Jacobian and proposal ratio of Birth/death step in RJMCMC of Gaussian mixture model

Question

I am asking questions regarding RJMCMC several times in this site. Some of my questions are answered and some are unanswered. It didn't clarify all of my unclear points but I am glad that I have better understanding of RJMCMC.

I hope this question also help to improve my understanding regarding RJMCMC. I am studying Richardson and Green (1997) paper (https://academic.oup.com/jrsssb/article/59/4/731/7083042) of RJMCMC of Bayesian mixture model. The acceptance probability is somewhat simple for birth/death case. Merge/split is more complicated.

The new parameter w_j*, μ_j*, σ_j* play role as u makes calculation of Jacobian straightforward. However, the calculation of proposal ratio here is not entirely clear for me. For the birth step, we have to create $w_j*$ and $(μ_j*, σ_j*)$ pair and death step, pair of $w_j*, (μ_j*, σ_j*)$ are deleted. I guess proposal probability discussed here, the death proposal probability is $d_{k+1}/(k0+1)$ and birth proposal probability is $b_k*g_{1,k}(w_j*)$ which make sense that death probability is computed from k0 which is number of zero component that is allowed to be deleted. and birth probability depend on the distribution of g_{1,k}(w_j*) which can be correspond to the probability of generating certain point of $w_j*$

However, I am confusing that "proposal ratio" of $(μ_j*, σ_j*)$ pair is not included in this equation. New component of $(μ_j*, σ_j*)$ are also generated with $w_j*$ with the some probability distribution and also deleted when one of w_j* is deleted. However, the "proposal ratio" of generating/deleting $(μ_j*, σ_j*)$ is not computed here because they are just generating and deleted randomly? Could you provide some more intuitive explanation for this?

Answer 1

For the birth step, we have to create $𝑤_{𝑗^∗}$ and $(μ_{𝑗^∗},σ_{𝑗^∗})$ pair and death step, pair of $𝑤_{𝑗^∗}$ and $(μ_{𝑗^∗},σ_{𝑗^∗})$ are deleted.

Correct. However, the probability $A$ in (12) is concerned with the move from $k$ to $k+1$ components, i.e., the birth move, while the death move is by construction (and in order to suit the justification for RJMCMC) completely determined as the symmetric.

the death proposal probability is $𝑑_{𝑘+1}/(𝑘_0+1)$ and birth proposal probability is $𝑏_𝑘\times 𝑔_{1,𝑘}(𝑤_{𝑗^∗})$ which make sense that death probability is computed from $k_0$ which is [the] number of zero [allocation] components that is allowed to be deleted and [the] birth probability depends on the distribution $g_{1,k}(w_{𝑗^∗})$ which corresponds to the probability of generating a certain realisation $w_{𝑗^∗}$

This is a product of the probability to propose a death/birth move when having $k+1$/$k$ components and of the density of the proposed new mixture, which consists of $1/k_0+1$ for the killed component and of the Jacobian for the renormalisation of the surviving weights, $(1-w_{𝑗^∗})^k$, for the death move, while the birth move involves $b_k$ times the proposal density of $(w_{𝑗^∗},μ_{𝑗^∗},σ_{𝑗^∗})$, which is $g_{1,k}(w_{𝑗^∗})$ times the prior on $(μ_{𝑗^∗},σ_{𝑗^∗})$ times again the Jacobian for the renormalisation of the other weights, $(1-w_{𝑗^∗})^k$. Hence it cancels in the ratio (12). [Imho, a simpler representation of the moves would be to express the mixture in terms of unnormalised weights divided by their sum.]

confused by the fact that "proposal ratio" of $(μ_{𝑗^∗},σ_{𝑗^∗})$ is not included in this equation. A new component $(μ_{𝑗^∗},σ{𝑗^∗})$ is also generated with $𝑤_{𝑗^∗}$ with the some probability distribution and also deleted when one of $w_{𝑗^∗}$ is deleted. However, the "proposal ratio" of generating/deleting $(μ_{𝑗^∗},σ_{𝑗^∗})$ is not computed here because they are just generating and deleted randomly?

The reason why the proposal density of $(μ_{𝑗^∗},σ_{𝑗^∗})$ does not appear in (12) is because the proposal is the prior. Therefore the ratio cancels in (12).