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For any ordering of the N received bits, consider the I bit positions I ⊆ { 1, 2, …, N }, and the I I 1 subsets I 1 ⊆ I of I1 bit positions, such that I1≤I≤N.
The expression ν ( A ) - ν ( A ∖ { k } ) is the marginal payoff of player k to the coalition A. The Shapley value can be interpreted as the expected marginal contribution made by a player to the value of a coalition, where the distribution of coalitions is such that any ordering of the players is equally likely.
It is likely that for many inputs, any ordering of these filters would yield indistinguishable performance.
The LCA classes were considered as dependent binomial variables without any ordering of the latent classes.
In particular, any ordering of Y that is compatible with ℭ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbaceaGae8xlHmeaaa@3882@ is also compatible with ω.
This allows a more efficient recursion over reconciliations, but at the cost of considering reconciliations that contain transfer events that are not consistent with any ordering of the species tree (Tofigh et al., 2011).
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This solves the maximum nonlinearity issue for 7-variable functions with any order of resiliency.
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