DiscretizeObsModel

   Discretizes the observation model on the observation side.

   Implementation of the lossless discretization of the observation model
   defined in section 6.2 of the paper. 
   The output is an observation model on a continuous state space but on a
   discrete a observation space. The dimensionality of this observation
   space is given by the number of elements in V (number of alpha elements
   in the previous iteration policy).
   This is only use in the experiment in Figure 11.

   See also MakeFigure11.

0001 function CS_DO_OM=DiscretizeObsModel(CS_CO_OM,V,ba,nSamples)
0002 %   Discretizes the observation model on the observation side.
0003 %
0004 %   Implementation of the lossless discretization of the observation model
0005 %   defined in section 6.2 of the paper.
0006 %   The output is an observation model on a continuous state space but on a
0007 %   discrete a observation space. The dimensionality of this observation
0008 %   space is given by the number of elements in V (number of alpha elements
0009 %   in the previous iteration policy).
0010 %   This is only use in the experiment in Figure 11.
0011 %
0012 %   See also MakeFigure11.
0013 
0014   nj=size(V); % number of synthetic observations
0015   oData=cell(1,nj); %Weights for the GMixtures in 's' (one for each element in 'V')
0016  
0017 
0018   % Sample the observations
0019   % Caution:
0020   %     To actually get a good discrete observation model
0021   %     we must have p(s)=\sum_o p(s|o) p(o) = ct
0022   %     This is a strong assumption make in Section 5.1, page 14.
0023   %     If this is not this way the whole system breaks.
0024   %     Even with acarefully defined p(o,s) in  the CS_CO_ObsModel
0025   %     p(s) would only be constant if we discretize the obs model with
0026   %     a uniform sampling on the observation space.
0027   %     This can only be achieved by carefully selecting the samples
0028   %     (this is what we do for one-dimensional obs. spaces) or using
0029   %     a very large number of uniformly sampled observations (that is
0030   %     what it should be done in higher dimensional observations).
0031   %     The number of observations to sample is probably so large that
0032   %     could make this approach not very useful.
0033   %
0034   if dim(CS_CO_OM.O)==1
0035     oMin=min(CS_CO_OM.O);
0036     oMax=max(CS_CO_OM.O);
0037     oStep=(oMax-oMin)/(nSamples-1);
0038     os=oMin:oStep:oMax;
0039   else
0040     DO=Discretize(CS_CO_OM.O,nSamples);
0041     os=zeros(dim(DO),nSamples);
0042     for i=1:nSamples
0043       os(:,i)=DO{i};
0044     end
0045   end
0046     
0047   nm=zeros(1,nj);
0048   po=UniformProbability(CS_CO_OM.O);  
0049 
0050   for i=1:nSamples
0051     o=os(:,i);
0052     
0053     [bao Os]=Update(CS_CO_OM,ba,o,CS_CO_OM.S);
0054     [v j]=max(Values(V,bao));
0055     
0056     if nm(j)==0
0057       oData{j}=get(Os,'w')/po;
0058     else
0059       oData{j}=oData{j}+get(Os,'w')/po;
0060     end
0061     nm(j)=nm(j)+1;
0062   end
0063   
0064   % Obs models are defined giving p(o,s) and
0065   % oData is a kind of p(o|s).
0066   % We use the fact that p(o,s)=p(o|s)*p(s)
0067   ps=UniformProbability(CS_CO_OM.S);  
0068   for i=1:nj
0069     if nm(i)==0
0070       oData{i}=GMixture();
0071     else
0072       oData{i}=GMixture(oData{i}*(ps/nSamples),CS_CO_OM.gS);
0073     end
0074   end
0075 
0076   CS_DO_OM=CS_DO_ObsModel(CS_CO_OM.S,DSpace(nj),oData);
0077