% approximations: Exact inference for Gaussian process classification is% intractable, and approximations are necessary. Different approximation% techniques have been implemented, which all rely on a Gaussian approximation% to the non-Gaussian posterior:%%   approxEP   the Expectation Propagation (EP) algorithm %   approxLA   Laplace's method%% which are used by the Gaussian process classification funtion binaryGP.m.% The interface to the approximation methods is the following:%%   function [alpha, sW, L, nlZ, dnlZ] = approx..(hyper, covfunc, lik, x, y)%% where:%%   hyper    is a column vector of hyperparameters%   covfunc  is the name of the covariance function (see covFunctions.m)%   lik      is the name of the likelihood function (see likelihoods.m)%   x        is a n by D matrix of training inputs %   y        is a (column) vector (of size n) of binary +1/-1 targets%   nlZ      is the returned value of the negative log marginal likelihood%   dnlZ     is a (column) vector of partial derivatives of the negative%               log marginal likelihood wrt each hyperparameter%   alpha    is a (sparse or full column vector) containing inv(K)*m, where K%               is the prior covariance matrix and m the approx posterior mean%   sW       is a (sparse or full column) vector containing diagonal of sqrt(W)%               the approximate posterior covariance matrix is inv(inv(K)+W)%   L        is a (sparse or full) matrix, L = chol(sW*K*sW+eye(n))%% Usually, the approximate posterior to be returned admits the form% N(m=K*alpha, V=inv(inv(K)+W)), where alpha is a vector and W is diagonal;% if not, then L contains instead -inv(K+inv(W)), and sW is unused.%% For more information on the individual approximation methods and their% implementations, see the separate approx??.m files. See also binaryGP.m%% Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2007-06-25.