/
incr_variance_logistic.m
1094 lines (899 loc) · 32.4 KB
/
incr_variance_logistic.m
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close all;
clear;
clc;
format long
rng(9)
save_file = 1;
date = floor(clock);
%% Check if export_fig-master folder is in this current folder,
% otherwise download it
% export_fig function is used to generate high quality plots in pdf or
% other formats
if ~exist('export_fig-master', 'dir')
url = 'https://github.com/altmany/export_fig/archive/refs/heads/master.zip';
outfilename = websave([pwd,'/export_fig-master'],url);
unzip('export_fig-master.zip')
end
addpath([pwd,'/export_fig-master'])
%%
nsimul = 50;
k = 100; %starting point -- to avoid burn-in period
m = 2*k;
Lmax = 11;
lmin = 4;
Ls = 10;
alpha = 0.9;
mcmc_maxiter = 1000;
sig = 3;
kap = sig^2/2;
M = 5200;
%% Logistic regression
n = 100;
d = 5;
% generate x
x = (rand(d,n)<.5)*2 - 1;
x = zscore(x,0,2);
inv_sigb = (1/n)*(x*x');
sigb = inv(inv_sigb);
mu = rand(1,d);
% generate y
p = 1./(1+exp(-mu*x));
y = binornd(1,p); %1xn
% term used in grad U
const = sum(y.*x,2)';
%% ML sampler
Al_h = zeros(Lmax-lmin,d,M);
parfor h = 1 :M
uf0 = [mvnrnd(zeros(1,d),sigb),mvnrnd(zeros(1,d),sig^2/kap * eye(d))];
% xf0_tild, vf0_tild
uf0_tilde=[mvnrnd(zeros(1,d),sigb),mvnrnd(zeros(1,d),sig^2/kap*eye(d))];
% xc0, vc0
uc0 = [mvnrnd(zeros(1,d),sigb),mvnrnd(zeros(1,d),sig^2/kap * eye(d))];
% xc0_tild, vc0_tild
uc0_tilde=[mvnrnd(zeros(1,d),sigb),mvnrnd(zeros(1,d),sig^2/kap*eye(d))];
Al = zeros(Lmax-lmin,d);
for l = lmin+1 : Lmax
[Al(l-lmin,:), ~] = pi_phi_coupled(d,mcmc_maxiter, uf0, ...
uf0_tilde, uc0,uc0_tilde, l,sig,...
kap,alpha,k,m,x,const,inv_sigb,Ls);
end
Al_h(:,:,h) = Al;
end
%%
coord = 1;
avg = mean(Al_h(:,coord,:),3);
second_moment_incr = mean(Al_h(:,coord,:).^2,3);
LL = (lmin+1:Lmax)';
%% Fit
beta1 = 1;
beta2 = 2;
%%% Variance
%%% fit the variance
log2_var = log2(second_moment_incr);
Polv = polyfit(LL, log2_var,1);
secondM_incr_fit2 = 2.^(Polv(1) *LL + Polv(2)); %Cv2 * 2.^(-beta2*LL);
% plot
set(0, 'defaultLegendInterpreter','latex');
set(0, 'defaultTextInterpreter','latex');
%set the background of the figure to be white
set(0,'defaultfigurecolor',[1 1 1])
figure('Position', [800 800 1050 600])
txt = sprintf('Fit $O(\\Delta_l^{%.3f})$',-Polv(1));
plot(LL,log2(second_moment_incr),'sk','DisplayName',...
'$E\left[\xi_l^2 \right]$','MarkerSize',35,...
'MarkerEdgeColor','k','MarkerFaceColor',[0.5 .5 .5])
hold on
plot(LL,log2(secondM_incr_fit2),'-r','DisplayName',txt, 'LineWidth',5)
hold off
xticks(LL)
title('$2^{nd}$ Moment of $\xi_l$ - Bayesian Logistic Model')
yticks = get(gca, 'YTick');
set(gca, 'YTickLabel', strcat('2^{',num2str(yticks'),'}'));
grid on
xlabel('Discretization Levels $l$','FontSize',40)
ylabel('Second Moment','FontSize',45)
legend show
legend('Location','northeast')
set(legend, 'FontSize', 55)
ax = gca;
ax.GridAlpha = 0.35;
ax.FontSize = 45;
export_fig('incr_var_Logist.pdf','-m3')
%% Functions
function [pi_phi_k0_m, cost] = pi_phi_coupled(d,mcmc_maxiter, uf0, ...
uf0_tilde, uc0,uc0_tilde, L,sig,...
kap,alpha,k,m,x,const,inv_sigb,Ls)
Uf = zeros(ceil(mcmc_maxiter/2),2*d);
Uf_tilde = zeros(ceil(mcmc_maxiter/2),2*d);
Uc = zeros(ceil(mcmc_maxiter/2),2*d);
Uc_tilde = zeros(ceil(mcmc_maxiter/2),2*d);
%initialize the chains
Uf(1,:) = uf0;
Uf_tilde(1,:) = uf0_tilde;
Uc(1,:) = uc0;
Uc_tilde(1,:) = uc0_tilde;
%
cost = 0;
identical_f = false;
identical_c = false;
for i = 2 : mcmc_maxiter
%(uf,uf_tilde,uc,uc_tilde) in D^2
if identical_f && identical_c
[ufQ,ufQ_tilde,ucQ,ucQ_tilde] = sample_from_Q_fc(uf0,uf0_tilde,...
