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

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?

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).

• Thanks for very intuitive explanation! Also in the earlier section of paper, prior on w will always be taken symmetric Dirichlet w|~D(δ+n1, δ+n2,....) Therefore, the first line of Eq(12) is the prior of newly added $w_j*$? I am also confused by the appearance of (k+1) multiplication in the first line. Commented Sep 1, 2023 at 16:11
• Also one more question is that if we want to solve problem that 𝑤∗ is not need to be normalized. So 𝑤∗ can be added by prior distribution and not normalized. And μ𝑗∗ is also can be added by proposal, which is same as prior distribution and can be deleted in the same rate. Then, in this case, both (proposal ratio)*(prior ratio) and Jacobian can be 1 (or unity). That can be correct? It might be not entirely clear for this line. I will post another question for the problem I want to solve by RJMCMC Commented Sep 1, 2023 at 16:25