Abstract
In this section, we treat the class of systems that can be described by piecewise bijective functions. We call a function piecewise bijective if every output value originates from at most countably many input values, i.e., if the preimage of every output value is an at most countable set.
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- 1.
Edwin T. Jaynes expressed his dislike of differential entropy with the following words: “[...] the entropy of a continuous probability distribution is not an invariant. This is due to the historical accident that in his original papers, Shannon assumed, without calculating, that the analog of \(\sum p_i\log p_i\) was \(\int w \log w \mathrm {d}x\) [...] we have realized that mathematical deduction from the uniqueness theorem, instead of guesswork, yields [an] invariant information measure” [Jay63, p. 202]. At this time, Rényi’s work had already been published, justifying differential entropy from a completely different point-of-view.
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This example appeared in slightly different forms in [GFK11, Sect. V.C] and [GK16, Sect. 5.5].
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Geiger, B.C., Kubin, G. (2018). Piecewise Bijective Functions and Continuous Inputs. In: Information Loss in Deterministic Signal Processing Systems. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-59533-7_2
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DOI: https://doi.org/10.1007/978-3-319-59533-7_2
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