Potential Refinement

Given the obtained interaction structure, the first approximation of potentials should be refined in order to determine the posterior probability distribution $Pr({g})$ in Eq. (6.2.5).

A stochastic approximation [85] algorithm is applied to iteratively refine the potential toward the MLE $\mathbf{V}^*$. Basically, the process constructs a Markov chain and updates the potential at each iteration $t$ based on the following equation [38],

$$\mathbf{V}^{[t]}=\mathbf{V}^{[t-1]}+\lambda^{[t]}{(\mathbf{F}({g}^{[0]})-\mathbf{F}({g}^{[t]}))}$$ (6.3.6)

$$\mathbf{V}^{[0]}=\mathbf{V}_a^{\circ}$$ (6.3.7)

Here, ${g}^{[t]}$ is an image generated at step $t$ by sampling the previous probability distribution, $Pr^{[t-1]}({g} \mid \mathbf{V}^{[t-1]})$, using Gibbs sampling [34] or Metropolis algorithm [77]. $\lambda^{[t]}=\lambda^{[0]}\frac{c_0+1}{c_1+c_2{t}}$ is the scale factor. The initial scale $\lambda^{[0]}$ and the control parameters $c_0$, $c_1$ and $c_2$ can be decided either analytically or empirically.

The termination condition of the process is given as follows,

$$\vert\mathbf{V}^{[t]}-\mathbf{V}^*\vert < \epsilon$$ (6.3.8)

where $\epsilon$ is a predefined threshold.

The stochastic approximation is rather computational-intensive, which usually takes more than a few hundred steps to attain convergence. Speed of convergence and the comparison with an alternative MCMC based technique, called Controllable Simulated Annealing(CSA), are discussed in [38],