33 research outputs found
Restricted Coding and Betting
One of the fundamental themes in the study of computability theory are oracle computations, i.e. the coding of one infinite binary sequence into another. A coding process where the prefixes of the coded sequence are coded such that the length difference of the coded and the coding prefix is bounded by a constant is known as cl-reducibility. This reducibility has received considerable attention over the last two decades due to its interesting degree structure and because it exhibits strong connections with algorithmic randomness. In the first part of this dissertation, we study a slightly relaxed version of cl-reducibility where the length difference is required to be bounded by some specific nondecreasing computable function~. We show that in this relaxed model some of the classical results about cl-reducibility still hold in case the function grows slowly, at certain particular rates. Examples are the Yu-Ding theorem, which states that there is a pair of left-c.e. sequences that cannot be coded simultaneously by any left-c.e. sequence, as well as the Barmpalias-Lewis theorem that states that there is a left-c.e. sequence which cannot be coded by any random left-c.e. sequence. In case the bounding function~ grows too fast, both results don't hold anymore.
Betting strategies, which can be formulated equivalently in terms of martingales, are one of the main tools in the area of algorithmic randomness. A betting strategy is usually determined by two factors, the guessed outcome at every stage and the wager on it. In the second part of this dissertation we study betting strategies where one of these factors is restricted. First we study single-sided strategies, where the guessed outcome either is always 0 or is always 1. For computable strategies we show that single-sided strategies and usual strategies have the same power for winning, whereas the latter does not hold for strongly left-c.e. strategies, which are mixtures of computable strategies, even if we extend the class of single-sided strategies to the more general class of decidably-sided strategies.
Finally, we study the case where the wagers are forced to have a certain granularity, i.e. must be multiples of some not necessarily constant betting unit. For usual strategies, wins can always be assumed to have the two following properties (a) ‘win with arbitrarily small initial capital’ and (b) ‘win by saving’. In a setting of variable granularity, where the betting unit shrinks over stages, we study how the shrinking rates interact with these two properties. We show that if the granularity shrinks fast, at certain particular rates,for such granular strategies both properties are preserved. For slower rates of shrinking, we show that neither property is preserved completely, however, a weaker version of property (a) still holds. In order to investigate property (b) in this case, we consider more restricted strategies where in addition the wager is bounded from above
Asymmetry of the Kolmogorov complexity of online predicting odd and even bits
Symmetry of information states that .
We show that a similar relation for online Kolmogorov complexity does not hold.
Let the even (online Kolmogorov) complexity of an n-bitstring
be the length of a shortest program that computes on input ,
computes on input , etc; and similar for odd complexity. We
show that for all n there exist an n-bit x such that both odd and even
complexity are almost as large as the Kolmogorov complexity of the whole
string. Moreover, flipping odd and even bits to obtain a sequence
, decreases the sum of odd and even complexity to .Comment: 20 pages, 7 figure
Randomized and Exchangeable Improvements of Markov's, Chebyshev's and Chernoff's Inequalities
We present simple randomized and exchangeable improvements of Markov's
inequality, as well as Chebyshev's inequality and Chernoff bounds. Our variants
are never worse and typically strictly more powerful than the original
inequalities. The proofs are short and elementary, and can easily yield
similarly randomized or exchangeable versions of a host of other inequalities
that employ Markov's inequality as an intermediate step. We point out some
simple statistical applications involving tests that combine dependent
e-values. In particular, we uniformly improve the power of universal inference,
and obtain tighter betting-based nonparametric confidence intervals.
Simulations reveal nontrivial gains in power (and no losses) in a variety of
settings
Harnessing The Collective Wisdom: Fusion Learning Using Decision Sequences From Diverse Sources
Learning from the collective wisdom of crowds enhances the transparency of
scientific findings by incorporating diverse perspectives into the
decision-making process. Synthesizing such collective wisdom is related to the
statistical notion of fusion learning from multiple data sources or studies.
However, fusing inferences from diverse sources is challenging since
cross-source heterogeneity and potential data-sharing complicate statistical
inference. Moreover, studies may rely on disparate designs, employ widely
different modeling techniques for inferences, and prevailing data privacy norms
may forbid sharing even summary statistics across the studies for an overall
analysis. In this paper, we propose an Integrative Ranking and Thresholding
(IRT) framework for fusion learning in multiple testing. IRT operates under the
setting where from each study a triplet is available: the vector of binary
accept-reject decisions on the tested hypotheses, the study-specific False
Discovery Rate (FDR) level and the hypotheses tested by the study. Under this
setting, IRT constructs an aggregated, nonparametric, and discriminatory
measure of evidence against each null hypotheses, which facilitates ranking the
hypotheses in the order of their likelihood of being rejected. We show that IRT
guarantees an overall FDR control under arbitrary dependence between the
evidence measures as long as the studies control their respective FDR at the
desired levels. Furthermore, IRT synthesizes inferences from diverse studies
irrespective of the underlying multiple testing algorithms employed by them.
While the proofs of our theoretical statements are elementary, IRT is extremely
flexible, and a comprehensive numerical study demonstrates that it is a
powerful framework for pooling inferences.Comment: 29 pages and 10 figures. Under review at a journa