nao  and  covariance  Qo.  All  variables are in general multidimensional.  The two challenges \n\n\f404 \n\nT  Briegel and V.  Tresp \n\nin NSSMs  are  the interrelated tasks of inference and  learning.  In  inference we try  to es(cid:173)\ntimate the states of unknown variables  Xs  given  some measurements  Yb ... , Yt  (typically \nthe states of past (s  < t), present (s  =  t) or future (s  > t)  values of Xt)  and in learning we \nwant to adapt some unknown parameters in the model  (i.e.  neural  network weight vectors \nwand v)  given  a set  of measurements.1  In the  special  case  of linear state space  models \nwith Gaussian  noise, efficient algorithms for inference and  maximum likelihood learning \nexist.  The  latter can  be  implemented  using EM  update equations in  which the E-step  is \nimplemented  using  forward-backward  Kalman  filtering  (Shumway  &  Stoffer,  1982).  If \nthe system is nonlinear,  however,  the problem of inference and learning leads to complex \nintegrals which  are  usually considered  intractable (Anderson &  Moore,  1979).  A  useful \napproximation  is  presented  in  section  3  where  we  show  how  the  learning  equations  for \nNSSMs can be implemented using two steps which are repeated until convergence.  First in \nthe (Monte-Carlo) E-step, random samples are generated from the unknown variables (e.g. \nthe hidden variables Xt)  given the measurements.  In the second step (a generalized M-step) \nthose  samples  are  treated  as  real  data  and  are  used  to  adapt  Iw (.)  and  gv (.)  using  some \nversion of the backpropagation algorithm. The problem lies in the first step, since it is dif(cid:173)\nficult to generate independent samples from a general multidimensional distribution. Since \nit is difficult to generate samples from the proper distribution the next best thing might be \nto  generate  samples  using an  approximation to the  proper distribution which  is  the  idea \npursued  in  this  paper.  The  first  thing  which might come  to mind  is  to  approximate  the \nposterior distribution of the hidden variables by  a multidimensional Gaussian distribution \nsince generating samples from such a distribution is simple. In the first approach we use the \nextended Kalman filter and smoother to obtain mode and covariance ofthis Gaussian. 2  Al(cid:173)\nternatively, we estimate the mode and the covariance of the posterior distribution using an \nefficient implementation of Fisher scoring derived by Fahrmeir and  Kaufmann (1991) and \nuse those as parameters of the Gaussian.  In some cases the approximation of the posterior \nmode by  a single Gaussian might be considered too crude.  Therefore,  as  a third solution, \nwe  approximate the posterior distribution by  a  sum  of Gaussians  (mixture of modes  ap(cid:173)\nproach).  Modes and covariances of those Gaussians are obtained using the Fisher scoring \nalgorithm.  The  weights of the Gaussians are derived from  the likelihood of the  observed \ndata given the individual Gaussian.  In the following section we derive the gradient of the \nlog-likelihood with respect  to the weights  in I w (.)  and  gv (.).  In  section  3,  we show that \nthe network weights can  be updated using a Monte-Carlo E-step and a generalized M-step. \nFurthermore, we derive the different Gaussian approximations to the posterior distribution \nand introduce the mixture of modes approach.  In section 4 we validate our algorithms using \na standard nonlinear stochastic time-series model. In section 5 we present conclusions. \n\n2  THE GRADIENTS FOR NONLINEAR STATE SPACE MODELS \n\nGiven  our  assumptions  we  can  write the joint probability of the  complete data for  t \n1, ... , T  as3 \n\np(Xr, Yr, Ur) = p(Ur) p(xo) II p(Xt IXt-l. ut} II p(Yt IXt, ut} \n\nr \n\nr \n\n(3) \n\nt=1 \n\nt=l \n\n1 In  this paper we focus  on the case s  :::;  t (smoothing and offline learning, respectively). \n2Independently from  our work,  a single Gaussian  approximation to  the  E-step  using the  EKFS \nhas  been proposed by Ghahramani  &  Roweis  (1998) for  the  special case of a RBF network.  