## 1 Introduction  
In applied geophysics, subsurface images are derived from non-invasive geophysical measurements, for example, direct current (DC) resistivity or small-loop electromagnetic (EM) measurements at the earth's surface. Geophysical data acquired from such measurements can be translated into a subsurface image of the investigated physical property by inversion of the data. Geological units often differ in terms of their physical properties; for example, ore bodies can usually be distinguished from their surroundings by identifying the spatial distribution of electrical conductivities in the subsurface. For this reason, inverse images can be useful to geologists, for example, in identifying mineral and hydrocarbon resources, groundwater reservoirs, and geological structures. Geophysical inverse problems are almost always ill-posed, i.e., no unique solution to the problem can be calculated [1]. This ill-posedness arises from the following general problems that are attributed to geophysical data collection: an inherent non-uniqueness (mostly problematic using potential methods), inaccuracies in data collection, possible inconsistencies in the data set, and, most importantly, an insufficient amount of data. Even for data sets acquired in the most precise manner (no noise or systematic errors), only limited information about the subsurface can be inferred, usually being insufficient for resolving the non-uniqueness of the inverse problem. One approach for improving geophysical inverse images is the so-called joint inversion of multiple geophysical data sets of different measurement responses [2]. Joint inversion usually refers to a simultaneous inversion of two or more data sets for a common subsurface parameter model. In general, joint inversions can be carried out for data sets informing two or more different sets of model parameters by using some kind of defined relation between the different types of parameters. However, joint inversions for methods that are sensitive to different subsurface properties will not be considered in this manuscript. Because different types of data contain complementary information about the subsurface, it is expected that the non-uniqueness of the inverse problem is mitigated (but not resolved) when data are inverted jointly. For the type of measurement data discussed in this paper, DC electrical resistivity and EM data, several approaches for joint inversion were presented in the literature (for example, by [3; 4; 5; 6]). In particular, [PERSON] and [PERSON] [5] summarize the use of joint in version of the two data types: EM inversion alone is useful for resolving thin conductive layers, but fails for detecting thin resistive layers in conductive surroundings. DC resistivity data on the other hand suffer from a general equivalence problem when resolving the thickness versus conductivity/resistivity ratio of a thin layer [7]. The joint inversions presented in the publications listed above applied deterministic gradient-based geophysical inversion approaches [8]. These studies showed the beneficial properties of joint inversions over the application of multiple separate inversions, for example the successful detection of thin middle layers was only achieved in joint results [4]. Two main problems are associated with the application of deterministic approaches in joint inversion frameworks. First, deterministic inversions have no measure for the theoretically expected reduction of uncertainty when applying joint inversion. The only available measure is the data misfit, which can generally be similar for different inversion approaches and, therefore, does not provide an objective assessment of the improvement brought by joint inversion. A model misfit computation is only available for synthetic studies and thus, unavailable for field data studies. Second, deterministic inversion schemes require largely arbitrary weights in the objective function, expressing the influence of the different data types on the objective function and the role of the regularization functional (e.g., [9]).