6.11 Cascaded Metropolis Algorithm

Let (x) be the homogeneous probability density and f ₁ (x), f ₂ (x), , f _p (x) be p probability densities. Our goal here is to develop a Metropolis random walk that samples the conjunction

We need to define the likelihood functions (or volumetric probabilities)

Assume that some random rules define a random walk that samples a probability density f ₁ (x). At a given step, the random walker is at point x _i , and the application of the rules would lead to a transition to point x _j. When all such proposed transitions x _i ? x _j are accepted, the random walker will sample the probability density f ₁ (x). Instead of always accepting the proposed transition x _i ? x _j, we reject it sometimes by using the following rules (to decide if the random walker is allowed to move to x _j or if it must stay at x _i) :

1. If ? ₂ (x _j ) ? ? ₂ (x _i ) , then go to step (c).

2. If ? ₂ (x _j ) < ? ₂ (x _i ), then decide randomly to go to step (c) or to reject the proposed move, with the following probability of going to step (c):

   
   

3. If ? ₃ (x _j ) ? ? ₃ (x _i ) , then go to...