**7. Non-Gaussian priors**

Very often, assuming that the noise is Gaussian is valid in many applications, but a Gaussian prior may not be adequate. Thus, the case of Non-Gaussian priors is of great importance. A very well known example is the case of Generalized Gaussian:

$$p(\mathbf{f}) \propto \exp\left[-\gamma \sum\_{j} \left| \mathbf{f}\_{j} \right|^{\beta} \right]. \tag{46}$$

The case of *β* ¼ 2 is the Gaussian case, *β* >2 gives the *Super-Gaussian* and *β* <1 is called *Sub-Gaussian*. Its particular case *β* ¼ 1 results to what is called *Double Exponential (DE)* prior law:

$$p(\boldsymbol{f}) \propto \exp\left[-\gamma \sum\_{j} |\boldsymbol{f}\_{j}|\right] \propto \exp\left[-\gamma \|\boldsymbol{f}\|\_{1}\right] \tag{47}$$

which, when using with a Gaussian likelihood, results to:

$$\begin{cases} p(\mathbf{f}|\mathbf{g}) & \propto \exp\left[-\frac{1}{2v\_c}J(\mathbf{f})\right] \quad \text{with} \\ J(\mathbf{f}) & = \frac{1}{2}||\mathbf{g} - \mathbf{H} \cdot \mathbf{f}||^2 + \lambda \|\mathbf{f}\|\_1, \quad \lambda = \eta v\_c. \end{cases} \tag{48}$$

From this, we can see that the computation of MAP solution needs an appropriate optimization algorithm and the computations of the Posterior Mean (PM) or Posterior Covariance (PCov) or any other expectations become more difficult. However, as we will see later, VBA can be used to do approximate computations.

Another example is the Total Variation (TV) regularization method [11–14] which can be interpreted as choosing the prior

$$p(\mathcal{f}) \propto \exp\left[-\gamma \sum\_{j} |f\_j - f\_{j-1}|\right] \propto \exp\left[-\gamma \|\mathbf{D}\mathcal{f}\|\_1\right] \tag{49}$$

where *D* is the first order difference matrix.

This prior with a Gaussian model for noise results to:

$$\begin{cases} p(\mathbf{f}|\mathbf{g}) & \text{as } \exp\left[-\frac{1}{2\nu\epsilon}J(\mathbf{f})\right] \text{ with} \\ J(\mathbf{f}) & \text{if } \|\mathbf{g} - \mathbf{H}\mathbf{f}\|^2 + \lambda \|\mathbf{D}\mathbf{f}\|\_1, \quad \lambda = \eta v\_{\epsilon}. \end{cases} \tag{50}$$

One last example is using the Cauchy or more generally the Student-t distribution as the prior:

$$p(\mathbf{f}) \propto \exp\left[-\gamma \sum\_{j} \ln\left(\mathbf{1} + f\_j^2\right)^{\nu/2}\right] \tag{51}$$

which results to:

$$\begin{cases} p(\mathbf{f}|\mathbf{g}) & \propto \exp\left[-\frac{1}{2v\_c}J(\mathbf{f})\right] \quad \text{with} \\ J(\mathbf{f}) &= \frac{1}{2}||\mathbf{g} - \mathbf{H}\mathbf{f}||^2 + \lambda \sum\_{j} \ln\left(1 + \mathbf{f}\_j^2\right)^{\nu/2}, \quad \lambda = \gamma v\_c. \end{cases} \tag{52}$$

These three examples are of great importance. They have been used in the framework of MAP estimation and thus the optimization of the criteria *J*(*f*) for many linear inverse problems. However, the computation of other Bayesian estimators and uncertainty quantification (UQ) need again specific approximate solutions.