uc0,uc0_tilde,d,L,sig,kap,x,const,inv_sigb,Ls);
Uf(i,:) = ufQ;
Uf_tilde(i,:) = ufQ_tilde;
Uc(i,:) = ucQ;
Uc_tilde(i,:) = ucQ_tilde;
uf0 = ufQ;
uf0_tilde = ufQ_tilde;
uc0 = ucQ;
uc0_tilde = ucQ_tilde;
cost = cost + 2*(2^L+2^(L-1));
else
if rand < alpha
[ufQ,ufQ_tilde,ucQ,ucQ_tilde] = sample_from_Q_fc(uf0,uf0_tilde,...
uc0,uc0_tilde,d,L,sig,kap,x,const,inv_sigb,Ls);
Uf(i,:) = ufQ;
Uf_tilde(i,:) = ufQ_tilde;
Uc(i,:) = ucQ;
Uc_tilde(i,:) = ucQ_tilde;
uf0 = ufQ;
uf0_tilde = ufQ_tilde;
uc0 = ucQ;
uc0_tilde = ucQ_tilde;
cost = cost + 2*(2^L+2^(L-1));
else
[ufP,ufP_tilde,ucP,ucP_tilde, identical_f, identical_c] = ...
sample_from_P_fc(uf0,uf0_tilde,uc0,uc0_tilde,...
d,L,sig,kap,x,const,inv_sigb,Ls,...
identical_f, identical_c);
Uf(i,:) = ufP;
Uf_tilde(i,:) = ufP_tilde;
Uc(i,:) = ucP;
Uc_tilde(i,:) = ucP_tilde;
uf0 = ufP;
uf0_tilde = ufP_tilde;
uc0 = ucP;
uc0_tilde = ucP_tilde;
cost = cost + 2*(2^L+2^(L-1))- 4;
end
end
if identical_f && identical_c
stop_time = i;
break
end
end
Uf = Uf(1:stop_time,:);
Uf_tilde = Uf_tilde(1:stop_time,:);
Uc = Uc(1:stop_time,:);
Uc_tilde = Uc_tilde(1:stop_time,:);
if k <= stop_time - 2
k0 = min(m,stop_time-1);
temp = min(1, ((k+1:stop_time-1)-k)/(k0-k+1))';
pi_phi_k0_m_f = 1/(k0-k+1) * sum(Uf(k:k0,1:d)) ...
+ sum(temp.* ( Uf(k+1:end-1,1:d) - Uf_tilde(k+1:end-1,1:d) ) );
pi_phi_k0_m_c = 1/(k0-k+1) * sum(Uc(k:k0,1:d)) ...
+ sum(temp.* ( Uc(k+1:end-1,1:d) - Uc_tilde(k+1:end-1,1:d) ) );
else
%start with k = 1;
k0 = min(m,stop_time-1);
k = min(ceil((stop_time-1)/2), ceil(m/2));
temp = min(1, ((k+1:stop_time-1)-k)/(k0-k+1))';
pi_phi_k0_m_f = 1/(k0-k+1) * sum(Uf(k:k0,1:d)) ...
+ sum(temp.* ( Uf(k+1:end-1,1:d) - Uf_tilde(k+1:end-1,1:d) ) );
pi_phi_k0_m_c = 1/(k0-k+1) * sum(Uc(k:k0,1:d)) ...
+ sum(temp.* ( Uc(k+1:end-1,1:d) - Uc_tilde(k+1:end-1,1:d) ) );
end
pi_phi_k0_m = pi_phi_k0_m_f - pi_phi_k0_m_c;
end
function [u,u_tilde] = sample_from_Q(u0,u0_tilde,d,L,sig,kap,x,const,inv_sigb,Ls)
%U(x) = 1/2 * (x-mu)^2
%kap = 1/2
%b = -grad U = -(x-mu)
np = 2^L;
h = 1/np;
sig_L = 2^(-(L+Ls));
%copy the first d columns of dB to d+1:end.
%Similar with Gamma
Gamma = repmat(sqrt(h) * randn(np,d),1,2);
dB = repmat(sqrt(h) * randn(np,d), 1,2);
X = [u0(1:d), u0_tilde(1:d)];
V = [u0(d+1:end),u0_tilde(d+1:end)];
X_old = X;
V_old = V;
for k = 1: np
b = [- gradU(X_old(1:d), x, const,inv_sigb),...
