Abstract
This chapter finally contains the definition of novelty, information and surprise for arbitrary covers and in particular for repertoires and some methods for their practical calculation. We give the broadest possible definitions of these terms for arbitrary covers, because we use it occasionally in Part VI. Practically it would be sufficient to define everything just for repertoires. It turns out that the theories of novelty and of information on repertoires are both proper extensions of classical information theory (where complementary theorems hold), which coincide with each other and with classical information theory, when the repertoires are partitions.
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- 1.
For arbitrary, possibly uncountable repertoires the max may be a sup (i.e., not attained). It may also be that the expectation does not exist or is infinite for some \(d \in D(\alpha )\); in this case, the max is defined as ∞. For finite repertoires α we will see that the max exists and is finite.
- 2.
For arbitrary, possible uncountable covers the min may be a inf (i.e., not attained). It may also be that the expectation does not exist or is infinite for all \(d \in D(\alpha )\); in this case, the min is defined as ∞. For finite repertoires α we will see that the min exists and is finite.
- 3.
If there is no \(d \in D(\alpha )\) with \(\mathcal{N}(d) = \mathcal{N}(\alpha )\), this definition is not reasonable. In this case, it should be replaced by \(\widehat{\mathcal{I}}(\alpha ) {=\lim }_{{a}_{n}\rightarrow \mathcal{N}(\alpha )}\min \{\mathcal{I}(d): d \in D(\alpha ),\mathcal{N}(d) \geq {a}_{n}\}\) and \(\widehat{\mathcal{S}}(\alpha ) {=\lim }_{{a}_{n}\rightarrow \mathcal{N}(\alpha )}\max \{\mathcal{S}(d): d \in D(\alpha ),\mathcal{N}(d) \geq {a}_{n}\}\).
- 4.
\({N}_{\alpha }\) is a random variable if α is at most countable. It may happen that \({N}_{\alpha }(\omega ) = \infty \) on a set of nonzero probability. In this case, of course, \(E({N}_{\alpha }) = \infty \).
- 5.
For this we need α and β to be finitary.
- 6.
This definition can be easily extended to countable repertoires with \(a: \mathbb{N} \rightarrow \alpha \).
- 7.
This proposition can be easily extended to countable repertoires,and even to arbitrary repertoires. This is because any “reasonable” description d has a countable range \(R(d) \subseteq \alpha \) (see Definition 2.3).
- 8.
This proposition actually holds for arbitrary repertoires in the same way as Proposition 10.7, if there is a description \(d \in D(\alpha )\) which satisfies the additional condition in Definition 2.3.
References
Palm, G. (1975). Entropie und Generatoren in dynamischen Verbänden. PhD Thesis, Tübingen.
Palm, G. (1976a). A common generalization of topological and measure-theoretic entropy. Astérisque, 40, 159–165.
Palm, G. (1976b). Entropie und Erzeuer in dynamischen Verbänden. Z. Wahrscheinlichkeitstheorie verw. Geb., 36, 27–45.
Palm, G. (1981). Evidence, information and surprise. Biological Cybernetics, 42(1), 57–68.
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© 2012 Springer-Verlag Berlin Heidelberg
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Palm, G. (2012). Novelty, Information and Surprise of Repertoires. In: Novelty, Information and Surprise. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29075-6_10
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