They \nshow  that  one obtains  a closed  form  M-step  when just  adapting  the  linear parameters by  holding \nthe nonlinear parameters fixed.  Although avoiding sampling, the computational load of their M-step \nseems to be significant. \n\n3In  the  following,  each  probability density is  conditioned on  the  current model.  For notational \n\nconvenience, we do not indicate this fact explicitly. \n\n\fFisher Scoring and Mixture of Modes for Inference and Learning in NSSM \n405 \nwhere UT  = {Ul,\"\"  UT}  is a set  of known  inputs which means that p( UT)  is irrelevant \nin  the following.  Since only YT  =  {Yl,\"\"  YT}  and UT  are observed,  the log-likelihood \nof the model is \n\nlog L = log J  p(XT, YTIUT)p(UT) dXT  ex  log J  p(XT, YTIUT ) dXT \n\n(4) \n\nwith  XT  =  {xo, ... , XT}.  By  inserting the  Gaussian  noise  assumptions  we  obtain  the \ngradients of the log-likelihood with respect  to the neural  network weight vectors wand v, \nrespectively (Tresp & Hofmann,  1995) \n\n810gL \n\n8w \n\n810gL \n\n8v \n\n8w \n\nT \n~J!8fw(Xt-l,Ut)( \nex  L.J \nt=1 \n~J8gv(Xt'Ut)( \nex  L.J \nt=1 \n\n8v \n\nXt  -fw(Xt-llUd  p(Xt,Xt-l YT,UT)dxt-ldxt \n\n) \n\nI \n\nYt-gv(Xt,ut}  p(Xt!YT,UT)dxt. \n\n) \n\n(5) \n\naIogL \n\n8w \n\naIogL \n\n8 \nv \n\nS  t=1  s=1 \n5 \n\nT \n\n3  APPROXIMATIONS TO THE E-STEP \n3.1  Monte-Carlo Generalized EM Learning \n\nThe integrals in the previous equations can be solved using Monte-Carlo integration which \nleads to the following learning algorithm. \n\n1.  Generate  S samples  {xo, ... , xr };=1  from P(XT \\YT , UT ) assuming the current \n2.  Treat those samples as  real  data and update wnew  =  wold + 1] &~! Land v new  = \n\nmodel is correct (Monte-Carlo E-Step). \n\nvoid  + 1]&I~~L with stepsize 1]  and \n\nT \n\n5 \n\nex  ~2:2:8fw(Xt-l,udl \n\n8w \n\nXt-l=t:_ 1 \n\n(x:-fW(X:_l,ud)  (6) \n\n~~~8gv (Xt,Ut)1 \n\nex  SL.JL.J \nt=1  s=1 \n\n8 \nV \n\nXt=i:; \n\n(_   (\"S \nYt \n\ngv  Xt,Ut \n\n)) \n\n(7) \n\n(generalized M-step).  Go back to step one. \n\nThe  second  step  is  simply  a  stochastic  gradient  step.  The  computational  difficulties  lie \nin  the first  step.  Methods which  produce samples  from  multivariate distributions such  as \nGibbs sampling and  other Markov  chain  Monte-Carlo methods have  (at  least) two prob(cid:173)\nlems.  First, the sampling process has  to \"forget\" its initial condition which means that the \nfirst  samples have  to  be discarded and there are no simple analytical tools available to de(cid:173)\ntermine how  many  samples must  be  discarded .  Secondly,  subsequent samples are  highly \ncorrelated which means that many samples have to be generated before a sufficient amount \nof independent  samples  is  available.  Since  it  is  so  difficult  to  sample  from  the  correct \nposterior distribution p(XT !YT, UT)  the  idea in this paper is to generate samples from an \napproximate distribution from  which  it  is easy  to draw  samples.  In  the  next sections  we \npresent approximations using a multivariate Gaussian and a mixture of Gaussians. \n\n3.2  Approximate Mode Estimation Using the Extended Kalman Filter \n\nWhereas  the  Kalman  filter is  an  optimal  state estimator for  linear state space  models  the \nextended Kalman filter is a suboptimal state estimator for NSSMs based on locallineariza(cid:173)\ntions of the nonlinearities.4  The extended Kalman filter and smoother (EKFS) algorithm is \n\n4 Note that we  do  not include the parameters in  the  NSSM as  additional states to be estimated as \n\ndone by other authors, e.g.  Puskorius &  Feldkamp (1994). \n\n\f406 \n\nT.  Briegel and V.  