-gradU(X_old(d+1:end), x, const,inv_sigb)];
X = X_old + V_old * h + sig_L * Gamma(k,:);
V = V_old + (b - kap*V_old)*h + sig*dB(k,:);
X_old = X;
V_old = V;
end
u = [X(1:d),V(1:d)]; %1x2d
u_tilde = [X(d+1:end),V(d+1:end)]; %1x2d
end
function [U,U_tilde,identical] = sample_from_P(u0,u0_tilde,d,L,sig,kap,x,const,...
inv_sigb,Ls)
%U(x) = 1/2 * (x-mu)^2
%kap = 1/2
%b = -grad U = x-mu
np = 2^L;
h = 1/np;
sig_L = 2^(-(L+Ls));
%copy the first d columns of dB to d+1:end.
%Similar with Gamma
Gamma = repmat(sqrt(h) * randn(np-1,d),1,2);
dB = repmat(sqrt(h) * randn(np-1,d), 1,2);
X = [u0(1:d), u0_tilde(1:d)];
V = [u0(d+1:end),u0_tilde(d+1:end)];
X_old = X;
V_old = V;
for k = 1: np - 1
b = [- gradU(X_old(1:d), x, const,inv_sigb),...
-gradU(X_old(d+1:end), x, const,inv_sigb)];
X = X_old + V_old * h + sig_L * Gamma(k,:);
V = V_old + (b - kap * V_old) * h + sig * dB(k,:);
X_old = X;
V_old = V;
end
b = -gradU(X(1:d), x, const,inv_sigb);
p_mu = [X(1:d),V(1:d)] + [V(1:d), b - kap*V(1:d)] * h;
b = -gradU(X(d+1:end), x, const,inv_sigb);
q_mu = [X(d+1:end),V(d+1:end)]+[V(d+1:end),...
b-kap*V(d+1:end)]* h;
[U,U_tilde, identical] = refl_max_coupl(d, p_mu,q_mu, h, sig_L,sig);
end
function [ufQ,ufQ_tilde,ucQ,ucQ_tilde] = sample_from_Q_fc(uf0,uf0_tilde,...
uc0,uc0_tilde,d,L,sig,kap,x,const,inv_sigb,Ls)
nf = 2^L;
hf = 1/nf;
nc = 2^(L-1);
hc = 1/nc;
sig_Lf = 2^(-(L+Ls));
sig_Lc = 2^(-(L-1+Ls));
%sig_Lc = sig_Lf;
%copy the first d columns of dB to d+1:end.
%Similar with Gamma
Gammaf = repmat(sqrt(hf) * randn(nf,d), 1,2);
dBf = repmat(sqrt(hf) * randn(nf,d), 1,2);
Xf = [uf0(1:d),uf0_tilde(1:d)];
Vf = [uf0(d+1:end),uf0_tilde(d+1:end)];
Xc = [uc0(1:d),uc0_tilde(1:d)];
Vc = [uc0(d+1:end),uc0_tilde(d+1:end)];
Xf_old = Xf;
Vf_old = Vf;
Xc_old = Xc;
Vc_old = Vc;
for k = 1: nc
Gammac = zeros(1,2*d);
dBc = zeros(1,2*d);
for j = 1 : 2
bf = [- gradU(Xf_old(1:d), x, const,inv_sigb),...
-gradU(Xf_old(d+1:end), x, const,inv_sigb)];
Xf = Xf_old + Vf_old * hf + sig_Lf * Gammaf(2*(k-1)+j,:);
Vf = Vf_old + (bf -kap*Vf_old)*hf+sig*dBf(2*(k-1)+j,:);
Xf_old = Xf;
Vf_old = Vf;
Gammac = Gammac + Gammaf(2*(k-1)+j,:);
dBc = dBc + dBf(2*(k-1)+j,:);
end
bc = [- gradU(Xc_old(1:d), x, const,inv_sigb),...
-gradU(Xc_old(d+1:end), x, const,inv_sigb)];
Xc = Xc_old + Vc_old * hc + sig_Lc * Gammac;
Vc = Vc_old + (bc - kap * Vc_old) * hc + sig * dBc;
Xc_old = Xc;
Vc_old = Vc;
end
ufQ = [Xf(1:d),Vf(1:d)];
ufQ_tilde = [Xf(d+1:end),Vf(d+1:end)];
ucQ = [Xc(1:d),Vc(1:d)];
ucQ_tilde = [Xc(d+1:end),Vc(d+1:end)];
end
function [Uf,Uf_tilde,Uc,Uc_tilde,identical_f, identical_c] = ...
sample_from_P_fc(uf0,uf0_tilde,...
uc0,uc0_tilde,d, L,sig,kap,x,const,inv_sigb,Ls,...
identical_f, identical_c)
nf = 2^L;
hf = 1/nf;
nc = 2^(L-1);
hc = 1/nc;
sig_Lf = 2^(-(L+Ls));
sig_Lc = 2^(-(L-1+Ls));
if ~identical_f && ~identical_c
%copy the first d columns of dB to d+1:end.
