wolffd@0: function [net, gamma, logev] = evidence_weighted(net, x, t, eso_w, num)
wolffd@0: %EVIDENCE Re-estimate hyperparameters using evidence approximation.
wolffd@0: %
wolffd@0: %	Description
wolffd@0: %	[NET] = EVIDENCE(NET, X, T) re-estimates the hyperparameters ALPHA
wolffd@0: %	and BETA by applying Bayesian re-estimation formulae for NUM
wolffd@0: %	iterations. The hyperparameter ALPHA can be a simple scalar
wolffd@0: %	associated with an isotropic prior on the weights, or can be a vector
wolffd@0: %	in which each component is associated with a group of weights as
wolffd@0: %	defined by the INDEX matrix in the NET data structure. These more
wolffd@0: %	complex priors can be set up for an MLP using MLPPRIOR. Initial
wolffd@0: %	values for the iterative re-estimation are taken from the network
wolffd@0: %	data structure NET passed as an input argument, while the return
wolffd@0: %	argument NET contains the re-estimated values.
wolffd@0: %
wolffd@0: %	[NET, GAMMA, LOGEV] = EVIDENCE(NET, X, T, NUM) allows the re-
wolffd@0: %	estimation  formula to be applied for NUM cycles in which the re-
wolffd@0: %	estimated values for the hyperparameters from each cycle are used to
wolffd@0: %	re-evaluate the Hessian matrix for the next cycle.  The return value
wolffd@0: %	GAMMA is the number of well-determined parameters and LOGEV is the
wolffd@0: %	log of the evidence.
wolffd@0: %
wolffd@0: %	See also
wolffd@0: %	MLPPRIOR, NETGRAD, NETHESS, DEMEV1, DEMARD
wolffd@0: %
wolffd@0: 
wolffd@0: %	Copyright (c) Ian T Nabney (1996-9)
wolffd@0: 
wolffd@0: errstring = consist(net, '', x, t);
wolffd@0: if ~isempty(errstring)
wolffd@0:   error(errstring);
wolffd@0: end
wolffd@0: 
wolffd@0: ndata = size(x, 1);
wolffd@0: if nargin == 4
wolffd@0:   num = 1;
wolffd@0: end
wolffd@0: 
wolffd@0: if isfield(net,'beta')
wolffd@0:     beta = net.beta;
wolffd@0: else 
wolffd@0:     beta = 1;
wolffd@0: end;
wolffd@0: 
wolffd@0: % Extract weights from network
wolffd@0: pakstr = [net.type, 'pak'];
wolffd@0: w = feval(pakstr, net);
wolffd@0: 
wolffd@0: % Evaluate data-dependent contribution to the Hessian matrix.
wolffd@0: [h, dh] = nethess_weighted(w, net, x, t, eso_w); 
wolffd@0: 
wolffd@0: % Now set the negative eigenvalues to zero.
wolffd@0: [evec, evl] = eig(dh);
wolffd@0: evl = evl.*(evl > 0);
wolffd@0: % safe_evl is used to avoid taking log of zero
wolffd@0: safe_evl = evl + eps.*(evl <= 0);
wolffd@0: 
wolffd@0: % Do the re-estimation. 
wolffd@0: for k = 1 : num
wolffd@0:   [e, edata, eprior] = neterr_weighted(w, net, x, t, eso_w);
wolffd@0:   h = nethess_weighted(w, net, x, t, eso_w, dh);
wolffd@0:   % Re-estimate alpha.
wolffd@0:   if size(net.alpha) == [1 1]
wolffd@0:     % Evaluate number of well-determined parameters.
wolffd@0:     if k == 1
wolffd@0:       % Form vector of eigenvalues
wolffd@0:       evl = diag(evl);
wolffd@0:       safe_evl = diag(safe_evl);
wolffd@0:     end
wolffd@0:     B = beta*evl;
wolffd@0:     gamma = sum(B./(B + net.alpha));       
wolffd@0:     net.alpha = 0.5*gamma/eprior;
wolffd@0:        
wolffd@0:     % Partially evaluate log evidence
wolffd@0:     logev = e - 0.5*sum(log(safe_evl)) + 0.5*net.nwts*log(net.alpha) - ...
wolffd@0:       0.5*ndata*log(2*pi);
wolffd@0:   else
wolffd@0:     ngroups = size(net.alpha, 1);
wolffd@0:     gams = zeros(1, ngroups);
wolffd@0:     logas = zeros(1, ngroups);
wolffd@0:     traces = zeros(1, ngroups);
wolffd@0:     % Reconstruct data hessian with negative eigenvalues set to zero.
wolffd@0:     dh = evec*evl*evec';
wolffd@0:     hinv = inv(nethess_weighted(w, net, x, t, eso_w, dh));
wolffd@0:     for m = 1 : ngroups
wolffd@0:       group_nweights = sum(net.index(:, m));
wolffd@0:       gams(m) = group_nweights - ...
wolffd@0: 	        net.alpha(m)*sum(diag(hinv).*net.index(:,m));
wolffd@0:       net.alpha(m) = real(gams(m)/(2*eprior(m)));
wolffd@0:       % Weight alphas by number of weights in group
wolffd@0:       logas(m) = 0.5*group_nweights*log(net.alpha(m));
wolffd@0:       % Compute sum of evalues corresponding to group
wolffd@0:       traces(m) = sum(log(safe_evl*net.index(:,m)));
wolffd@0:     end 
wolffd@0:     gamma = sum(gams, 2);
wolffd@0:     logev = e - 0.5*sum(traces) + sum(logas) - 0.5*ndata*log(2*pi);
wolffd@0:   end
wolffd@0:   % Re-estimate beta.
wolffd@0:   if isfield(net, 'beta')
wolffd@0:       net.beta = 0.5*(net.nout*ndata - gamma)/edata;
wolffd@0:   end
wolffd@0:   logev = logev + 0.5*ndata*log(beta);
wolffd@0: end
wolffd@0: