Probabilistic inference by means of a massive probabilistic model usually has exponential-order computational complexity. For such massive probabilistic model, loopy belief propagation was proposed as a scheme to obtain the approximate inference. It is known that the generalized loopy belief propagation is constructed by using a cluster variation method. However, it is difficult to calculate the correlation in every pair of nodes which are not connected directly to each other by means of the generalized loopy belief propagation. In the present paper, we propose a general scheme for calculating an approximate correlation in every pair of nodes in a probabilistic model for probabilistic inference. The general scheme is formulated by combining a cluster variation method with a linear response theory.

The C-oscillation due to Martin-Löf shows that {α| ∀ n [C(α n)≥ n-O(1)]=, which also follows {α| ∀ n [K(α n)≥ n+K(n)-O(1)]=. By generalizing them, we show that there does not exist a real α such that ∀ n (K(α n)≥ n+λ K(n)-O(1)) for any λ>0.

We propose a new solvable Markov random field model for Bayesian image processing and give the exact expressions of the marginal likelihood and the restored image by using the multi-dimensional Gaussian formula and the discrete Fourier transform. The proposed Markov random field model includes the conditional autoregressive model and the simultaneous autoregressive model as a special case. The estimates of hyperparameters are obtained by maximizing the marginal likelihood. We study some statistical properties of the solvable Markov random field model. In some numerical experiments for standard images, we show that the proposed Markov random field model is useful for practical applications in image restorations. The investigation of probabilistic information processing by means of a solvable probabilistic model is recently in progress not only for image processing but also for error correcting codes and so on. The solvable probabilistic model gives us some important aspects for the availability of probabilistic computational systems.

In this paper, we consider Bayesian image denoising based on a Gaussian Markov random field (GMRF) model, for which we propose an new algorithm. Our method can solve Bayesian image denoising problems, including hyperparameter estimation, in O(n)-time, where n is the number of pixels in a given image. From the perspective of the order of the computational time, this is a state-of-the-art algorithm for the present problem setting. Moreover, the results of our numerical experiments we show our method is in fact effective in practice.