Experience shows that many data processing problems are difficult to solve, and some of these problems have even been proven to be computationally intractable. Human experts successfully solve many such problems by using a hierarchical, multi-resolution approach. These multi-resolution methods are, in several cases, provably optimal. However, due to the computational intractability of the problem itself, the multi-resolution approach can only work if the systems that we are analyzing are themselves hierarchical. We show that, first, due to (inevitable) measurement inaccuracies, an arbitrary input data is consistent with the hierarchical model, and second, that in many cases, the actual physical world is indeed fundamentally hierarchical.
Since traditional statistical methods have been designed primarily for non-hierarchical models, their direct application to multi-resolution data processing can lead to biased estimates. On a simple example, we show how these methods can be corrected to avoid this bias. Surprisingly, the analysis of this problem leads to new unexpected symmetries.