83,616 research outputs found
Measurability Is Not about Information
We comment on the relation between models of information based on signals/partitions, and those based on sigma-algebras. We show that more informative signals need not generate finer sigma-algebras, hence that Blackwell's theorem fails if information is modeled as sigma-algebras. The reason is that the sigma-algebra generated by a partition does not contain all the events that can be known from the information provided by the signal. We also show that there is a non-conventional sigma-algebra that can be associated to a signal which does preserve its information content. Further, expectations and conditional expectations may depend on the choice of sigma-algebra that is associated to a signal. We provide a simple characterization of when the model is robust to changes in the sigma-algebras.Blackwell's theorem, measurability, models of information, partitions, information- preserving sigma-algebras
Information is not about measurability
We present a simple example where the use of σ-algebras as a model of information leads to a paradoxical conclusion: a decisionmaker prefers less information to more. We then explain that the problem arises because the use of σ-algebras as the informational content of a signal is inadequate. We provide a characterization of the different models of information in the literature in terms of Blackwell’s theorem
Towers of recollement and bases for diagram algebras: planar diagrams and a little beyond
The recollement approach to the representation theory of sequences of
algebras is extended to pass basis information directly through the
globalisation functor. The method is hence adapted to treat sequences that are
not necessarily towers by inclusion, such as symplectic blob algebras (diagram
algebra quotients of the type-\hati{C} Hecke algebras).
By carefully reviewing the diagram algebra construction, we find a new set of
functors interrelating module categories of ordinary blob algebras (diagram
algebra quotients of the type- Hecke algebras) at {\em different} values
of the algebra parameters. We show that these functors generalise to determine
the structure of symplectic blob algebras, and hence of certain two-boundary
Temperley-Lieb algebras arising in Statistical Mechanics.
We identify the diagram basis with a cellular basis for each symplectic blob
algebra, and prove that these algebras are quasihereditary over a field for
almost all parameter choices, and generically semisimple. (That is, we give
bases for all cell and standard modules.)Comment: 61 page
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