# Do the following normalizing constants cancel out in Reversible Jump ratio?

We know that a Strauss Point Process has density

$$p(x_{1}, x_{2},..., x_{K})\propto \prod_{i=1}^{K}\phi(x_{i};\theta)\prod_{1\leq i\leq j \leq K}a^{1(\left | x_{i}-x_{j} \right |\leq \delta)}$$

where $$\phi$$ is a positive function, and $$a\in(0,1)$$ and $$\delta \in (0,\infty)$$.

Suppose that we have the Strauss Point Process defined above for $$K$$ points, then the normalizing constant is computed as

$$Z_{\theta}=\int \prod_{i=1}^{K}\phi(x_{i};\theta)\prod_{1\leq i\leq j \leq K}a^{1(\left | x_{i}-x_{j} \right |\leq \delta)} dx_{1}...dx_{K}$$

Now let's assume that we have a second Strauss Point Process but this time with $$K+1$$ points, i.e. the same $$K$$ points as in the previous point process but we also have one additional $$x_{K+1}$$.

$$p(x_{1}, x_{2},..., x_{K}, x_{K+1})\propto \prod_{i=1}^{K+1}\phi(x_{i};\theta)\prod_{1\leq i\leq j \leq K+1}a^{1(\left | x_{i}-x_{j} \right |\leq \delta)}$$

This process has normalizing constant

$$Z_{\theta}^{'}=\int \prod_{i=1}^{K+1}\phi(x_{i};\theta)\prod_{1\leq i\leq j \leq K+1}a^{1(\left | x_{i}-x_{j} \right |\leq \delta)} dx_{1}...dx_{K+1}$$

Now if we are interested in calculating a Reversible Jump acceptance ratio, which includes the following ratio

$$\frac{p(x_{1}, x_{2},..., x_{K}, x_{K+1})}{p(x_{1}, x_{2},..., x_{K})} = \frac{\frac{\prod_{i=1}^{K+1}\phi(x_{i};\theta)\prod_{1\leq i\leq j \leq K+1}a^{1(\left | x_{i}-x_{j} \right |\leq \delta)}}{Z_{\theta}^{'}}}{\frac{\prod_{i=1}^{K}\phi(x_{i};\theta)\prod_{1\leq i\leq j \leq K}a^{1(\left | x_{i}-x_{j} \right |\leq \delta)}}{Z_{\theta}}}$$ ideally we would like the normalizing constants $$Z_{\theta}$$ and $$Z_{\theta}^{'}$$ to cancel out, but are they actually equal? Since the one depends on $$K$$ points and the other one on $$K+1$$. Am I calculating something wrong?

• Sorry for that I meant Reversible Jump ratio, Ill update it Apr 6, 2022 at 11:46
• What do you mean as probability weights of different models? Apr 6, 2022 at 12:25
• What I mean is that in RJ situations, the target distribution is either defined as a global density across models ($K$), in which case model-dependent ($K$) normalising constants do not matter or as a weighted sum of model-dependent ($K$) densities, in which case model-dependent ($K$) normalising constants matter. Apr 6, 2022 at 14:44
• So, in the case where we have $p(x_{1},x_{2},...;\theta)=\sum_{k=1}^{\infity}\pi_{k}p(x_{1},...,x_{k}|k)$ the normalizing constant matters, on the other hand in the case where we let the target distribution is $p(x_{1},x_{2},...,x_{K};\theta)$ updated with Birth\Death movements we are in the global case and the normalizing constant doesn't matter? I feel that I'm missing something Apr 6, 2022 at 15:45

$$\require{graphicx}$$The full model is not defined in the question, as it should include $$K$$ in the target. For instance, when $$p(K,x_{1}, x_{2},..., x_{K})\propto \prod_{i=1}^{K}\phi(x_{i};\theta)\prod_{1\leq i\leq j \leq K}a^{1(\left | x_{i}-x_{j} \right |\leq \delta)}$$ with a "flat prior" on $$K$$, the proportionality constant is $$Z_\theta=\sum_K \int \prod_{i=1}^{K}\phi(x_{i};\theta)\prod_{1\leq i\leq j \leq K}a^{1(\left | x_{i}-x_{j} \right |\leq \delta)} \text dx_{1}...\text dx_{K}$$ and the $$K$$ dependent normalising constants $$Z^K_\theta = \int \prod_{i=1}^{K}\phi(x_{i};\theta)\prod_{1\leq i\leq j \leq K}a^{1(\left | x_{i}-x_{j} \right |\leq \delta)} \text dx_{1}...\text dx_{K}$$ do not appear in the reversible jump ratio. If in the opposite $$p(K,x_{1}, x_{2},..., x_{K}) \underbrace{=}_{\text{normalised}}\frac{\varrho_K}{Z^K_\theta}\prod_{i=1}^{K}\phi(x_{i};\theta)\prod_{1\leq i\leq j \leq K}a^{1(\left | x_{i}-x_{j} \right |\leq \delta)}$$ with a prior $$\pi(K)=\varrho_K$$ on $$K$$, then the $$K$$ dependent normalising constants $$Z^K_\theta$$ will be involved in the reversible jump ratio, unless an exchange algorithm à la Murray et al. (2006) is introduced.