Tresp \n\na forward-backward algorithm and can  be derived as  an  approximation to posterior mode \nestimation for Gaussian error sequences (Sage & Melsa, 1971). Its application to our frame(cid:173)\nwork amounts to approximating x~ode ~ x~KFS where  x~KFS is  the smoothed estimate \nof Xt  obtained from forward-backward extended Kalman filtering over the set of measure(cid:173)\nments YT  and x~ode is the mode of the posterior distribution p( Xt I YT , UT).  We use x~KFS \nas  the center of the  approximating Gaussian.  The EKFS  also  provides an  estimate of the \nerror covariance of the state vector at each time step t which can be used to form the covari(cid:173)\nance matrix of the approximating Gaussian. The EKFS equations can be found in Anderson \n& Moore (1979). To generate samples we recursively apply the following algorithm. Given \nxLI  is a sample from  the Gaussian approximation of p(xt-IIYT, UT)  at time t  - 1 draw \na  sample xt  from  p(XtIXt-1  =  X:_I' YT, UT).  The last  conditional density is  Gaussian \nwith mean  and covariance calculated from the EKFS  approximation and  the lag-one error \ncovariances derived in Shumway & Stoffer (1982), respectively. \n\n3.3  Exact Mode Estimation Using the Fisher Scoring Algorithm \n\nIf the  system  is  highly  nonlinear,  however,  the  EKFS  can  perform  badly  in  finding  the \nposterior mode due to the fact  that it uses a first  order Taylor series expansion of the non(cid:173)\nlinearities fw (.)  and gv(.)  (for an illustration, see Figure 1).  A u!:>cful- and computationally \ntractable - alternative to the EKFS is to compute the \"exact\" posterior mode by maximizing \nlogp(XT IYrr, UT)  with  respect  to XT.  A  suitable way  to determine a stationary point of \nthe log posterior, or equivalently, of P(XT, YTIUT)  (derived from  (3) by dropping P(UT)) \n. \nIS  to  app y  rlS  er scormg. \nt  e  current  estImate  T'  we  get  a  etter  estImate \nX~s,new =  X;S,old + 1] J for the unknown state sequence XT  where J is the solution of \n\n.  W'  h  h \n\nX FS  old \n\nI  D'  h \n\nb \n\n. \n\nIt \n\n. \n\n(8) \n\nT \n\nT \n\nwith the  score function  s(XT )  =  alogp(::~YTIUT)  and  the expected  information matrix \nS(XT)  = E[_a210~1X;{fIUT'J.5  By  extending  the  arguments  given  in  Fahrmeir  & \nKaufmann  (1991) to nonlinear state space models it turns out that solving equation  (8) -\ne.g.  to  compute  the  inverse  of the expected  information  matrix - can  be  performed  by \nCholesky decomposition in one forward and backward pass.6  The forward-backward steps \ncan  be implemented as  a fast  EKFS-Iike algorithm which has  to  be  iterated to obtain the \nmaximum posterior estimates x~ode = x;S  (see Appendix).  Figure 1 shows the estimate \nobtained by  the  Fisher scoring procedure for  a  bimodal  posterior density.  Fisher scoring \nis  successful  in  finding  the  \"exact\"  mode,  the  EKFS  algorithm  is  not.  Samples  of the \napproximating Gaussian are generated in the same way as  in the last section. \n\n3.4  The Mixture of Modes Approach \n\nThe previous two approaches to posterior mode smoothing can  be viewed as  single Gaus(cid:173)\nsian  approximations of the  mode  of p(XTIYT, UT).  In  some cases  the  approximation of \nthe posterior density by  a  single Gaussian  might be  considered too crude,  in  particular if \nthe posterior distribution is multimodal.  In this section we approximate the posterior by a \nweighted sum of m  Gaussians p(XT IYT, UT)  ~ :E~I okp(XT Ik)  where p(XT Ik) is the \nk-th Gaussian.  