255,465 research outputs found
Characterizing predictable classes of processes
The problem is sequence prediction in the following setting. A sequence
of discrete-valued observations is generated according to
some unknown probabilistic law (measure) . After observing each outcome,
it is required to give the conditional probabilities of the next observation.
The measure belongs to an arbitrary class \C of stochastic processes.
We are interested in predictors whose conditional probabilities converge
to the "true" -conditional probabilities if any \mu\in\C is chosen to
generate the data. We show that if such a predictor exists, then a predictor
can also be obtained as a convex combination of a countably many elements of
\C. In other words, it can be obtained as a Bayesian predictor whose prior is
concentrated on a countable set. This result is established for two very
different measures of performance of prediction, one of which is very strong,
namely, total variation, and the other is very weak, namely, prediction in
expected average Kullback-Leibler divergence
On Characterizing Spector Classes
We study in this paper characterizations of various interesting classes of relations arising in recursion theory. We first determine which Spector classes on the structure of arithmetic arise from recursion in normal type 2 objects, giving a partial answer to a problem raised by Moschovakis [8], where the notion of Spector class was first
essentially introduced. Our result here was independently discovered by S. G. Simpson (see [3]). We conclude our study of Spector classes by examining two simple relations between them and a natural hierarchy to which they give rise
Topologies for intermediate logics
We investigate the problem of characterizing the classes of Grothendieck
toposes whose internal logic satisfies a given assertion in the theory of
Heyting algebras, and introduce natural analogues of the double negation and De
Morgan topologies on an elementary topos for a wide class of intermediate
logics.Comment: 21 page
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