Hi folks,
I have a 2D bimodal posterior distribution and I’m trying to sample from this posterior.
Your help would be gratefully appreciated if you could guide me whether there is any way i could sampling from both modes using Stan.
Here is the model information:
Given two correlated Gaussian random variables, u1 and u2 with zero mean, unit variance and correlation coefficient rho equal to 0.9, consider the following transformation:
x = u1* a
y = (u2/a)+b*(u1^2+a^2)
where a = 1.15 and b=0.5.
I used the standard normal bivariate distribution “z” as my prior.
and i believe I’ve done all of the transformations correctly.
I used 8 different chains for my simulation.
The problem is that in all chains, it is just sampling from one mode at the time and cannot sample from both modes together.
I would gratefully appreciate it if you could guide me in this regard.
I attach the simulation results for 8 chains as well. (The plot axis are “x” and “y” variables and the curve line in plot is the “g” function)
Here is my Stan code:
data{
vector[2] mu;
matrix[2, 2] Sigma;
real a;
real b;
matrix[2, 2] L;
real J;}
parameters {
vector[2] z;
}transformed parameters {
// z is bivariarte standard normal, u is correlated bivariate normal and L is lower triangular Cholesky matrix.
vector[2] u = L*z;
real x = a * u[1];
real y = (u[2]/a) + (b * (u[1]^2+(a^2)));
real g = (10^2) - (0.5 * (x+y)^2) - (((x-y)^2)/0.5);
}model {
//prior (Bivariate standard normal)
z ~ multi_normal(mu, Sigma);
//likelihood
target += normal_lpdf(g | 0,0.15)// J is Jacobian transformation but it has constant value.
target += log(J);
}generated quantities {
int I = 0;
{
real gg = (10^2) - (0.5 * (x+y)^2) - (((x-y)^2)/0.5);
if (gg <= 0) I = 1;
}
}
Thank you so much for your time and consideration.
Hamed
