Construction of a RJMCMC dynamics

 

The different movements defining the proposal

 

Simulation of Candy model

RJMCMC dynamics : adding, deleting, moving segments in the image. The movements (the transition kernels) need to guarantee :

• the irreducibility, the aperiodicity and the reversibility of the Markov chain
• the computation of the acceptance ratio must be possible

Some movements :

$\cdot$ birth and death of a free segment
$\cdot$ birth and death of a simple connected segment
$\cdot$ birth and death of a double connected segment
$\cdot$ modifying the orientation of a single or simple connected segment
$\cdot$ modifying the length of a single or simple connected segment
$\cdot$ modifying the position of a single connected segment
$\cdot$ modifying the position of a simple connected segment

 

Example of the acceptance ratio

Birth and death of a free segment : $S^{\prime}=S \cup \zeta$.

Let be P_n the probability to choose the birth of a free segment and P_m the probability to choose to kill one.

The acceptance ratio is :

$$R=\frac {f(S^{\prime})Q(S^{\prime} \rightarrow S)}{f(S)Q(S \rightarrow S^{\prime})}$$ (22)

We have :

$$Q(S^{\prime} \rightarrow S)=P_{m} \times \frac{1}{n_{d}+1}$$ (23)

n_d is the number of free segments in the configuration S.

And :

$$Q(S \rightarrow S^{\prime})=P_{n} \times \frac{1}{\nu(T)} \times \frac{1}{2\pi(l_{max}-l_{min})(w_{max}-w_{min})}$$ (24)

We obtain :

$$R=\frac{P_{m}}{P_{n}} \times \frac{2\pi\nu(T)(l_{max}-l_{min})(w_{max}-w_{min})}{n_{d}+1} \times \frac{f(S \cup \zeta)}{f(S)}$$ (25)

 

Realizations of the Candy model

 

Figure: Realizations of the prior model with different densities : a) $\beta = 0.25$ b) $\beta =0.5$

a)

b)

Statistics of the Candy model

 

Figure 10: Statistics of the segments

a)	b)
c)	d)