114 research outputs found
Quantifying dimensionality: Bayesian cosmological model complexities
We demonstrate a measure for the effective number of parameters constrained
by a posterior distribution in the context of cosmology. In the same way that
the mean of the Shannon information (i.e. the Kullback-Leibler divergence)
provides a measure of the strength of constraint between prior and posterior,
we show that the variance of the Shannon information gives a measure of
dimensionality of constraint. We examine this quantity in a cosmological
context, applying it to likelihoods derived from Cosmic Microwave Background,
large scale structure and supernovae data. We show that this measure of
Bayesian model dimensionality compares favourably both analytically and
numerically in a cosmological context with the existing measure of model
complexity used in the literature.Comment: 14 pages, 9 figures. v2: updates post peer-review. v3: typographical
correction to equation 3
Quantifying tensions in cosmological parameters: Interpreting the DES evidence ratio
We provide a new interpretation for the Bayes factor combination used in the
Dark Energy Survey (DES) first year analysis to quantify the tension between
the DES and Planck datasets. The ratio quantifies a Bayesian confidence in our
ability to combine the datasets. This interpretation is prior-dependent, with
wider prior widths boosting the confidence. We therefore propose that if there
are any reasonable priors which reduce the confidence to below unity, then we
cannot assert that the datasets are compatible. Computing the evidence ratios
for the DES first year analysis and Planck, given that narrower priors drop the
confidence to below unity, we conclude that DES and Planck are, in a Bayesian
sense, incompatible under LCDM. Additionally we compute ratios which confirm
the consensus that measurements of the acoustic scale by the Baryon Oscillation
Spectroscopic Survey (SDSS) are compatible with Planck, whilst direct
measurements of the acceleration rate of the Universe by the SHOES
collaboration are not. We propose a modification to the Bayes ratio which
removes the prior dependency using Kullback-Leibler divergences, and using this
statistical test find Planck in strong tension with SHOES, in moderate tension
with DES, and in no tension with SDSS. We propose this statistic as the optimal
way to compare datasets, ahead of the next DES data releases, as well as future
surveys. Finally, as an element of these calculations, we introduce in a
cosmological setting the Bayesian model dimensionality, which is a
parameterisation-independent measure of the number of parameters that a given
dataset constrains.Comment: 16 pages, 9 figures. v2 & v3: updates post peer-review. v4:
typographical correction to the reported errors in the log S column of Table
II. v5: typographical correction to equation 2
Analytical approximations for curved primordial power spectra
We extend the work of Contaldi et al. and derive analytical approximations
for primordial power spectra arising from models of inflation which include
primordial spatial curvature. These analytical templates are independent of any
specific inflationary potential and therefore illustrate and provide insight
into the generic effects and predictions of primordial curvature, manifesting
as cut-offs and oscillations at low multipoles and agreeing with numerical
calculations. We identify through our analytical approximation that the effects
of curvature can be mathematically attributed to shifts in the wavevectors
participating dynamically.Comment: 11 pages, 2 figures, supplementary material available at
https://doi.org/10.5281/zenodo.4024321. v1: As submitted to PRD. v2: As
published in PRD (with only minor additions between v1 and v2
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