n-Category Cafe May 10, 2011 's entry:

*Entropies vs. Means*(Tom Leinster) as a start, which then takes us to Entropy as a functor and specifically A characterization of entropy in terms of information loss.

This takes us neatly to Reyni's paper

*A. Rényi, Measures of information and entropy, in Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability 1960, pp. 547–561.*

The thesis being:

Given an ontology, we can calculate the entropy of a set of attributes/classes from that ontology, ie, the entropy of a function that selects a sub-set of that ontology. This is of course complicated by dependencies between attributes within that ontology, eg: star-sign <---> date-of-birth etc.

For each sub-set we can construct a further set of functions that partition the ontology as above - thus creating a partial ordering of functions (and a semi-lattice where all functions can be ground to a bottom value by a function f O -> _|_ which effectively removes everything giving an entropy of 1, ie: pure randomness and loss of all information.

A privacy preserving function is one that introduces more entropy, ie: obfusicates or anonymised any data passing through. However, There are certain other properties that need to be investigated, such as aspects over the original ontology - not all information is PII etc. Does this imply some weighting in the entropy calculation or something more exotic such as a matrix structure.

This might fit in nicely with some earlier work on trajectories of information....

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