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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?

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  • $\begingroup$ Sorry for that I meant Reversible Jump ratio, Ill update it $\endgroup$
    – Fiodor1234
    Apr 6 at 11:46
  • $\begingroup$ What do you mean as probability weights of different models? $\endgroup$
    – Fiodor1234
    Apr 6 at 12:25
  • $\begingroup$ 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. $\endgroup$
    – Xi'an
    Apr 6 at 14:44
  • $\begingroup$ 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 $\endgroup$
    – Fiodor1234
    Apr 6 at 15:45

1 Answer 1

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$\require{graphicx}$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.

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