276,403 research outputs found
A pitfall of piecewise-polytropic equation of state inference
The only messenger radiation in the Universe which one can use to
statistically probe the Equation of State (EOS) of cold dense matter is that
originating from the near-field vicinities of compact stars. Constraining
gravitational masses and equatorial radii of rotating compact stars is a major
goal for current and future telescope missions, with a primary purpose of
constraining the EOS. From a Bayesian perspective it is necessary to carefully
discuss prior definition; in this context a complicating issue is that in
practice there exist pathologies in the general relativistic mapping between
spaces of local (interior source matter) and global (exterior spacetime)
parameters. In a companion paper, these issues were raised on a theoretical
basis. In this study we reproduce a probability transformation procedure from
the literature in order to map a joint posterior distribution of Schwarzschild
gravitational masses and radii into a joint posterior distribution of EOS
parameters. We demonstrate computationally that EOS parameter inferences are
sensitive to the choice to define a prior on a joint space of these masses and
radii, instead of on a joint space interior source matter parameters. We focus
on the piecewise-polytropic EOS model, which is currently standard in the field
of astrophysical dense matter study. We discuss the implications of this issue
for the field.Comment: 16 pages, 9 figures. Accepted for publication in MNRA
Hall Normalization Constants for the Bures Volumes of the n-State Quantum Systems
We report the results of certain integrations of quantum-theoretic interest,
relying, in this regard, upon recently developed parameterizations of Boya et
al of the n x n density matrices, in terms of squared components of the unit
(n-1)-sphere and the n x n unitary matrices. Firstly, we express the normalized
volume elements of the Bures (minimal monotone) metric for n = 2 and 3,
obtaining thereby "Bures prior probability distributions" over the two- and
three-state systems. Then, as an essential first step in extending these
results to n > 3, we determine that the "Hall normalization constant" (C_{n})
for the marginal Bures prior probability distribution over the
(n-1)-dimensional simplex of the n eigenvalues of the n x n density matrices
is, for n = 4, equal to 71680/pi^2. Since we also find that C_{3} = 35/pi, it
follows that C_{4} is simply equal to 2^{11} C_{3}/pi. (C_{2} itself is known
to equal 2/pi.) The constant C_{5} is also found. It too is associated with a
remarkably simple decompositon, involving the product of the eight consecutive
prime numbers from 2 to 23.
We also preliminarily investigate several cases, n > 5, with the use of
quasi-Monte Carlo integration. We hope that the various analyses reported will
prove useful in deriving a general formula (which evidence suggests will
involve the Bernoulli numbers) for the Hall normalization constant for
arbitrary n. This would have diverse applications, including quantum inference
and universal quantum coding.Comment: 14 pages, LaTeX, 6 postscript figures. Revised version to appear in
J. Phys. A. We make a few slight changes from the previous version, but also
add a subsection (III G) in which several variations of the basic problem are
newly studied. Rather strong evidence is adduced that the Hall constants are
related to partial sums of denominators of the even-indexed Bernoulli
numbers, although a general formula is still lackin
Optimal compromise between incompatible conditional probability distributions, with application to Objective Bayesian Kriging
Models are often defined through conditional rather than joint distributions,
but it can be difficult to check whether the conditional distributions are
compatible, i.e. whether there exists a joint probability distribution which
generates them. When they are compatible, a Gibbs sampler can be used to sample
from this joint distribution. When they are not, the Gibbs sampling algorithm
may still be applied, resulting in a "pseudo-Gibbs sampler". We show its
stationary probability distribution to be the optimal compromise between the
conditional distributions, in the sense that it minimizes a mean squared misfit
between them and its own conditional distributions. This allows us to perform
Objective Bayesian analysis of correlation parameters in Kriging models by
using univariate conditional Jeffreys-rule posterior distributions instead of
the widely used multivariate Jeffreys-rule posterior. This strategy makes the
full-Bayesian procedure tractable. Numerical examples show it has near-optimal
frequentist performance in terms of prediction interval coverage
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