%Similar with Gamma
Gammaf = repmat(sqrt(hf) * randn(nf-1,d),1,2);
dBf = repmat(sqrt(hf) * randn(nf-1,d),1,2);
Xf = [uf0(1:d),uf0_tilde(1:d)];
Vf = [uf0(d+1:end),uf0_tilde(d+1:end)];
Xc = [uc0(1:d),uc0_tilde(1:d)];
Vc = [uc0(d+1:end),uc0_tilde(d+1:end)];
Xf_old = Xf;
Vf_old = Vf;
Xc_old = Xc;
Vc_old = Vc;
for k = 1: nc - 1
Gammac = zeros(1,2*d);
dBc = zeros(1,2*d);
for j = 1 : 2
bf = [-gradU(Xf_old(1:d), x, const,inv_sigb),...
-gradU(Xf_old(d+1:end), x, const,inv_sigb)];
Xf = Xf_old + Vf_old * hf + sig_Lf * Gammaf(2*(k-1)+j,:);
Vf = Vf_old + (bf -kap*Vf_old)*hf+sig*dBf(2*(k-1)+j,:);
Xf_old = Xf;
Vf_old = Vf;
Gammac = Gammac + Gammaf(2*(k-1)+j,:);
dBc = dBc + dBf(2*(k-1)+j,:);
end
bc = [-gradU(Xc_old(1:d), x, const,inv_sigb),...
-gradU(Xc_old(d+1:end), x, const,inv_sigb)];
Xc = Xc_old + Vc_old * hc + sig_Lc * Gammac;
Vc = Vc_old + (bc - kap * Vc_old) * hc + sig * dBc;
Xc_old = Xc;
Vc_old = Vc;
end
bf = -gradU(Xf(1:d), x, const,inv_sigb);
muf = [Xf(1:d),Vf(1:d)] + [Vf(1:d), bf-kap*Vf(1:d)] * hf;
bf_tilde = -gradU(Xf(d+1:end), x, const,inv_sigb);
muf_tilde = [Xf(d+1:end),Vf(d+1:end)]+...
[Vf(d+1:end),bf_tilde-kap*Vf(d+1:end)]*hf;
bc = -gradU(Xc(1:d), x, const,inv_sigb);
muc = [Xc(1:d),Vc(1:d)] + [Vc(1:d), bc-kap*Vc(1:d)] * hc;
bc_tilde = -gradU(Xc(d+1:end), x, const,inv_sigb);
muc_tilde = [Xc(d+1:end),Vc(d+1:end)] + ...
[Vc(d+1:end),bc_tilde-kap*Vc(d+1:end)]*hc;
elseif identical_f && ~identical_c
%copy the first d columns of dB to d+1:end.
%Similar with Gamma
Gammaf = repmat(sqrt(hf) * randn(nf-1,d),1,2);
dBf = repmat(sqrt(hf) * randn(nf-1,d),1,2);
Xf = uf0(1:d);
Vf = uf0(d+1:end);
Xc = [uc0(1:d),uc0_tilde(1:d)];
Vc = [uc0(d+1:end),uc0_tilde(d+1:end)];
Xf_old = Xf;
Vf_old = Vf;
Xc_old = Xc;
Vc_old = Vc;
for k = 1: nc - 1
Gammac = zeros(1,2*d);
dBc = zeros(1,2*d);
for j = 1 : 2
bf = -gradU(Xf_old, x, const,inv_sigb);
Xf = Xf_old + Vf_old * hf + sig_Lf * Gammaf(2*(k-1)+j,1:d);
Vf = Vf_old + (bf -kap*Vf_old)*hf+sig*dBf(2*(k-1)+j,1:d);
Xf_old = Xf;
Vf_old = Vf;
Gammac = Gammac + Gammaf(2*(k-1)+j,:);
dBc = dBc + dBf(2*(k-1)+j,:);
end
bc = [-gradU(Xc_old(1:d), x, const,inv_sigb),...
-gradU(Xc_old(d+1:end), x, const,inv_sigb)];
Xc = Xc_old + Vc_old * hc + sig_Lc * Gammac;
Vc = Vc_old + (bc - kap * Vc_old) * hc + sig * dBc;
Xc_old = Xc;
Vc_old = Vc;
end
bf = -gradU(Xf, x, const,inv_sigb);
muf = [Xf,Vf] + [Vf, bf-kap*Vf] * hf;
muf_tilde = muf;
bc = -gradU(Xc(1:d), x, const,inv_sigb);
muc = [Xc(1:d),Vc(1:d)] + [Vc(1:d), bc-kap*Vc(1:d)] * hc;
bc_tilde = -gradU(Xc(d+1:end), x, const,inv_sigb);
muc_tilde = [Xc(d+1:end),Vc(d+1:end)] +...
[Vc(d+1:end),bc_tilde-kap*Vc(d+1:end)]*hc;
elseif ~identical_f && identical_c
%copy the first d columns of dB to d+1:end.
