12,314 research outputs found
Bayesian astrostatistics: a backward look to the future
This perspective chapter briefly surveys: (1) past growth in the use of
Bayesian methods in astrophysics; (2) current misconceptions about both
frequentist and Bayesian statistical inference that hinder wider adoption of
Bayesian methods by astronomers; and (3) multilevel (hierarchical) Bayesian
modeling as a major future direction for research in Bayesian astrostatistics,
exemplified in part by presentations at the first ISI invited session on
astrostatistics, commemorated in this volume. It closes with an intentionally
provocative recommendation for astronomical survey data reporting, motivated by
the multilevel Bayesian perspective on modeling cosmic populations: that
astronomers cease producing catalogs of estimated fluxes and other source
properties from surveys. Instead, summaries of likelihood functions (or
marginal likelihood functions) for source properties should be reported (not
posterior probability density functions), including nontrivial summaries (not
simply upper limits) for candidate objects that do not pass traditional
detection thresholds.Comment: 27 pp, 4 figures. A lightly revised version of a chapter in
"Astrostatistical Challenges for the New Astronomy" (Joseph M. Hilbe, ed.,
Springer, New York, forthcoming in 2012), the inaugural volume for the
Springer Series in Astrostatistics. Version 2 has minor clarifications and an
additional referenc
Frequentist Estimation of Cosmological Parameters from the MAXIMA-1 Cosmic Microwave Background Anisotropy Data
We use a frequentist statistical approach to set confidence intervals on the
values of cosmological parameters using the MAXIMA-1 and COBE measurements of
the angular power spectrum of the cosmic microwave background. We define a
statistic, simulate the measurements of MAXIMA-1 and COBE,
determine the probability distribution of the statistic, and use it and the
data to set confidence intervals on several cosmological parameters. We compare
the frequentist confidence intervals to Bayesian credible regions. The
frequentist and Bayesian approaches give best estimates for the parameters that
agree within 15%, and confidence interval-widths that agree within 30%. The
results also suggest that a frequentist analysis gives slightly broader
confidence intervals than a Bayesian analysis. The frequentist analysis gives
values of \Omega=0.89{+0.26\atop -0.19}, \Omega_{\rm B}h^2=0.026{+0.020\atop
-0.011} and n=1.02{+0.31\atop -0.10}, and the Bayesian analysis gives values of
\Omega=0.98{+0.14\atop -0.19}, \Omega_{\rm B}h^2=0.0.029{+0.015\atop-0.010},
and , all at the 95% confidence level.Comment: 10 pages, 9 Postscript figures, changes made to reflect published
versio
Comment on `Tainted evidence: cosmological model selection versus fitting', by Eric V. Linder and Ramon Miquel (astro-ph/0702542v2)
In astro-ph/0702542v2, Linder and Miquel seek to criticize the use of
Bayesian model selection for data analysis and for survey forecasting and
design. Their discussion is based on three serious misunderstandings of the
conceptual underpinnings and application of model-level Bayesian inference,
which invalidate all their main conclusions. Their paper includes numerous
further inaccuracies, including an erroneous calculation of the Bayesian
Information Criterion. Here we seek to set the record straight.Comment: 6 pages RevTeX
Chain ladder method: Bayesian bootstrap versus classical bootstrap
The intention of this paper is to estimate a Bayesian distribution-free chain
ladder (DFCL) model using approximate Bayesian computation (ABC) methodology.
We demonstrate how to estimate quantities of interest in claims reserving and
compare the estimates to those obtained from classical and credibility
approaches. In this context, a novel numerical procedure utilising Markov chain
Monte Carlo (MCMC), ABC and a Bayesian bootstrap procedure was developed in a
truly distribution-free setting. The ABC methodology arises because we work in
a distribution-free setting in which we make no parametric assumptions, meaning
we can not evaluate the likelihood point-wise or in this case simulate directly
from the likelihood model. The use of a bootstrap procedure allows us to
generate samples from the intractable likelihood without the requirement of
distributional assumptions, this is crucial to the ABC framework. The developed
methodology is used to obtain the empirical distribution of the DFCL model
parameters and the predictive distribution of the outstanding loss liabilities
conditional on the observed claims. We then estimate predictive Bayesian
capital estimates, the Value at Risk (VaR) and the mean square error of
prediction (MSEP). The latter is compared with the classical bootstrap and
credibility methods
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