A 2D sampling example

We show a 2D sampling example. The dimension of the model space is 2. We assume the ground truth is at the origin and the initial error range is R1 = (8, 2).

Let delta=1.7, sigmamax=1000.

Based on equation (11) in the paper, we have:

R1 = (8, 2)           Sigma1 = (4.71 1000)    

R2=(2.77 2.00)    Sigma2 = (1.63 1.63)     

R3=(0.96 0.96)    Sigma3 = (0.56 0.56)     

 

The plot of feasible regions defined by R1(black) , R2(red), and R3(green).  

The level 1 sampling results. R1 = (8, 2)  and Sigma1 = (4.71 1000).  The target function  q1(C|I) is the normal distribution N(0, Sigma1).  The sampling density function is the gradient magnitude of q1(C|I).  The metropolis sampling is used to sample training examples.

The plot of q1(C|I) The gradient magnitude of q1(C|I) The sampled points using Metropolis algorithm.

 

The level 2 sampling results. R2 = (2.77 2.00)  and Sigma2 = (1.63 1.63).  The target function  q2(C|I) is the normal distribution N(0, Sigma2).  The sampling density function is the gradient magnitude of q2(C|I).  The metropolis sampling is used to sample training examples.

The plot of q2(C|I) The gradient magnitude of q2(C|I) The sampled points using Metropolis algorithm.

 

The level 3 sampling results. R3 = (0.96 0.96)  and Sigma3 = (0.56 0.56).  The target function  q3(C|I) is the normal distribution N(0, Sigma3).  The sampling density function is the gradient magnitude of q3(C|I).  The metropolis sampling is used to sample training examples.

The plot of q3(C|I) The gradient magnitude of q3(C|I) The sampled points using Metropolis algorithm.