%Similar with Gamma
Gammaf = repmat(sqrt(hf) * randn(nf-1,d),1,2);
dBf = repmat(sqrt(hf) * randn(nf-1,d),1,2);
Xf = [uf0(1:d),uf0_tilde(1:d)];
Vf = [uf0(d+1:end),uf0_tilde(d+1:end)];
Xc = uc0(1:d);
Vc = uc0(d+1:end);
Xf_old = Xf;
Vf_old = Vf;
Xc_old = Xc;
Vc_old = Vc;
for k = 1: nc - 1
Gammac = zeros(1,d);
dBc = zeros(1,d);
for j = 1 : 2
bf = [-gradU(Xf_old(1:d), x, const,inv_sigb),...
-gradU(Xf_old(d+1:end), x, const,inv_sigb)];
Xf = Xf_old + Vf_old * hf + sig_Lf * Gammaf(2*(k-1)+j,:);
Vf = Vf_old + (bf -kap*Vf_old)*hf+sig*dBf(2*(k-1)+j,:);
Xf_old = Xf;
Vf_old = Vf;
Gammac = Gammac + Gammaf(2*(k-1)+j,1:d);
dBc = dBc + dBf(2*(k-1)+j,1:d);
end
bc = -gradU(Xc_old, x, const,inv_sigb);
Xc = Xc_old + Vc_old * hc + sig_Lc * Gammac;
Vc = Vc_old + (bc - kap * Vc_old) * hc + sig * dBc;
Xc_old = Xc;
Vc_old = Vc;
end
bf = -gradU(Xf(1:d), x, const,inv_sigb);
muf = [Xf(1:d),Vf(1:d)] + [Vf(1:d), bf-kap*Vf(1:d)] * hf;
bf_tilde = -gradU(Xf(d+1:end), x, const,inv_sigb);
muf_tilde = [Xf(d+1:end),Vf(d+1:end)]+...
[Vf(d+1:end),bf_tilde-kap*Vf(d+1:end)]*hf;
bc = -gradU(Xc(1:d), x, const,inv_sigb);
muc = [Xc,Vc] + [Vc, bc-kap*Vc] * hc;
muc_tilde = muc;
elseif identical_f && identical_c
%copy the first d columns of dB to d+1:end.
%Similar with Gamma
Gammaf = sqrt(hf) * randn(nf-1,d);
dBf = sqrt(hf) * randn(nf-1,d);
Xf = uf0(1:d);
Vf = uf0(d+1:end);
Xc = uc0(1:d);
Vc = uc0(d+1:end);
Xf_old = Xf;
Vf_old = Vf;
Xc_old = Xc;
Vc_old = Vc;
for k = 1: nc - 1
Gammac = zeros(1,d);
dBc = zeros(1,d);
for j = 1 : 2
bf = -gradU(Xf_old, x, const,inv_sigb);
Xf = Xf_old + Vf_old * hf + sig_Lf * Gammaf(2*(k-1)+j,:);
Vf = Vf_old + (bf -kap*Vf_old)*hf+sig*dBf(2*(k-1)+j,:);
Xf_old = Xf;
Vf_old = Vf;
Gammac = Gammac + Gammaf(2*(k-1)+j,:);
dBc = dBc + dBf(2*(k-1)+j,:);
end
bc = -gradU(Xc_old(1:d), x, const,inv_sigb);
Xc = Xc_old + Vc_old * hc + sig_Lc * Gammac;
Vc = Vc_old + (bc - kap * Vc_old) * hc + sig * dBc;
Xc_old = Xc;
Vc_old = Vc;
end
bf = -gradU(Xf, x, const,inv_sigb);
muf = [Xf,Vf] + [Vf, bf-kap*Vf] * hf;
muf_tilde = muf;
bc = -gradU(Xc, x, const,inv_sigb);
muc = [Xc,Vc] + [Vc, bc-kap*Vc] * hc;
muc_tilde = muc;
end
[Uf,Uf_tilde,Uc,Uc_tilde,identical_f, identical_c] = ...
synch_refl_max_coupl(d, muf, muf_tilde, muc, muc_tilde, sig_Lf,...
sig_Lc, sig, hf, hc);
%
% [Uf,Uf_tilde, identical_f] =max_coupl(d, muf,muf_tilde, hf, sig_Lf,sig);
% [Uc,Uc_tilde, identical_c] =max_coupl(d, muc,muc_tilde, hc, sig_Lc,sig);
end
function [uf,uc] = sample_from_Kfc(uf0,uc0,d,L,...
sig,kap,x,const,inv_sigb,Ls)
nf = 2^L;
hf = 1/nf;
nc = 2^(L-1);
hc = 1/nc;
sig_Lf = 2^(-(L+Ls));
sig_Lc = 2^(-(L-1+Ls));
%sig_Lc = sig_Lf;
%copy the first d columns of dB to d+1:end.