If the individual Gaussians model the different modes we are able to model \nmultimodal posterior distributions accurately. The approximations of the individual modes \nare local maxima of the Fisher scoring algorithm which are f~)Und by starting the algorithm \nusing different initial conditions. Given the different Gaussians, the optimal weighting fac(cid:173)\ntors  are  ok  = p(YTlk)p(k)jp(YT)  where  p(YTlk)  = jp(YTIXT)P(XTlk)dXT  is  the \nSNote that the difference between the Fisher scoring and  the Gauss-Newton update is that in the \n\nfonner we take the expectation of the information matrix. \n\n6The expected information matrix  is a positive definite blOCk-tridiagonal matrix. \n\n\fFisher Scoring and Mixture of Modes for Inference and Learning in NSSM \n\n407 \n\nlikelihood of the data given mode k.  If we approximate that integral by inserting the Fisher \nscoring solutions x;S,k  for each time step t  and  linearize the nonlinearity gv (.)  about the \nFisher scoring solutions, we  obtain a closed form solution for computing the ok (see Ap(cid:173)\npendix). The resulting estimator is a weighted sum of the m single Fisher scoring estimates \nx~M = L::=l ok x;s,k.  The mixture of modes algorithm can  be found in the Appendix. \nFor the learning task samples of the mixture of Gaussians are based on samples of each of \nthe m  single Gaussians which are obtained the same way as in subsection 3.2. \n\n4  EXPERIMENTAL RESULTS \nIn the first experiment we want to test how well the different approaches can approximate \nthe posterior distribution of a nonlinear time series (inference).  As a time-series model  we \nchose \n\n( \n\n) \n\n1 \n\n2 \n\nXt-l \n\n1 + x t _ 1 \n\nf(Xt-l, Ut}  = 0.5 Xt-l + 25 \n\n2  + 8eas  1.2(t -I},  g(xt}  = 20Xt, \n\n(9) \nthe  covariances  Qt  = 10,  lit  = 1 and  initial  conditions ao  = 0 and  Qo  = 5  which  is \nconsidered a hard inference problem (Kitagawa,  1987). At each time step we calculate the \nexpected  value of the hidden  variables  Xt, t  =  1, ... , 400 based  on a set of measurements \nY400  =  {Yl, ... , Y400}  (which is the optimal estimator in the mean squared sense) and based \non the different approximations presented in the last section.  Note that for the single mode \napproximation, x~ode is the best estimate of Xt  based on the approximating Gaussian.  For \nthe mixture of modes approach, the best estimate is L:~l ok x;S,k  where x;S,k  is the mode \nof the k-th Gaussian in the dimension of Xt.  Figure 2 (left) shows the mean  squared error \n(MSE) of the smoothed estimates using the different approaches.  The Fisher scoring (FS) \nalgorithm is significantly better than the EKFS approach.  In this experiment, the mixture of \nmodes (MM) approach is significantly better than both the EKFS and Fisher scoring.  The \nreason is that the posterior probability is multimodal as shown in Figure 1. \n\nIn  the  second  experiment  we  used  the  same  time-series  model  and  trained  a  neural  net(cid:173)\nwork to  approximate  fw (.)  where  all  covariances  were  assumed  to  be  fixed  and  known. \nFor  adaptation  we  used  the  learning  rules  of section  3  using  the  various  approximations \nto  the  posterior distribution of XT .  Figure 2  (right) shows  the results.  The  experiments \nshow that truly sampling from  the approximating Gaussians  gives significantly better re(cid:173)\nsults  than  using  the  expected  value  as  a point estimate.  Furthermore,  using  the  mixture \nof modes approach in conjunction with sampling gave  significantly better results than the \napproximations using a single Gaussian. When used for inference, the network trained us(cid:173)\ning the mixture of modes  approach  was  not significantly  worse  than  the true model  (5% \nsignificance level, based on 20 experiments). \n\n5  CONCLUSIONS \nIn  our paper  we  presented  novel  approaches  for  inference  and  learning in NSSMs.  The \napplication of Fisher scoring and  the mixture of modes  approach  to  nonlinear models  as \npresented  in  our paper is  new.  Also  the idea of sampling  from  an  approximation  to the \nposterior distribution of the hidden variables is presented here for the first time.  