This tutorial addresses geometrical issues that are often misunder- stood in Bayesian inference or inversion. These issues concern the specification of sampling distributions that are consistent with, but not necessarily identical to, high-dimensional prior information. We illustrate that prior perceptions of simple geometrical concepts used to construct distributions in 2- or 3-dimensions may be completely distorted (and indeed untrue) in higher dimensions. If, for convenience, the initial sampling distribution is constructed in such a way and is different from the prior distribution, these effects can be very expensive to correct and may make a nonlinear inversion computationally intractable when this need not be the case. A crucial factor in Bayesian inversion is therefore whether one firmly believes in a particular prior distribution. If so, its form is exactly known and may constitute the most e.cient sampling distribution, even in cases where it is not straightforward to draw samples from that distribution. Sampling artifacts such as those above, while interesting, then become irrelevant since they represent true prior beliefs.

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