%Similar with Gamma
Gammaf = sqrt(hf) * randn(nf,d);
dBf = sqrt(hf) * randn(nf,d);
Xf = uf0(1:d);
Vf = uf0(d+1:end);
Xc = uc0(1:d);
Vc = uc0(d+1:end);
Xf_old = Xf;
Vf_old = Vf;
Xc_old = Xc;
Vc_old = Vc;
for k = 1: nc
Gammac = zeros(1,d);
dBc = zeros(1,d);
for j = 1 : 2
bf = - gradU(Xf_old, x, const,inv_sigb);
Xf = Xf_old + Vf_old * hf + sig_Lf * Gammaf(2*(k-1)+j,:);
Vf = Vf_old + (bf -kap*Vf_old)*hf+sig*dBf(2*(k-1)+j,:);
Xf_old = Xf;
Vf_old = Vf;
Gammac = Gammac + Gammaf(2*(k-1)+j,:);
dBc = dBc + dBf(2*(k-1)+j,:);
end
bc = - gradU(Xc_old, x, const,inv_sigb);
Xc = Xc_old + Vc_old * hc + sig_Lc * Gammac;
Vc = Vc_old + (bc - kap * Vc_old) * hc + sig * dBc;
Xc_old = Xc;
Vc_old = Vc;
end
uf = [Xf,Vf];
uc = [Xc,Vc];
end
function [X, Y, identical] = refl_max_coupl(d, p_mu,q_mu, h, sig_L,sig)
%The reflection maximal coupling is only for two Gaussians with the
%same covariance
% cov = [sig_L^2 * h * I_{dxd}, 0_{dxd} ;
% 0_{dxd} , sig^2 * h * I_{dxd} ]
% cov is a diagonal matrix
% A below is the diagonal of chol(cov) = sqrt(diag(cov))
d1 = 2*d;
A = [sig_L * sqrt(h) * ones(1,d), sig * sqrt(h) * ones(1,d)];
z = (p_mu - q_mu) ./ A; %Sig^{-1/2} (mu_1 - mu_2)
e = z/norm(z);
%draw Xdot from s = N(0,I_{2*d}) (which is spherically symmetrical)
Xdot = randn(1,d1);
ratio = log_stand_mvnpdf(d1, Xdot + z) - log_stand_mvnpdf(d1, Xdot);
if log(rand) <= ratio
X = p_mu + A .* Xdot;
Y = X;
identical = true;
else
Ydot = Xdot - 2*(e*Xdot')*e;
X = p_mu + A .* Xdot;
Y = q_mu + A .* Ydot;
identical = false;
end
end
function [Uf,Uf_tilde, Uc, Uc_tilde, identical_f, identical_c] = ...
synch_refl_max_coupl(d,muf, muf_tilde, muc, muc_tilde,...
sig_Lf, sig_Lc, sig,hf, hc)
%Synchronous pairwise reflection maximal couplings%
d1 = 2*d;
Af = [sig_Lf * sqrt(hf) * ones(1,d), sig * sqrt(hf) * ones(1,d)];
Ac = [sig_Lc * sqrt(hc) * ones(1,d), sig * sqrt(hc) * ones(1,d)];
%draw v from s = N(0,I_{2*d}) (which is spherically symmetrical)
v = randn(1,d1);
%1)
vf = v;
Uf = muf + Af .* vf;
vc = v;
Uc = muc + Ac .* vc;
%2)
uf = (muf - muf_tilde) ./ Af;
ef = uf / norm(uf);
uc = (muc - muc_tilde) ./ Ac;
ec = uc / norm(uc);
%3)
if log(rand) < log_stand_mvnpdf(d1, vf+uf) - log_stand_mvnpdf(d1, vf)
Uf_tilde = Uf;
identical_f = true;
else
vfdot = vf - 2*(ef * vf')*ef;
Uf_tilde = muf_tilde + Af .* vfdot;
identical_f = false;
end
if log(rand) < log_stand_mvnpdf(d1, vc+uc) - log_stand_mvnpdf(d1, vc)
Uc_tilde = Uc;
identical_c = true;
else
vcdot = vc - 2*(ec * vc')*ec;
Uc_tilde = muc_tilde + Ac .* vcdot;
identical_c = false;
end
end
function [X,Y,identical] = max_coupl(d, p_mu,q_mu, h, sig_L,sig)
% Note that this very specific to the case
% where p & q are 2d - dimensional Gaussian with the same covariance
% cov = [sig_L^2 * h * I_{dxd}, 0_{dxd} ;
% 0_{dxd} , sig^2 * h * I_{dxd} ]
% A below is the diagonal of chol(cov) = sqrt(diag(cov))
d1 = 2*d;
A = [sig_L * sqrt(h) * ones(1,d), sig * sqrt(h) * ones(1,d)];
X = randn(1,d1) .* A + p_mu; % mvnrnd(p_mu, cov);
W = log(rand) + log_mvnpdf1(d1, X, p_mu, A);
%W = rand * mvnpdf(X,p_mu,p_sig);
%if W < mvnpdf(X,q_mu,q_sig)
if W < log_mvnpdf1(d1,X,q_mu,A)
Y = X;
identical = true;
return
else
while 1
%Y = mvnrnd(q_mu,q_sig);
Y = randn(1,d1) .* A + q_mu;
%W = rand * mvnpdf(Y,q_mu,q_sig);
W = log(rand) + log_mvnpdf1(d1,Y,q_mu,A);
%if W > mvnpdf(Y,p_mu,p_sig)
if W > log_mvnpdf1(d1,Y,p_mu,A)
identical = false;
return
end
end
end
end
function [Uf1,Uf2,Uc1,Uc2,identical_f, identical_c] = ...
