**4. Bayesian inference for inverse problems**

As described before, in inverse problems, the unknown *f* is a function (of time, space, wavelength, … ) and the observable quantity *g* is also another function which is related to *f* via an operator *g* ¼ Hð Þþ *f ϵ:* When discretized, they can be represented by the great dimensional vectors *f*, *g* and *g* ¼ *H f*ð Þþ *ϵ*, where *ϵ* represents all the errors (measurement, model and discretization). When the operator is a linear one, we have:

**Figure 6.**

*Illustration of the Bayesian inference for inverse problems.*

$$\mathbf{g} = H\mathbf{f} + \mathbf{e},\tag{20}$$

where *f* is a vector of length *n*, *H* the known forward model matrix of size *m* � *n*, and *g* and *ϵ* two vectors of size *m*.

The Bayes rule for this case is written as:

$$p(\mathbf{f}|\mathbf{g}, \mathcal{M}) = \frac{p(\mathbf{g}|\mathbf{f}, \mathcal{M})p(\mathbf{f}|\mathcal{M})}{p(\mathbf{g}, \mathcal{M})} \tag{21}$$

where we introduce M to represent the model, *p*ð Þ *g*j*f*,M , called commonly the *likelihood*, is obtained using the forward model (20) and the assigned probability law of the noise *p*ð Þ*ϵ* , *p*ð Þ *f*j,M is the assigned prior model and *p*ð Þ *f*j*g*,M the posterior probability law. **Figure 6** shows in a schematic way the main ingredients of the Bayesian inference for inverse problems.

This even very simple linear model has been used in many areas: linear inverse problems, compressed sensing, curve fitting and linear regression, machine learning, etc.

In inverse problems such as deconvolution, image restoration, *f* represent the input or original image, *g* represents the blurred and noisy image and *H* is the convolution operator matrix. In image reconstruction in Computed Tomography (CT), *f* represents the distribution of some internal property of an object, for example the density of the material and *g* represents the radiography data and *H* is the radiographic projection operator (discretized Radon transform operator).

In Compress Sensing, *g* is the compressed data, *f* is the uncompressed image and *H* the compressing matrix. In machine learning, *g* are the data, *H* is a dictionary and *f* represents the sparse coefficients of the projections of the data on that dictionary.
