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| Original file line number | Diff line number | Diff line change |
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| % This demo re-visits the mountain car problem to show that adaptive | ||
| % (desired) behaviour can be prescribed in terms of loss-functions (i.e. | ||
| % reward functions of state-space). | ||
| % It exploits the fact that under the free-energy formulation, loss is | ||
| % divergence. This means that priors can be used to make certain parts of | ||
| % state-space costly (i.e. with high divergence) and others rewarding (low | ||
| % divergence). Active inference under these priors will lead to sampling of | ||
| % low cost states and (apparent) attractiveness of those states. | ||
| % | ||
| % This is version four; that includes a drive state. | ||
| %__________________________________________________________________________ | ||
| % Copyright (C) 2008 Wellcome Trust Centre for Neuroimaging | ||
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| % Karl Friston | ||
| % $Id: ADEM_mountaincar_loss_4.m 3333 2009-08-25 16:12:44Z karl $ | ||
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| % generative process (mountain car terrain) | ||
| %========================================================================== % switch for demo | ||
| clear | ||
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| % parameters of generative process | ||
| %-------------------------------------------------------------------------- | ||
| P.a = 0; | ||
| P.b = [0 0]; | ||
| P.c = [0 0 0 0]; | ||
| P.d = 1; % action on | ||
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| fx = inline('spm_mc_fxa_4(x,v,a,P)','x','v','a','P'); | ||
| gx = inline('[x.x; x.v; x.d]','x','v','a','P'); | ||
| x0.x = 0; | ||
| x0.v = 0; | ||
| x0.p = 0; | ||
| x0.d = 0; | ||
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| % level 1 | ||
| %-------------------------------------------------------------------------- | ||
| G(1).x = x0; | ||
| G(1).f = fx; | ||
| G(1).g = gx; | ||
| G(1).pE = P; | ||
| G(1).V = exp(16); % error precision | ||
| G(1).W = exp(16); % error precision | ||
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| % level 2 | ||
| %-------------------------------------------------------------------------- | ||
| G(2).a = 0; % action | ||
| G(2).v = 0; % inputs | ||
| G(2).V = exp(16); | ||
| G = spm_ADEM_M_set(G); | ||
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| % generative model | ||
| %========================================================================== | ||
| clear P x0 | ||
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| % parameters (previously learned) and equations of motion | ||
| %-------------------------------------------------------------------------- | ||
| P = [2.7 1.7 0.74 -0.51 -0.85 0.08 -0.23 -1.15]; | ||
| np = length(P); | ||
| fx = inline('spm_mc_fx_4(x,v,P)','x','v','P'); | ||
| gx = inline('[x.x; x.v; x.d]','x','v','P'); | ||
| x0.x = 0; | ||
| x0.v = 0; | ||
| x0.c = 0; | ||
| x0.d = 0; | ||
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| % level 1 | ||
| %-------------------------------------------------------------------------- | ||
| M(1).x = x0; | ||
| M(1).f = fx; | ||
| M(1).g = gx; | ||
| M(1).pE = P; | ||
| M(1).V = exp(8); % error precision | ||
| M(1).W = diag(exp([8 4 16 16])); % error precision | ||
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| % level 2 | ||
| %-------------------------------------------------------------------------- | ||
| M(2).v = 0; % inputs | ||
| M(2).V = exp(16); | ||
| M = spm_DEM_M_set(M); | ||
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| % learn gradients with a flat loss-functions (priors on divergence) | ||
| %========================================================================== | ||
| N = 1600; | ||
| U = sparse(N,M(1).m); | ||
| DEM.U = U; | ||
| DEM.C = U; | ||
| DEM.G = G; | ||
| DEM.M = M; | ||
| DEM = spm_ADEM(DEM); | ||
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| % show dynamics | ||
| %========================================================================== | ||
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| % inference | ||
| %-------------------------------------------------------------------------- | ||
| spm_figure('GetWin','Graphics'); | ||
| spm_DEM_qU(DEM.qU) | ||
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| % true and inferred position | ||
| %-------------------------------------------------------------------------- | ||
| subplot(2,2,3) | ||
| plot(DEM.pU.x{1}(1,:),DEM.pU.x{1}(2,:)),hold on | ||
| plot( 1,0,'r.','Markersize',32), hold on | ||
| plot(-1/2,0,'g.','Markersize',16), hold off | ||
| xlabel('position','Fontsize',14) | ||
| ylabel('velcitiy','Fontsize',14) | ||
| title('trajectories','Fontsize',16) | ||
| axis([-1 1 -1 1]*3) | ||
| axis square | ||
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| % true position | ||
| %-------------------------------------------------------------------------- | ||
| subplot(2,2,3) | ||
| plot3(DEM.pU.x{1}(1,:),DEM.pU.x{1}(2,:),1:N), hold on | ||
| plot3( 1,0,1:64:N,'r.','Markersize',8), hold on | ||
| plot3(-1/2,0,1:64:N,'g.','Markersize',8), hold off | ||
| xlabel('position','Fontsize',14) | ||
| ylabel('velocity','Fontsize',14) | ||
| zlabel('time','Fontsize',14) | ||
| title('trajectories','Fontsize',16) | ||
| axis([-2 2 -2 2 0 N]) | ||
| axis square | ||
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| % real states | ||
| %========================================================================== | ||
| spm_figure('GetWin','DEM'); | ||
| spm_DEM_qU(DEM.pU) | ||
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| subplot(2,2,1) | ||
| plot3( 1,0,1:1/8:2,'r.'), hold on | ||
| plot3(-1/2,0,1:1/8:2,'g.'), hold on | ||
| plot3(DEM.pU.x{1}(1,:),DEM.pU.x{1}(2,:),DEM.pU.x{1}(4,:)), hold off | ||
| xlabel('position','Fontsize',14) | ||
| ylabel('velocity','Fontsize',14) | ||
| zlabel('satiety','Fontsize',14) | ||
| title('trajectories','Fontsize',16) | ||
| axis([-2 2 -2 2 0 8]) | ||
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| % cost function (see spm_mc_fx_4.m) | ||
| %-------------------------------------------------------------------------- | ||
| subplot(2,1,2) | ||
| x = -2:1/64:2; | ||
| d = 0:1/64:2; | ||
| for i = 1:length(x) | ||
| for j = 1:length(d) | ||
| D = spm_phi((1 - d(j))*8); | ||
| A = 2 - 32*exp(-(x(i) - 1).^2*32); | ||
| C(i,j) = A*D - 1; | ||
| end | ||
| end | ||
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| surf(d,x,C) | ||
| shading interp | ||
| xlabel('drive','Fontsize',14) | ||
| ylabel('position','Fontsize',14) | ||
| title('cost-function','Fontsize',16) | ||
| axis square | ||
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| return | ||
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| % NOTES for graphics | ||
| %-------------------------------------------------------------------------- | ||
| clear C | ||
| r = 0:1/64:1; | ||
| d = 0:1/64:4; | ||
| for i = 1:length(r) | ||
| for j = 1:length(d) | ||
| D = spm_phi((1 - d(j))*8); | ||
| A = 2 - 32*r(i); | ||
| C(i,j) = A*D - 1; | ||
| end | ||
| end | ||
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| imagesc(d,r,C) | ||
| shading interp | ||
| xlabel('satiety','Fontsize',14) | ||
| ylabel('reward','Fontsize',14) | ||
| axis xy | ||
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