Our results \nindicate that  the Fisher scoring  algorithm gives better estimates  of the expected  value  of \nthe hidden variable than the EKFS based approximations.  Note that the Fisher scoring al(cid:173)\ngorithm is more complex in requiring typically 5 forward-backward passes instead of only \none forward-backward pass  for the EKFS  approach.  Our experiments also showed that  if \nthe posterior distribution is multi modal, the mixture of modes approach gives significantly \nbetter estimates if compared to the approaches based on a single Gaussian approximation. \nOur learning experiments  show that  it  is  important to  sample  from  the  approximate dis(cid:173)\ntributions and  that  it  is  not sufficient to  simply  substitute point estimates.  Based  on  the \n\n\f408 \n\nT.  Briegel and V.  Tresp \n\n0 .2 , - - -- -- - - - - - - - - - - - - - - - .  \n\n0.4,---------------------, \n\n0 . 18 \n\n0 . 16  \n\n0 .14  \n\n,  , \n\n,  . \n,  . \n\n0 . 35 \n\n0.3 \n\n1t 0.25 \n.8  0.2 \n1.0.15 \n\n0.05 \n\n,,- .... \n\no -'o---~~~-=-=--~o--~--~~-~ \n\n, \n\n, \n\nt =-= 295  \n\nFigure 1:  Approximations to the posterior distribution p( x t iY400, U 400) for t  =  294 and t  = \n295.  The continuous line shows the posterior distribution based on Gibbs sampling using \n1000 samples and can be considered a close approximation to the true posterior. The EKFS \napproximation (dotted) does  not converge  to  a  mode.  The Fisher scoring solution (dash(cid:173)\ndotted)  finds  the largest  mode.  The mixture of modes  approach with 50 modes  (dashed) \ncorrectly finds the two modes. \n\nsampling approach it is also possible to estimate hyperparameters (e.g.  the covariance ma(cid:173)\ntrices)  which  was  not done  in  this  paper.  The  approaches  can  also  be  extended  towards \nonline learning and estimation in various ways (e.g.  missing data problems). \n\nAppendix:  Mixture of Modes Algorithm \nThe mixture of modes estimate x~M is derived as a weighted sum of k  = 1, . .. ,m individual Fisher \nscoring (mode) estimates x;S ,k. For m  =  1 we obtain the Fisher scoring algorithm of subsection 3.3. \nFirst, one performs the set of forward recursions (t  =  1, ... , T) for each single mode estimator k. \n\n\",k \n\"\"'tit-I \nBtk \n\n=  p(~FS , k)\"'k \n\ntXt_I  \"\"'t-Ilt-I  tXt_I \n\npT( , FS ,k)+Q \nt \n\n\",k \n\"\"'t-llt-I  tXt_I \n\npT('FS ,k)(\",k \n\n\"\"'tit-I \n\n)-1 \n\n(10) \n\n(11 ) \n\n(12) \n\nk \n'Yt \n\n( ,FS ,k) \n\n+ B t \n\nk  T \n\nk \n\nSt  x t \n\n(13) \nwith the initialization  :E~lo = Qo, 'Yo  = So (X~S , k). Then, one performs the set of backward smooth(cid:173)\ning recursions (t = T , .. . , 1) \n(Dk \n\nBk:Ek \n\nBk T \n\n'Yt-l \n\n)-1 \n\nt-I \n\n:E~_1 \n0:_1  =  (  k \n\nt \n\ntit-I \n\nt-llt-l  -\n\n:Ek \nt \n(D~_d-l + B;:E~ B; T \nBkok \nDt- 1 \nt \n\n)-1  k \n\n'Yt-l  +  t \n( \nXt=Z,  St  Z \n\nt  Z \n\n-\n\n&Xt_l \n\nXt_l=Z' \n\nG  (  ) - &YdXt,Utll \n\nwith Pt(z) = 8fw(Xt_l ' U') I \na  d \nXt=Z  n \ninitialization o} = :E}'Y}.  The k  individual mode estimates x;S ,k are obtained by iterative applica(cid:173)\ntion of the update rule X~S , k  := '7 Ok  + X~S ,k  with  stepsize '7  where X~S , k =  {X~S , k, .. . ,X~S,k} \nand Ok  =  {o~ , ... , o} }.  After convergence we obtain the mixture of modes estimate as the weighted \nsum  X t  = 6k=1  0'  x t \n0) \nare computed recursively starting with  a uniform  prior O'}  =  .k  (N(xlp,:E)  stands for  a Gaussian \nwith  center p  and covariance :E  evaluated at x): \n\n:= 0'0  were O't  t  =  - ,  \"' , \n\n)  - &logp(Xr,YrIUT) I \n\n'  WIt  welg  ttng coe  Clents 0' \n\nk~FS k \n\nk  h \n\n.  h' \n\nffi  . \n\n,MM \n\n\",m \n\n. h \n\nk( \n\n&Xt \n\n&Xt \n\nk \n\nT \n\n1 \n\n-\n\nO'~+IN(Ytlgv(xfs,k, ur), nn \n\nk \n\nO't  = \n\n(14) \n\n(15) \n\n(16) \n\n(17) \n\n(18) \n\n\fFisher Scoring and Mixture of Modes for Inference and Learning in NSSM \n\n409 \n\ne \n\n8 \n\n7 \n\nf: \n\n';\"4 \n~3 \n\n0.8 \n\n0.7 \n\no.s \n\n0.5 \n\n~ \n~O.4 \n~ \n\n0.3 \n\n2 \n\n0 \n\n0.2 \n\n0 . \"1 \n\n0 \n\nFigure 2:  Left (inference):  The heights of the bars indicate the mean squared error between \nthe  true  Xt  (which  we  know since  we  simulated  the system) and  the estimates using the \nvarious approximations.  