fourWay_max_coupl(d, muf1, muf2, muc1, muc2, sig_Lf, sig_Lc,...
sig, hf, hc,identical_f, identical_c)
% Note that this very specific to the case
% where pf & qf are 2d - dimensional Gaussians with the same covariance
% covf = [sig_Lf^2 * hf * I_{dxd}, 0_{dxd} ;
% 0_{dxd} , sig^2 * hf * I_{dxd} ]
% Af below is the diagonal of chol(cov) = sqrt(diag(covf))
% Similar with pc & qc
% a = norm(muf1 - muc1)
% b = norm(muf1 - muf2)
% c = norm(muf1 - muc2)
% dd = norm(muc1 - muc2)
d1 = 2*d;
Af = [sig_Lf * sqrt(hf) * ones(1,d), sig * sqrt(hf) * ones(1,d)];
Ac = [sig_Lc * sqrt(hc) * ones(1,d), sig * sqrt(hc) * ones(1,d)];
Uc1 = randn(1,d1) .* Ac + muc1; %mvnrnd(pc_mu, pc_sig);
% lncf = -0.5 * d1*log(2*pi) - d*(log(sig_Lf*sqrt(hf)) + log(sig*sqrt(hf)));
% lncc = -0.5 * d1*log(2*pi) - d*(log(sig_Lc*sqrt(hc)) + log(sig*sqrt(hc)));
% inv_pq_f_sig = inv(diag(Af.^2)); %pf_sig = qf_sig
% inv_pq_c_sig = inv(diag(Ac.^2)); %pc_sig = qc_sig
%
% la1 = lncf -0.5 * sum(((U-pf_mu)./Af).^2,2); %q_l(xf,U)
% la2 = lncf -0.5 * ((U-qf_mu)*inv_pq_f_sig*(U-qf_mu)'); %q_l(wf,U)
% la3 = lncc -0.5 * ((U-pc_mu)*inv_pq_c_sig*(U-pc_mu)'); %q_{l-1}(xc,U)
% la4 = lncc -0.5 * ((U-qc_mu)*inv_pq_c_sig*(U-qc_mu)'); %q_{l-1}(wc,U)
%
la1 = log_mvnpdf1(d1,Uc1, muc1, Ac);
la2 = log_mvnpdf1(d1,Uc1, muc2, Ac);
la3 = log_mvnpdf1(d1,Uc1, muf1, Af);
la4 = log_mvnpdf1(d1,Uc1, muf2, Af);
la = min([la1,la2,la3,la4],[],'all') - la1;
fprintf('outside if la = %.3e\n',la)
if log(rand) <= la
Uf1 = Uc1;
Uf2 = Uc1;
Uc2 = Uc1;
identical_f = true;
identical_c = true;
else
if identical_c
% if coarse chains have met: sample common proposal\
reject = true;
while reject
Uc1 = randn(1,d1) .* Ac + muc1;
la1 = log_mvnpdf1(d1,Uc1, muc1, Ac);
la2 = log_mvnpdf1(d1,Uc1, muc2, Ac);
la3 = log_mvnpdf1(d1,Uc1, muf1, Af);
la4 = log_mvnpdf1(d1,Uc1, muf2, Af);
la = min([la1,la2,la3,la4],[],'all') - la1;
fprintf('inside if identc inside while la = %.3e\n',la)
reject = (log(rand) < la);
end
Uc2 = Uc1;
identical_c = true;
else
% if coarse chains have not met: sample proposals independently
identical_c = false;
reject = true;
while reject
% sample first proposal of coarse chain
Uc1 = randn(1,d1) .* Ac + muc1;
la1 = log_mvnpdf1(d1,Uc1, muc1, Ac);
la2 = log_mvnpdf1(d1,Uc1, muc2, Ac);
la3 = log_mvnpdf1(d1,Uc1, muf1, Af);
la4 = log_mvnpdf1(d1,Uc1, muf2, Af);
la = min([la1,la2,la3,la4],[],'all') - la1;
fprintf('inside if not identc inside while1 la = %.3e\n',la)
reject = (log(rand) < la);
end
% sample proposal for second coarse chain
reject = true;
while reject
% sample first proposal of coarse chain
Uc2 = randn(1,d1) .* Ac + muc2;
la1 = log_mvnpdf1(d1,Uc2, muc1, Ac);
la2 = log_mvnpdf1(d1,Uc2, muc2, Ac);
la3 = log_mvnpdf1(d1,Uc2, muf1, Af);
la4 = log_mvnpdf1(d1,Uc2, muf2, Af);
la = min([la1,la2,la3,la4],[],'all') - la2;
fprintf('inside if not identc inside while2 la = %.3e\n',la)
reject = (log(rand) < la);
end
end
if identical_f
% if fine chains have met: sample common proposal
reject = true;
while reject
Uf1 = randn(1,d1) .* Af + muf1;
la1 = log_mvnpdf1(d1,Uf1, muc1, Ac);
la2 = log_mvnpdf1(d1,Uf1, muc2, Ac);
la3 = log_mvnpdf1(d1,Uf1, muf1, Af);
la4 = log_mvnpdf1(d1,Uf1, muf2, Af);
