变分推断matlab程序

作为一个AI语言模型，我并不具备编写Matlab程序的能力。但是我可以为您提供一些变分推断的Matlab程序示例，供您参考。

1. 高斯混合模型的变分推断

%生成数据
n = 1000; % 数据量
k = 3; % 高斯分量个数
d = 2; % 数据维度
% 生成高斯分量的均值和协方差矩阵
mu = rand(k,d)*10-5; % 均值矩阵
sigma = zeros(d,d,k); % 协方差矩阵
for i=1:k
sigma(:,:,i) = diag(rand(d,1)*5); % 对角线矩阵
end
% 生成数据
data = zeros(n,d);
for i=1:n
k_index = randi(k);
data(i,:) = mvnrnd(mu(k_index,:),sigma(:,:,k_index));
end

% 变分推断
% 初始化参数
alpha = ones(1,k)./k; % 高斯分量的先验概率
mu_init = rand(k,d)*10-5; % 均值矩阵
sigma_init = zeros(d,d,k); % 协方差矩阵
for i=1:k
sigma_init(:,:,i) = diag(rand(d,1)*5); % 对角线矩阵
end
q_alpha = alpha; % 变分分布的先验概率
q_mu = mu_init; % 均值矩阵
q_sigma = sigma_init; % 协方差矩阵
% 迭代计算
max_iter = 100;
for iter=1:max_iter
% 更新q_alpha
E_z = zeros(n,k);
for i=1:n
for j=1:k
E_z(i,j) = log(alpha(j)) + log_mvnpdf(data(i,:),q_mu(j,:),q_sigma(:,:,j));
end
E_z(i,:) = exp(E_z(i,:) - max(E_z(i,:))); % 防止指数爆炸
E_z(i,:) = E_z(i,:) ./ sum(E_z(i,:));
end
q_alpha = alpha + sum(E_z,1);
% 更新q_mu
for j=1:k
q_mu(j,:) = sum(E_z(:,j).*data,1) ./ sum(E_z(:,j));
end
% 更新q_sigma
for j=1:k
diff = data - q_mu(j,:);
q_sigma(:,:,j) = diff'*diag(E_z(:,j))*diff ./ sum(E_z(:,j));
end
end

% 计算后验概率
posterior = zeros(n,k);
for i=1:n
for j=1:k
posterior(i,j) = log(q_alpha(j)) + log_mvnpdf(data(i,:),q_mu(j,:),q_sigma(:,:,j));
end
posterior(i,:) = exp(posterior(i,:) - max(posterior(i,:))); % 防止指数爆炸
posterior(i,:) = posterior(i,:) ./ sum(posterior(i,:));
end

% 显示结果
figure;
hold on;
scatter(data(:,1),data(:,2),10,posterior(:,1),'filled');
scatter(q_mu(:,1),q_mu(:,2),100,'k','filled');
scatter(mu(:,1),mu(:,2),100,'r','filled');
hold off;
title('GMM with Variational Inference');
legend('Cluster 1','Cluster 2','Cluster 3');
xlabel('Feature 1');
ylabel('Feature 2');

2. 隐马尔可夫模型的变分推断

% 生成数据
n = 1000; % 数据量
k = 3; % 隐状态个数
d = 2; % 数据维度
% 生成隐状态转移矩阵和观测矩阵
A = rand(k,k); % 隐状态转移矩阵
A = A ./ sum(A,2);
B = rand(k,d)*10-5; % 观测矩阵
% 生成数据
data = zeros(n,d);
z = zeros(n,1);
z(1) = randi(k);
data(1,:) = mvnrnd(B(z(1),:),eye(d));
for i=2:n
z(i) = randsample(k,1,true,A(z(i-1),:));
data(i,:) = mvnrnd(B(z(i),:),eye(d));
end

% 变分推断
% 初始化参数
alpha = ones(1,k)./k; % 隐状态的先验概率
A_init = rand(k,k); % 隐状态转移矩阵
A_init = A_init ./ sum(A_init,2);
B_init = rand(k,d)*10-5; % 观测矩阵
q_alpha = alpha; % 变分分布的先验概率
q_A = A_init; % 隐状态转移矩阵
q_B = B_init; % 观测矩阵
% 迭代计算
max_iter = 100;
for iter=1:max_iter
% 更新q_alpha
E_z = zeros(n,k);
for i=1:n
for j=1:k
E_z(i,j) = log(alpha(j)) + log(A(z(i-1),j)) + log_mvnpdf(data(i,:),q_B(j,:),eye(d));
end
E_z(i,:) = exp(E_z(i,:) - max(E_z(i,:))); % 防止指数爆炸
E_z(i,:) = E_z(i,:) ./ sum(E_z(i,:));
end
q_alpha = alpha + E_z(1,:);
% 更新q_A
for i=1:k
for j=1:k
q_A(i,j) = sum(E_z(1:end-1,i).*E_z(2:end,j)) ./ sum(E_z(1:end-1,i));
end
end
% 更新q_B
for j=1:k
q_B(j,:) = sum(E_z(:,j).*data,1) ./ sum(E_z(:,j));
end
end

% 计算后验概率
posterior = zeros(n,k);
for i=1:n
for j=1:k
if i==1
posterior(i,j) = log(q_alpha(j)) + log_mvnpdf(data(i,:),q_B(j,:),eye(d));
else
posterior(i,j) = log(A(z(i-1),j)) + log_mvnpdf(data(i,:),q_B(j,:),eye(d));
end
end
posterior(i,:) = exp(posterior(i,:) - max(posterior(i,:))); % 防止指数爆炸
posterior(i,:) = posterior(i,:) ./ sum(posterior(i,:));
end

% 显示结果
figure;
hold on;
scatter(data(:,1),data(:,2),10,posterior(:,1),'filled');
scatter(q_B(:,1),q_B(:,2),100,'k','filled');
scatter(B(:,1),B(:,2),100,'r','filled');
hold off;
title('HMM with Variational Inference');
legend('State 1','State 2','State 3');
xlabel('Feature 1');
ylabel('Feature 2');