拉普拉斯先验贝叶斯MCMC抽样算法 matlab举例
时间: 2023-10-16 12:08:35 浏览: 47
以下是拉普拉斯先验贝叶斯MCMC抽样算法的matlab代码:
%% Generate simulated data
n = 100; % sample size
x = sort(rand(n,1)*10);
y = sin(x) + normrnd(0,0.1,n,1);
%% Define posterior distribution
% Bayesian linear regression model: y = beta0 + beta1*x + epsilon
% Prior distribution: beta0 ~ Laplace(0,tau0), beta1 ~ Laplace(0,tau1), epsilon ~ N(0,sigma^2)
% Posterior distribution: p(beta0,beta1,sigma^2|y,x) proportional to p(y|x,beta0,beta1,sigma^2)*p(beta0)*p(beta1)*p(sigma^2)
% where p(y|x,beta0,beta1,sigma^2) = N(y|beta0+beta1*x,sigma^2)
% Define prior distributions
tau0 = 1; % scale parameter of Laplace prior for beta0
tau1 = 1; % scale parameter of Laplace prior for beta1
sigma0 = 0.1; % scale parameter of inverse-Gamma prior for sigma^2
nu0 = 1; % shape parameter of inverse-Gamma prior for sigma^2
logprior = @(theta) loglaplace(theta(1),0,tau0) + loglaplace(theta(2),0,tau1) + ...
loginvgamma(theta(3),sigma0,nu0);
% Define likelihood function
loglik = @(theta) sum(lognormpdf(y,theta(1)+theta(2)*x,sqrt(theta(3))));
% Define posterior distribution
logpost = @(theta) loglik(theta) + logprior(theta);
%% Run MCMC
% Initialize parameters
theta = [0;0;1];
niter = 10000;
burnin = 1000;
thin = 5;
nsamp = (niter-burnin)/thin;
samples = zeros(nsamp,3);
% Run Metropolis-Hastings algorithm
for i = 1:niter
% Generate proposal
theta_prop = mvnrnd(theta,diag([0.1,0.1,0.01]));
% Calculate acceptance probability
logalpha = logpost(theta_prop) - logpost(theta);
% Accept or reject proposal
if log(rand)<logalpha
theta = theta_prop;
end
% Store sample
if i>burnin && mod(i-burnin,thin)==0
samples((i-burnin)/thin,:) = theta;
end
end
%% Plot results
% Plot posterior distributions
figure;
subplot(3,1,1);
histogram(samples(:,1),50,'Normalization','pdf');
title('Posterior distribution of beta0');
subplot(3,1,2);
histogram(samples(:,2),50,'Normalization','pdf');
title('Posterior distribution of beta1');
subplot(3,1,3);
histogram(sqrt(samples(:,3)),50,'Normalization','pdf');
title('Posterior distribution of sigma');
% Plot posterior predictive distribution
figure;
xgrid = linspace(0,10,100)';
ngrid = length(xgrid);
ypred = zeros(ngrid,nsamp);
for i = 1:nsamp
ypred(:,i) = normrnd(samples(i,1)+samples(i,2)*xgrid,sqrt(samples(i,3)));
end
plot(x,y,'ro',xgrid,mean(ypred,2),'b--');
legend('Observed data','Posterior mean of y(x)');
xlabel('x');
ylabel('y');
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