11,389 research outputs found
Convex Rank Tests and Semigraphoids
Convex rank tests are partitions of the symmetric group which have desirable
geometric properties. The statistical tests defined by such partitions involve
counting all permutations in the equivalence classes. Each class consists of
the linear extensions of a partially ordered set specified by data. Our methods
refine existing rank tests of non-parametric statistics, such as the sign test
and the runs test, and are useful for exploratory analysis of ordinal data. We
establish a bijection between convex rank tests and probabilistic conditional
independence structures known as semigraphoids. The subclass of submodular rank
tests is derived from faces of the cone of submodular functions, or from
Minkowski summands of the permutohedron. We enumerate all small instances of
such rank tests. Of particular interest are graphical tests, which correspond
to both graphical models and to graph associahedra
When does aggregation reduce uncertainty aversion?
We study the problem of uncertainty sharing within a household: "risk sharing," in a context of Knightian uncertainty. A household shares uncertain prospects using a social welfare function. We characterize the social welfare functions such that the household is collectively less averse to uncertainty than each member, and satises the Pareto principle and an independence axiom. We single out the sum of certainty equivalents as the unique member of this family which provides quasiconcave rankings over risk-free allocations
Process, System, Causality, and Quantum Mechanics, A Psychoanalysis of Animal Faith
We shall argue in this paper that a central piece of modern physics does not
really belong to physics at all but to elementary probability theory. Given a
joint probability distribution J on a set of random variables containing x and
y, define a link between x and y to be the condition x=y on J. Define the {\it
state} D of a link x=y as the joint probability distribution matrix on x and y
without the link. The two core laws of quantum mechanics are the Born
probability rule, and the unitary dynamical law whose best known form is the
Schrodinger's equation. Von Neumann formulated these two laws in the language
of Hilbert space as prob(P) = trace(PD) and D'T = TD respectively, where P is a
projection, D and D' are (von Neumann) density matrices, and T is a unitary
transformation. We'll see that if we regard link states as density matrices,
the algebraic forms of these two core laws occur as completely general theorems
about links. When we extend probability theory by allowing cases to count
negatively, we find that the Hilbert space framework of quantum mechanics
proper emerges from the assumption that all D's are symmetrical in rows and
columns. On the other hand, Markovian systems emerge when we assume that one of
every linked variable pair has a uniform probability distribution. By
representing quantum and Markovian structure in this way, we see clearly both
how they differ, and also how they can coexist in natural harmony with each
other, as they must in quantum measurement, which we'll examine in some detail.
Looking beyond quantum mechanics, we see how both structures have their special
places in a much larger continuum of formal systems that we have yet to look
for in nature.Comment: LaTex, 86 page
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