Exact Calculation of Typical Hyper-Parameter Posterior Distribution of Gaussian Markov Random Field model.*

We investigated a hyper-parameter estimation method using posterior distributions for a Gaussian Markov random field (GMRF) model. GMRF is a modelling tool of gray scale images. Our GMRF model has hyper-parameters which are related to physical quantities such as a diffusion coefficient [1]. We analyzed the negative logarithm of posterior distributions called free-energy based on an analogy with statistical mechanics and exactly calculated the configurational average of free energy with respect to data. We found that the contour lines of free energy typically shrink as the amount of data increases and posterior distributions work well in evaluating the confidence of estimated values of hyper-parameters [2].
[1] Y. Nakanishi-Ohno, K. Nagata, H. Shouno, and M. Okada, J. Phys. A: Mathematical and Theoretical, 47, 045001, (2014).
[2] H. Sakamoto, Y. Nakanishi-Ohno and M. Okada, J. Phys. Soc. Jpn., 85, [6], 063801, (2016).