## 2 Methods
### Bayesian image modeling  
[PERSON] theorem follows directly from the standard axioms of probability relating the conditional probabilities of events. Probabilities may have two meanings: (i) to describe frequencies of outcomes in random experiments, or more generally, (ii) to describe degrees of belief in propositions (hypotheses) that do not involve random variables and repetitive events. The degrees of belief can be mapped onto probabilities if they satisfy simple consistency rules known as the Cox axioms [16], so that probabilities allow describing hypotheses as well. Bayesian methods deal with explicit assumptions and provide rulesfor reasoning consistently given those assumptions. Bayesian inferences are subjective in the sense that it is not plausible to reason about data without making assumptions based on some theoretical and/or experimental background. Yet Bayesian inferences are objective since following the same assumptions on a model, then identical inferences are drawn [12]. The fundamental concept of Bayesian analysis is that the plausibility of alternative hypotheses is represented by probabilities and inference is performed by evaluating these probabilities. Since plausibility is itself conditioned by some sort of prior assumptions, all Bayesian probabilities are regarded as _conditional_ on some collective background information. A set of events is termed _exhaustive_ if one or more of them must occur. Suppose that hypotheses \(H_{1},H_{2},...,H_{N}\) are mutually exclusive and exhaustive, then the rule of total probability [19] gives for any event \(D\) in the hypothesis space (Figure 2):  
\[P(D)=\sum_{i=1}^{N}P(H_{i})\,P(D|H_{i}) \tag{10}\]  
Further, assume that the probabilities \(\{P(H_{i})\}_{i}\) and \(\{P(D|H_{i})\}_{i}\), \(i=1,2,...,N\) are known. [PERSON] rule makes use of these probabilities to determine the conditional probabilities:  
\[P(H_{i}|D)=\frac{P(H_{i})\,P(D|H_{i})}{\sum_{i=1}^{N}P(H_{i})\,P(D|H_{i})},\,i =1,2,...,N \tag{11}\]  
[PERSON] theorem makes no reference to any sample or hypothesis space, nor it determines the numerical value of any probability directly from available information. As a prerequisite to apply [PERSON] theorem, a principle to cast available information into numerical values is needed. It explicitly shows that the observations giving \(\{P(D|H_{i})\}\) are not enough to determine the results \(\{P(H_{i}|D)\}\) since priors \(\{P(H_{i})\}\) need to be assigned. In statistical restoration of gray-level digital images, the basic assumption is that a scene is adequately represented by an orderly array of \(N\) pixels. The task is to infer reliable statistical descriptions of images, which are gray-scale digitized pictures and stored as an array of integers representing the intensity of gray level in each pixel. Then the shape of any positive, additive image can be directly identified with a  
Figure 2: \(N\) mutually exclusive and exhaustive hypotheses \(\{H_{i}\}_{i}\) (left); an event \(D\) in the hypothesis space (right). probability distribution. The image is conceived as an outcome of a random vector \(\mathbf{f}=\{f_{1},f_{2},...,f_{N}\}\) given in the form of a positive, additive probability density function (pdf). Likewise, the measured data \(\mathbf{g}=\{g_{1},g_{2},...,g_{M}\}\) are expressed in the form of a probability distribution. Further assumption refers to image data as a linear function of physical intensity, and that the noise \(\mathbf{e}\) is data independent, additive, and Gaussian with zero mean and known standard deviation \(\sigma_{m}\) in each measured pixel \(m,\,m=1,2,...,M\). A model is conceived as means to make inferences, predictions, and decisions. Any model depends on some free parameters. There are two levels of inference in any data modeling process. The first concerns fitting each model to the data, that is, to infer what the free parameters of each model might be given the data and some background information.