The error bars show the standard deviation derived from  20 rep(cid:173)\netitions of the experiment.  Based on the paired t-test, Fisher scoring is significantly better \nthan the EKFS and all mixture of modes approaches are significantly better than both EKFS \nand Fisher scoring based  on a  1 % rejection region.  The mixture of modes approximation \nwith 50 modes (MM 50) is significantly better than the approximation using 20 modes. The \nimprovement of the approximation using 20 modes (MM 20) is not significantly better than \nthe approximation with 10 (MM 10) modes using a 5% rejection region. \nRight (learning):  The heights of the bars indicate the mean squared error between the true \nfw (.)  (which is known) and the approximations using a multi-layer perceptron with 3 hid(cid:173)\nden units and T  =  200.  Shown are results using the EKFS approximation, (left) the Fisher \nscoring approximation (center) and the mixture of modes approximation (right). There are \ntwo bars  for each  experiment:  The left bars show results  where the expected  value of x t \ncalculated using the approximating Gaussians are used as  (single) samples for the general(cid:173)\nized M-step - in other words - we use a point estimate for Xt.  Using the point estimates, the \nresults of all three approximations are not significantly different based on a 5% significance \nlevel.  The right  bars shows the result where S  =  50 samples  are  generated  for approxi(cid:173)\nmating the gradient using the Gaussian approximations. The results using sampling are all \nsignificantly better than the results using point estimates (l % significance level).  The sam(cid:173)\npling approach  using the mixture of modes approximation is significantly better than  the \nother two sampling-based approaches (l % significance level).  If compared to the inference \nresults of the experiments shown on the left, we achieved a mean squared error of 6.02 for \nthe mixture of modes  approach  with  10 modes  which is  not significantly  worse than  the \nresults the with the true model of 5.87 (5% significance level). \n\nReferences \n\nAnderson, B.  and Moore, J.  (1979) Optimal Filtering, Prentice-Hall, New Jersey. \nFahnneir, L.  and Kaufmann, H. (1991) On  Kalman  Filtering.  Posterior Mode Estimation and Fisher \nScoring in Dynamic Exponential Family Regression, Metrika, 38, pp.  37-60. \nGhahramani, Z.  and Roweis,  S.  (1999) Learning Nonlinear Stochastic  Dynamics using  the Gener(cid:173)\nalized EM ALgorithm,  Advances in  Neural Infonnation Processing Systems  11, eps.  M.  Keams, S. \nSolla, D. Cohn, MIT Press, Cambridge, MA. \nKitagawa, G.  (1987) Non-Gaussian State Space Modeling of Nonstationary Time Series (with  Com(cid:173)\nments), JASA 82, pp.  1032-1063. \nPuskorius, G.  and Feldkamp, L.  (1994) NeurocontroL of Nonlinear Dynamical Systems with KaLman \nFilter Trained Recurrent Networks, IEEE Transactions on Neural Networks, 5:2, pp.  279-297. \nSage, A.  and Melsa, J. (1971) Estimation Theory with Applications to Communications and Control, \nMcGraw-Hill, New York. \nShumway, R.  and Stoffer,  D. (1982)  Time  Series Smoothing and Forecasting  Using  the  EM Algo(cid:173)\nrithm, Technical Report No.  27, Division of Statistics, UC Davis. \nTresp,  V.  and  Hofmann,  R.  (1995)  Missing  and Noisy Data  in  NonLinear  Time-Series  Prediction, \nNeural Networks for Signal Processing 5, IEEE Sig.  Proc. Soc., pp.  1-10. \n\n\f", "award": [], "sourceid": 1539, "authors": [{"given_name": "Thomas", "family_name": "Briegel", "institution": null}, {"given_name": "Volker", "family_name": "Tresp", "institution": null}]}