la = min([la1,la2,la3,la4],[],'all') - la3;
fprintf('inside if identf inside while la = %.3e\n',la)
reject = (log(rand) < la);
end
Uf2 = Uf1;
identical_f = true;
else
% if fine chains have not met: sample proposals independently
identical_f = false;
reject = true;
while reject
% sample first proposal of fine chain
Uf1 = randn(1,d1) .* Af + muf1;
la1 = log_mvnpdf1(d1,Uf1, muc1, Ac);
la2 = log_mvnpdf1(d1,Uf1, muc2, Ac);
la3 = log_mvnpdf1(d1,Uf1, muf1, Af);
la4 = log_mvnpdf1(d1,Uf1, muf2, Af);
la = min([la1,la2,la3,la4],[],'all') - la3;
fprintf('inside if not identf inside while1 la = %.3e\n',la)
reject = (log(rand) < la);
end
reject = true;
while reject
% sample second proposal of fine chain
Uf2 = randn(1,d1) .* Af + muf2;
la1 = log_mvnpdf1(d1,Uf2, muc1, Ac);
la2 = log_mvnpdf1(d1,Uf2, muc2, Ac);
la3 = log_mvnpdf1(d1,Uf2, muf1, Af);
la4 = log_mvnpdf1(d1,Uf2, muf2, Af);
la = min([la1,la2,la3,la4],[],'all') - la4;
fprintf('inside if not identf inside while2 la = %.3e\n',la)
reject = (log(rand) < la);
end
end
end
end
function ly = log_mvnpdf1(d,X,mu,sig_sqrt)
% evaluate the Multivariate Normal Distribution at X with mean mu and a
% a cov that is a diagonal matrix.
% Inputs: d - dimension
% X - 1xd to evaluate the PDF at
% mu - 1xd vector (mean)
% sigd - 1xd vector (sqrt of diag of Sigma)
X0 = X - mu;
xSigSqrtinv = X0./sig_sqrt;
logSqrtDetSigma = sum(log(sig_sqrt));
quadform = sum(xSigSqrtinv.^2, 2);
ly = -0.5*quadform - logSqrtDetSigma - d*log(2*pi)/2;
end
function ly = log_stand_mvnpdf(d,X)
% evaluate the log Multivariate Standard Normal Distribution at X
ly = -0.5*sum(X.^2, 2) - d*log(2*pi)/2;
end
function g = gradU(beta, x, const, inv_sigb)
%beta is 1xd
%x is dxn
%const = sum(y.*x,2)', where y is 1xn; const is 1xd
%inv_sigb is dxd
xi = 0.5;
t = exp(-beta * x); %1xn
g = -const + sum((1./(1+t)) .* x,2)' + 2*xi* beta*inv_sigb;
end
function U = comput_U(beta, x, inv_sigb,y)
%beta is 1xd
%x is dxn
xi = 0.5;
t = beta * x; %1xn
U = sum(-y.*t +log(1+exp(t)),2) + xi* (beta*inv_sigb)*beta';
end
function m = ULA(beta0, burn_in, N, step, x, const, inv_sigb)
d = length(beta0);
beta = beta0;
sqrtstep = sqrt(2*step);
m = zeros(size(beta0));
for i = 2 : burn_in + N
b = gradU(beta, x, const, inv_sigb);
beta = beta - step * b + sqrtstep * randn(1,d);
if i > burn_in
m = m + beta;
end
end
m = m /(burn_in + N);
end
function m = TULA(beta0, burn_in, N, step, x, const, inv_sigb)
d = length(beta0);
beta = beta0;
sqrtstep = sqrt(2*step);
m = zeros(size(beta0));
for i = 2 : burn_in + N
b = gradU(beta, x, const, inv_sigb);
bTamed = b / (1 + step*norm(b));
%bTamed = b ./ (1 + step*abs(b));
beta = beta - step * bTamed + sqrtstep * randn(1,d);
if i > burn_in
m = m + beta;
end
end
m = m /(burn_in + N);
end
function [m,ar] = RWMH(beta0, burn_in, N, step, x, y, inv_sigb)
d = length(beta0);
beta = beta0;
sqrts = sqrt(step);
m = zeros(size(beta0));
lb = -comput_U(beta, x, inv_sigb,y);
count = 0;
for i = 2 : burn_in + N
betap = beta + sqrts * randn(1,d);
la = -comput_U(betap, x, inv_sigb,y);
if log(rand) <= (la-lb)
beta = betap;
count = count + 1; %accept
lb = la;
end
if i > burn_in
m = m + beta;
end
end
m = m / (N+burn_in);
ar = count/(N+burn_in);
end