An important property of MML (and SMML) is that it is invariant under
isomorphic transforms [#!Dowe:1999!#,#!Dowe:private!#]. This is the same as a
continuous bijection (ie continuous, one-to-one and onto mapping), with the
additional constraint that its inverse is also continuous. This means that
nearby points in the original domain remain nearby points (using the standard
definition) in the final domain and vice versa. Depending on
the specific approximation technique, there may be other assumptions required.
As shown later in the thesis, certain approximation techniques can only be applied to extremely simplified versions of the original problem. Although the simplified versions are interesting academically, they do not necessarily correspond to real-life problems and the assumptions made are too extreme to be considered reasonable. However, provided there exists an isomorphic transformation from the original problem-space to a simplified problem space, then the invariance of MML guarantees the correct solution when the inverse transformation is applied.
When required, any approximation technique can be made invariant requiring that every problem is solved using a uniform prior [#!Dowe:private!#]. This process of re-parameterisation is easily applied to any univariate distributions. However, there are still problems of applying this technique to multivariate distributions [#!Dowe:private!#] as there are multiple different uniform priors [#!Dowe:private!#].
Practically, invariance provides an easier path when applying MML to a range of problems. There is no need to derive a new set of equations every time the problem changed slightly. Instead, the solution to a similar problem can be utilised with the help of an appropriate mapping. Furthermore, invariance implies greater computational efficiency as researchers can optimise the algorithm for a single problem with the understanding that these improvements will transfer over to a wide range of problems.