overview.txt

------------- RJMCMC algorithm overview ------------- 

i: current model
j: proposed model
theta_*: model parameters
u_*: auxiliary transformation variables
y: observed data
k: proposition kernel
pi(.|i,theta_i,y): possibly configuration and data-dependent kernel choice probability
g_pki: bijective transformation of the p^th variant of kernel k from model i to j=k(i) with (theta_j,u_j)=g_pki(theta_i,u_i)
RJ-MCMC step: (i,theta_i)->(j,theta_j)


//////// SAMPLER ////////////
1) sampler 
- ask each kernel k for an unnormalized probability pi(.|i,theta_i,y)
- normalize pi(.|i,theta_i,y) (optional)
- chooses k~pi(.|i,theta_i,y), and thus a proposed model j=k(i)

//////// KERNEL ////////////
2) kernel k
- samples discrete parameters   p   ~ q_k (p  | i, theta_i, y) (e.g. decide the dying or modified object or its place of insertion)
- samples continuous parameters u_i ~ q_pk(u_i| i, theta_i, y) (e.g. create the new object parameters)
- computes (theta_j,u_j)=g_pki(theta_i,u_i)					   (e.g. create the new object)
- R  = pi(k^-1|j,theta_j)/pi(k|i,theta_i)
- R *= q_k^-1 (p   | j, theta_j, y)/q_k (p   | i, theta_i, y)
- R *= q_pk^-1(u_j | j, theta_j, y)/q_pk(u_i | i, theta_i, y)
- R *= |J_g_pki(theta_i,u_i)|
- returns R and the proposed modification from (i,theta_i) to (j,theta_j)

//////// DENSITY  ////////////  R *= f(j,theta_j,y)/f(i,theta_i,y)
3) reference process :  R *= f(j,theta_j)/f(i,theta_i)
- R *= f(j)/f(i)
- R *= f(theta_j|j)/f(theta_i|i)

4) simulated annealing (metropolis, szu, dueker ...)
- R *= f(y|j,theta_j)/f(y|i,theta_i) 
- [optional threshold variant] : R = (R>threshold)?1:0

//////// SAMPLER ////////////
5) acceptation 
-    x~[0,1] if (x<R) { i=j, theta_i=theta_j }


e.g: homogeneous poisson distribution of n objects, V=measure of the parameter space of one object
R_birth = pi_death/pi_birth * (1/(n+1))/(1/(n+1)) * (1)/(1/V) * lambda/(n+1) * (1/V) * exp(-E/T)
        = pi_death/pi_birth * lambda/(n+1) * exp(-E/T)

R_death = pi_birth/pi_death * (1/n)/(1/n) * (1/V) * n/lambda * (1)/(1/V) * exp(-E/T)
        = pi_death/pi_birth * n/lambda * exp(-E/T)


------------------------ syntax wish list -----------------------------------------

energy        = sum_1(m_defaultEnergy -  m_ponderation_dsm*integrated_flux(m_dsm)) - m_coefSurface* sum_2(intersection_area_binary_energy)
kernel        = 0.3*uniform_birth + 0.1*translate_edge;
reference_pdf = poisson(100)*rectangle_gen + poisson(10)*circle_gen;
reference_pdf = poisson(200)*(0.4*rectangle_gen + 0.6*circle_gen);