Daniel@0: function [net, gamma, logev] = evidence(net, x, t, num)
Daniel@0: %EVIDENCE Re-estimate hyperparameters using evidence approximation.
Daniel@0: %
Daniel@0: %	Description
Daniel@0: %	[NET] = EVIDENCE(NET, X, T) re-estimates the hyperparameters ALPHA
Daniel@0: %	and BETA by applying Bayesian re-estimation formulae for NUM
Daniel@0: %	iterations. The hyperparameter ALPHA can be a simple scalar
Daniel@0: %	associated with an isotropic prior on the weights, or can be a vector
Daniel@0: %	in which each component is associated with a group of weights as
Daniel@0: %	defined by the INDEX matrix in the NET data structure. These more
Daniel@0: %	complex priors can be set up for an MLP using MLPPRIOR. Initial
Daniel@0: %	values for the iterative re-estimation are taken from the network
Daniel@0: %	data structure NET passed as an input argument, while the return
Daniel@0: %	argument NET contains the re-estimated values.
Daniel@0: %
Daniel@0: %	[NET, GAMMA, LOGEV] = EVIDENCE(NET, X, T, NUM) allows the re-
Daniel@0: %	estimation  formula to be applied for NUM cycles in which the re-
Daniel@0: %	estimated values for the hyperparameters from each cycle are used to
Daniel@0: %	re-evaluate the Hessian matrix for the next cycle.  The return value
Daniel@0: %	GAMMA is the number of well-determined parameters and LOGEV is the
Daniel@0: %	log of the evidence.
Daniel@0: %
Daniel@0: %	See also
Daniel@0: %	MLPPRIOR, NETGRAD, NETHESS, DEMEV1, DEMARD
Daniel@0: %
Daniel@0: 
Daniel@0: %	Copyright (c) Ian T Nabney (1996-2001)
Daniel@0: 
Daniel@0: errstring = consist(net, '', x, t);
Daniel@0: if ~isempty(errstring)
Daniel@0:   error(errstring);
Daniel@0: end
Daniel@0: 
Daniel@0: ndata = size(x, 1);
Daniel@0: if nargin == 3
Daniel@0:   num = 1;
Daniel@0: end
Daniel@0: 
Daniel@0: % Extract weights from network
Daniel@0: w = netpak(net);
Daniel@0: 
Daniel@0: % Evaluate data-dependent contribution to the Hessian matrix.
Daniel@0: [h, dh] = nethess(w, net, x, t); 
Daniel@0: clear h;  % To save memory when Hessian is large
Daniel@0: if (~isfield(net, 'beta'))
Daniel@0:   local_beta = 1;
Daniel@0: end
Daniel@0: 
Daniel@0: [evec, evl] = eig(dh);
Daniel@0: % Now set the negative eigenvalues to zero.
Daniel@0: evl = evl.*(evl > 0);
Daniel@0: % safe_evl is used to avoid taking log of zero
Daniel@0: safe_evl = evl + eps.*(evl <= 0);
Daniel@0: 
Daniel@0: [e, edata, eprior] = neterr(w, net, x, t);
Daniel@0: 
Daniel@0: if size(net.alpha) == [1 1]
Daniel@0:   % Form vector of eigenvalues
Daniel@0:   evl = diag(evl);
Daniel@0:   safe_evl = diag(safe_evl);
Daniel@0: else
Daniel@0:   ngroups = size(net.alpha, 1);
Daniel@0:   gams = zeros(1, ngroups);
Daniel@0:   logas = zeros(1, ngroups);
Daniel@0:   % Reconstruct data hessian with negative eigenvalues set to zero.
Daniel@0:   dh = evec*evl*evec';
Daniel@0: end
Daniel@0: 
Daniel@0: % Do the re-estimation. 
Daniel@0: for k = 1 : num
Daniel@0:   % Re-estimate alpha.
Daniel@0:   if size(net.alpha) == [1 1]
Daniel@0:     % Evaluate number of well-determined parameters.
Daniel@0:     L = evl;
Daniel@0:     if isfield(net, 'beta')
Daniel@0:       L = net.beta*L;
Daniel@0:     end
Daniel@0:     gamma = sum(L./(L + net.alpha));
Daniel@0:     net.alpha = 0.5*gamma/eprior;
Daniel@0:     % Partially evaluate log evidence: only include unmasked weights
Daniel@0:     logev = 0.5*length(w)*log(net.alpha);
Daniel@0:   else
Daniel@0:     hinv = inv(hbayes(net, dh));
Daniel@0:     for m = 1 : ngroups
Daniel@0:       group_nweights = sum(net.index(:, m));
Daniel@0:       gams(m) = group_nweights - ...
Daniel@0: 	        net.alpha(m)*sum(diag(hinv).*net.index(:,m));
Daniel@0:       net.alpha(m) = real(gams(m)/(2*eprior(m)));
Daniel@0:       % Weight alphas by number of weights in group
Daniel@0:       logas(m) = 0.5*group_nweights*log(net.alpha(m));
Daniel@0:     end 
Daniel@0:     gamma = sum(gams, 2);
Daniel@0:     logev = sum(logas);
Daniel@0:   end
Daniel@0:   % Re-estimate beta.
Daniel@0:   if isfield(net, 'beta')
Daniel@0:       net.beta = 0.5*(net.nout*ndata - gamma)/edata;
Daniel@0:       logev = logev + 0.5*ndata*log(net.beta) - 0.5*ndata*log(2*pi);
Daniel@0:       local_beta = net.beta;
Daniel@0:   end
Daniel@0:   
Daniel@0:   % Evaluate new log evidence
Daniel@0:   e = errbayes(net, edata);
Daniel@0:   if size(net.alpha) == [1 1]
Daniel@0:     logev = logev - e - 0.5*sum(log(local_beta*safe_evl+net.alpha));
Daniel@0:   else
Daniel@0:     for m = 1:ngroups  
Daniel@0:       logev = logev - e - ...
Daniel@0: 	  0.5*sum(log(local_beta*(safe_evl*net.index(:, m))+...
Daniel@0: 	  net.alpha(m)));
Daniel@0:     end
Daniel@0:   end
Daniel@0: end
Daniel@0: