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function [w, V, invV, logdetV, E_a, L,predprob,i] = bayes_logit_fit_nonzero_mean(X, y, muprior, a0, b0, max_iter, criterion) % BAYES_LOGIT_FIT(X, y, V_prior) returns parpameters of a fitted logit % model p(y = 1 | x, w) = 1 / (1 + exp(- w' * x)). % The arguments are: % X - input matrix, inputs x as row vectors % y - output vector, containing either 1 or -1 % The function returns the posterior p(w1 | X, y) = N(w1 | w, V), and % additionally the inverse of V and ln|V| (just in case). The prior on % p(w1) is determined by assigning it a hyperprior p(w1_i | a_i) = % N(w1_i | muprior_i, a_i^-1) on each of its elements, with % a_i = Gam(a_i | a0, b0), with parameters that make it % uninformative. The returned vector E_a is the expectations of a_i. % L is the final variational bound, which is a lower bound on the log-model % evidence. % hyperprior parameters if nargin<3 a0 = 1e-2; b0 = 1e-4; end; % equations from Bishop (2006) PRML Book + errata (!) + new stuff % constants [num_datapts, dim_x] = size(X); %max_iter = 100; an = a0 + 0.5; % start first iteration kind of here, with xi = 0 -> lam_xi = 1/8 % xi=ones(N, 1)*10; % % lam_xi = lam(xi); %ones(N, 1) / 8; % E_a = ones(dim_x, 1) * a0 / b0; % w_fixedpart = 0.5 * sum(X .* repmat(y, 1, dim_x), 1)'; % invV = diag(E_a) + 2 * X' * (X .* repmat(lam_xi, 1, dim_x));%inv(sigmaprior); % % V = inv(invV); % sigmaprior; % % w = V * w_fixedpart; %V*diag(E_a)*muprior + V * w_fixedpart % bn = b0 + 0.5 * (w .^ 2 + diag(V)); % L_last = - sum(log(1 + exp(- xi))) + sum(lam_xi .* xi .^ 2) ... % + 0.5 * (w' * invV * w - logdet(invV) - sum(xi)) ... % + dim_x * (gammaln(an) + an) - sum((b0 * an) ./ bn) - sum(an * log(bn)); xi=ones(num_datapts, 1)*0.1; lam_xi = lam(xi); %ones(num_datapts, 1) / 8; E_a = ones(dim_x, 1) * a0 / b0; w_fixedpart = 0.5 * sum(X .* repmat(y, 1, dim_x), 1)'; invV = diag(E_a) + 2 * X' * (X .* repmat(lam_xi, 1, dim_x));% + inv(sigmaprior)*eye(size(sigmaprior)); %inv(sigmaprior); % V = inv(invV); % sigmaprior; % w = V * (diag(E_a) * muprior + w_fixedpart); %V*diag(E_a)*muprior + V * w_fixedpart bn = b0 + 0.5 * ((w-muprior) .^ 2 +diag(V));% L_last = - sum(log(1 + exp(- xi))) + sum(lam_xi .* xi .^ 2) ... + 0.5 * (w' * invV * w - logdet(invV) - sum(xi)) ... + dim_x * (gammaln(an) + an) - sum((b0 * an) ./ bn) - sum(an * log(bn)); for i = 1:max_iter; %fprintf('i in loop: %d\n', i); % update xi by EM-algorithm xi = sqrt(sum(X .* (X * (V + w * w')), 2)); lam_xi = lam(xi); % update posterior parameters of a based on xi bn = b0 + 0.5 * ((w-muprior) .^ 2 +diag(V));% E_a = an ./ bn; % recompute posterior parameters of w invV = diag(E_a) + 2 * X' * (X .* repmat(lam_xi, 1, dim_x)); V = inv(invV); logdetV = - logdet(invV); w = V * (diag(E_a) * muprior + w_fixedpart); %V V*diag(E_a)*muprior + * w_fixedpart; % variational bound L = - sum(log(1 + exp(- xi))) + sum(lam_xi .* xi .^ 2) ... + 0.5 * (w' * invV * w + logdetV - sum(xi)) ... + dim_x * (gammaln(an) + an) - sum(b0 * E_a) - sum(an * log(bn)); % either stop if variational bound grows or change is < 0.001% % HACK ALARM: theoretically, the bound should never grow, and it doing % so points to numerical instabilities. As it seems, these start to % occur close to the optimal bound, which already points to a good % approximation. % if (L_last > L) %& i>5 % fprintf('warning: variational bound should grow, L_last > L, i=%d\n',i); % % error('warning: variational bound should grow, L_last > L'); % % break % end if abs(L_last - L) < abs(criterion * L) & i>20 break end L_last = L; Lmat(i) = L; end; % fprintf('final i: %d\n', i); if i == max_iter warning('Bayes:maxIter', ... 'Bayesian logistic regression reached maximum number of iterations.'); end % add constant terms to variational bound L = L - dim_x * (gammaln(a0) + a0 * log(b0)); predprob = exp(L); function out = lam(xi) % returns 1 / (4 * xi) * tanh(xi / 2) divby0_w = warning('query', 'MATLAB:divideByZero'); warning('off', 'MATLAB:divideByZero'); out = tanh(xi ./ 2) ./ (4 .* xi); warning(divby0_w.state, 'MATLAB:divideByZero'); % fix values where xi = 0 out(isnan(out)) = 1/8;
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