27,924 research outputs found
A categorical characterization of relative entropy on standard Borel spaces
We give a categorical treatment, in the spirit of Baez and Fritz, of relative
entropy for probability distributions defined on standard Borel spaces. We
define a category suitable for reasoning about statistical inference on
standard Borel spaces. We define relative entropy as a functor into Lawvere's
category and we show convexity, lower semicontinuity and uniqueness.Comment: 16 page
Multiwinner Analogues of Plurality Rule: Axiomatic and Algorithmic Perspectives
We characterize the class of committee scoring rules that satisfy the
fixed-majority criterion. In some sense, the committee scoring rules in this
class are multiwinner analogues of the single-winner Plurality rule, which is
uniquely characterized as the only single-winner scoring rule that satisfies
the simple majority criterion. We define top--counting committee scoring
rules and show that the fixed majority consistent rules are a subclass of the
top--counting rules. We give necessary and sufficient conditions for a
top--counting rule to satisfy the fixed-majority criterion. We find that,
for most of the rules in our new class, the complexity of winner determination
is high (that is, the problem of computing the winners is NP-hard), but we also
show examples of rules with polynomial-time winner determination procedures.
For some of the computationally hard rules, we provide either exact FPT
algorithms or approximate polynomial-time algorithms
Local proper scoring rules of order two
Scoring rules assess the quality of probabilistic forecasts, by assigning a
numerical score based on the predictive distribution and on the event or value
that materializes. A scoring rule is proper if it encourages truthful
reporting. It is local of order if the score depends on the predictive
density only through its value and the values of its derivatives of order up to
at the realizing event. Complementing fundamental recent work by Parry,
Dawid and Lauritzen, we characterize the local proper scoring rules of order 2
relative to a broad class of Lebesgue densities on the real line, using a
different approach. In a data example, we use local and nonlocal proper scoring
rules to assess statistically postprocessed ensemble weather forecasts.Comment: Published in at http://dx.doi.org/10.1214/12-AOS973 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A Smooth Transition from Powerlessness to Absolute Power
We study the phase transition of the coalitional manipulation problem for
generalized scoring rules. Previously it has been shown that, under some
conditions on the distribution of votes, if the number of manipulators is
, where is the number of voters, then the probability that a
random profile is manipulable by the coalition goes to zero as the number of
voters goes to infinity, whereas if the number of manipulators is
, then the probability that a random profile is manipulable
goes to one. Here we consider the critical window, where a coalition has size
, and we show that as goes from zero to infinity, the limiting
probability that a random profile is manipulable goes from zero to one in a
smooth fashion, i.e., there is a smooth phase transition between the two
regimes. This result analytically validates recent empirical results, and
suggests that deciding the coalitional manipulation problem may be of limited
computational hardness in practice.Comment: 22 pages; v2 contains minor changes and corrections; v3 contains
minor changes after comments of reviewer
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