## 2 Methods
### Bayesian image modeling
The second refers to model comparison that ranks alternative models on the basis of their plausibility with respect to the data. Bayesian methods deal consistently and quantitatively with both inductive inference levels in data modeling (Figure 3). Image models are derived on the basis of intuitive ideas and observations of real images, and have to comply with certain criteria of invariance, that is, operations on images should not affect their likelihood. Each model comprises a hypothesis \(H\) with some free parameters \(\mathbf{w}=\{\alpha,\beta,...\}\) that assign a probability density \(P(\mathbf{f}|\mathbf{w},H)\) over the entire image space and normalized to integrate to unity. Prior beliefs about the validity of \(H\) before data acquisition are embedded in \(P(H)\). Extreme choices for \(P(H)\) only may exceed the evidence \(P(\mathbf{f}|H)\), thus the plausibility \(P(H|\mathbf{f})\) of \(H\) is given essentially by the evidence \(P(\mathbf{f}|H)\) of the image \(\mathbf{f}\). Initially, the free parameters \(\mathbf{w}=\{\alpha,\beta,...\}\) are either unknown or they are assigned very wide prior distributions. The task is to search for the best fit parameter set \(\mathbf{w}_{MP}\), which has the largest likelihood given the image. Following [PERSON] theorem:  
\[P(\mathbf{w}|\mathbf{f},H)=\frac{P(\mathbf{f}|\mathbf{w},H)\,P(\mathbf{w}|H)}{P(\mathbf{f}|H)} \tag{12}\]  
where \(P(\mathbf{f}|\mathbf{w},H)\) is the likelihood of the image \(\mathbf{f}\) given \(\mathbf{w}\), \(P(\mathbf{w}|H)\) is the prior distribution of \(\mathbf{w}\), and  
Figure 3: Statistical inference levels in data modeling. \(P(\mathbf{f}|H)\) is the evidence for the image \(\mathbf{f}\). A prior \(P(\mathbf{w}|H)\) has to be assigned quite subjectively based on our beliefs about images. For completeness, [PERSON] theorem should include the background knowledge denoted hereafter by \(I\):  
\[P(\mathbf{w}|\mathbf{f},H,I)=\frac{P(\mathbf{f}|\mathbf{w},H,I)\,P(\mathbf{w}|H,I)}{P(\mathbf{f}|H,I)} \tag{13}\]  
If some additional information, say \(H^{\prime}\), becomes available, the estimate probability \(P(\mathbf{w}|\mathbf{f},H,I)\) can be refined by simply substituting the background information \(I\) with \(H^{\prime}I\) in eq. (13). Elementary manipulations of the product rule of probabilities lead to:  
\[P(\mathbf{w}|\mathbf{f},H^{\prime}H,I)=\frac{P(\mathbf{f}|\mathbf{w},H^{\prime}H,I)\,P(\mathbf{w} |H^{\prime}H,I)}{P(\mathbf{f}|H^{\prime}H,I)} \tag{14}\]  
It turns out that the product \(H^{\prime}H\) behaves like \(H^{\prime}\) and \(H\) were taken together. As such, the Bayesian analysis reports newly acquired information on the image space, the so-called \"posterior bubble\", but it does not suggest any prior. As the noiseless limit of the likelihood \(P(\mathbf{f}|\mathbf{w},H)\) is approaching a delta function, the prior becomes irrelevant because data are forcing the correct result, irrespective of theory. Yet images are usually digitized on thousands or even million of cells, each of which has an intensity to be estimated. The prior \(P(\mathbf{w}|H)\) plays a crucial role in such complex cases. A nontrivial prior knowledge \(P(\mathbf{w}|H^{\prime}H,I)\) is necessary to resolve the degeneracy of the problem and to obtain a unique and stable solution, since the number of parameters (components of \(\mathbf{f}\)) to be estimated is comparable or larger than the number of measured data (components of \(\mathbf{g}\)). A reasonable choice for prior knowledge is that the image \(\mathbf{f}\) is a positive-valued vector. The ME methods have historically been designed to make use of this prior knowledge in order to solve the inverse problem of image restoration. Since the probability of the image \(\mathbf{f}\) after observing the data \(\mathbf{g}\), that is, \(P(\mathbf{w}|\mathbf{f},H)\) is normalized to unity, then the denominator in eq. (12) ought to ensure \(P(\mathbf{f}|H)=\int_{\mathbf{w}}P(\mathbf{f}|\mathbf{w},H)\,P(\mathbf{w}|H)\,\mathrm{d}\mathbf{w}\).