Backup

   Backup for a given belief (continuous state version).

   Belief backup function for POMDP defined on continuous state spaces.
   The backup operator is introduced in Section 4 (page 12) but without any
   assumption on the type of state space.
   This function corresponds to the particularization of the backup for
   continuous state spaces given on Section 5.2 of the
   paper (pages 15 and 16). 

   This function basically computes one Alpha-element for each action and
   selects the one that gives higher value for the given belief.

   Parameters:
     P: The POMDP
     b: The belief
     V: The previous policy/value function/set of alpha-elements.
     Alphas_j_a_o: The set of pre-compute alpha_j_a_o elements. If empty,
                   the alpha_j_a_o elements are computed on the fly.

0001 function [alpha a v]=Backup(P,b,V,Alphas_j_a_o)
0002 %   Backup for a given belief (continuous state version).
0003 %
0004 %   Belief backup function for POMDP defined on continuous state spaces.
0005 %   The backup operator is introduced in Section 4 (page 12) but without any
0006 %   assumption on the type of state space.
0007 %   This function corresponds to the particularization of the backup for
0008 %   continuous state spaces given on Section 5.2 of the
0009 %   paper (pages 15 and 16).
0010 %
0011 %   This function basically computes one Alpha-element for each action and
0012 %   selects the one that gives higher value for the given belief.
0013 %
0014 %   Parameters:
0015 %     P: The POMDP
0016 %     b: The belief
0017 %     V: The previous policy/value function/set of alpha-elements.
0018 %     Alphas_j_a_o: The set of pre-compute alpha_j_a_o elements. If empty,
0019 %                   the alpha_j_a_o elements are computed on the fly.
0020 
0021   A=get(P,'ActionSpace'); % This is suposed to be discrete
0022   if isa(A,'CSpace')
0023     error('The Action Space must be discrete at this point');
0024   end
0025   Elements_a=arrayfun(@(a)(ComputeAlpha_a(P,V,b,a,Alphas_j_a_o)),1:dim(A),'UniformOutput',false);
0026   fprintf('a');
0027   [v na]=max(cellfun(@(ea)(Expectation(b,ea)),Elements_a));
0028   fprintf('c');
0029   alpha=Compress(Elements_a{na},P.maxAlphaC);
0030   v=Expectation(b,alpha);
0031   a=A(na);