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Subglacial water influences the dynamics of ice masses. The state of subglacial pore water, whether liquid or frozen, is associated with differences in electrical resistivity that span several orders of magnitude, hence liquid water can be inferred from electrical resistivity depth profiles. Such profiles can be obtained from inversions of time domain electromagnetics (TEM) soundings, but these are often non-unique. Here, we adapt an existing Bayesian transdimensional algorithm ("MuLTI") to the inversion of TEM data constrained by independent depth constraints, to provide statistical properties and uncertainty analysis of the resistivity profile with depth. The method was applied to ground-based TEM data acquired on the terminus of the Norwegian glacier Midtdalsbreen, with depth constraints provided by co-located ground penetrating radar data. Our inversion shows that the glacier bed is directly underlain by material of resistivity 10² Ωm ± 100 %, with thickness 5-40 m, in turn underlain by a highly conductive basement (100 Ωm ± 15 %). High resistivity material, 5 x 104 Ωm ± 25 %, exists at the front of the glacier. All uncertainties are defined by the interquartile range of the posterior resistivity distribution. Combining these resistivity profiles with co-located seismic shear-wave velocity inversions to further reduce ambiguity in the hydro-geological interpretation of the subsurface, we propose a new 3D interpretation of the Midtdalsbreen subglacial material partitioned into partially frozen sediment, frozen sediment/permafrost and weathered/fractured bedrock with saline water. [ABSTRACT FROM AUTHOR]