Daniel@0: function [net, gamma, logev] = evidence_weighted(net, x, t, eso_w, 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 (1996-9)
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 == 4
Daniel@0:   num = 1;
Daniel@0: end
Daniel@0: 
Daniel@0: if isfield(net,'beta')
Daniel@0:     beta = net.beta;
Daniel@0: else 
Daniel@0:     beta = 1;
Daniel@0: end;
Daniel@0: 
Daniel@0: % Extract weights from network
Daniel@0: pakstr = [net.type, 'pak'];
Daniel@0: w = feval(pakstr, net);
Daniel@0: 
Daniel@0: % Evaluate data-dependent contribution to the Hessian matrix.
Daniel@0: [h, dh] = nethess_weighted(w, net, x, t, eso_w); 
Daniel@0: 
Daniel@0: % Now set the negative eigenvalues to zero.
Daniel@0: [evec, evl] = eig(dh);
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: % Do the re-estimation. 
Daniel@0: for k = 1 : num
Daniel@0:   [e, edata, eprior] = neterr_weighted(w, net, x, t, eso_w);
Daniel@0:   h = nethess_weighted(w, net, x, t, eso_w, dh);
Daniel@0:   % Re-estimate alpha.
Daniel@0:   if size(net.alpha) == [1 1]
Daniel@0:     % Evaluate number of well-determined parameters.
Daniel@0:     if k == 1
Daniel@0:       % Form vector of eigenvalues
Daniel@0:       evl = diag(evl);
Daniel@0:       safe_evl = diag(safe_evl);
Daniel@0:     end
Daniel@0:     B = beta*evl;
Daniel@0:     gamma = sum(B./(B + net.alpha));       
Daniel@0:     net.alpha = 0.5*gamma/eprior;
Daniel@0:        
Daniel@0:     % Partially evaluate log evidence
Daniel@0:     logev = e - 0.5*sum(log(safe_evl)) + 0.5*net.nwts*log(net.alpha) - ...
Daniel@0:       0.5*ndata*log(2*pi);
Daniel@0:   else
Daniel@0:     ngroups = size(net.alpha, 1);
Daniel@0:     gams = zeros(1, ngroups);
Daniel@0:     logas = zeros(1, ngroups);
Daniel@0:     traces = zeros(1, ngroups);
Daniel@0:     % Reconstruct data hessian with negative eigenvalues set to zero.
Daniel@0:     dh = evec*evl*evec';
Daniel@0:     hinv = inv(nethess_weighted(w, net, x, t, eso_w, 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:       % Compute sum of evalues corresponding to group
Daniel@0:       traces(m) = sum(log(safe_evl*net.index(:,m)));
Daniel@0:     end 
Daniel@0:     gamma = sum(gams, 2);
Daniel@0:     logev = e - 0.5*sum(traces) + sum(logas) - 0.5*ndata*log(2*pi);
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:   end
Daniel@0:   logev = logev + 0.5*ndata*log(beta);
Daniel@0: end
Daniel@0: