2,802 research outputs found
Learning about a Categorical Latent Variable under Prior Near-Ignorance
It is well known that complete prior ignorance is not compatible with
learning, at least in a coherent theory of (epistemic) uncertainty. What is
less widely known, is that there is a state similar to full ignorance, that
Walley calls near-ignorance, that permits learning to take place. In this paper
we provide new and substantial evidence that also near-ignorance cannot be
really regarded as a way out of the problem of starting statistical inference
in conditions of very weak beliefs. The key to this result is focusing on a
setting characterized by a variable of interest that is latent. We argue that
such a setting is by far the most common case in practice, and we show, for the
case of categorical latent variables (and general manifest variables) that
there is a sufficient condition that, if satisfied, prevents learning to take
place under prior near-ignorance. This condition is shown to be easily
satisfied in the most common statistical problems.Comment: 15 LaTeX page
Limits of Learning about a Categorical Latent Variable under Prior Near-Ignorance
In this paper, we consider the coherent theory of (epistemic) uncertainty of
Walley, in which beliefs are represented through sets of probability
distributions, and we focus on the problem of modeling prior ignorance about a
categorical random variable. In this setting, it is a known result that a state
of prior ignorance is not compatible with learning. To overcome this problem,
another state of beliefs, called \emph{near-ignorance}, has been proposed.
Near-ignorance resembles ignorance very closely, by satisfying some principles
that can arguably be regarded as necessary in a state of ignorance, and allows
learning to take place. What this paper does, is to provide new and substantial
evidence that also near-ignorance cannot be really regarded as a way out of the
problem of starting statistical inference in conditions of very weak beliefs.
The key to this result is focusing on a setting characterized by a variable of
interest that is \emph{latent}. We argue that such a setting is by far the most
common case in practice, and we provide, for the case of categorical latent
variables (and general \emph{manifest} variables) a condition that, if
satisfied, prevents learning to take place under prior near-ignorance. This
condition is shown to be easily satisfied even in the most common statistical
problems. We regard these results as a strong form of evidence against the
possibility to adopt a condition of prior near-ignorance in real statistical
problems.Comment: 27 LaTeX page
A Noninformative Prior on a Space of Distribution Functions
In a given problem, the Bayesian statistical paradigm requires the
specification of a prior distribution that quantifies relevant information
about the unknowns of main interest external to the data. In cases where little
such information is available, the problem under study may possess an
invariance under a transformation group that encodes a lack of information,
leading to a unique prior---this idea was explored at length by E.T. Jaynes.
Previous successful examples have included location-scale invariance under
linear transformation, multiplicative invariance of the rate at which events in
a counting process are observed, and the derivation of the Haldane prior for a
Bernoulli success probability. In this paper we show that this method can be
extended, by generalizing Jaynes, in two ways: (1) to yield families of
approximately invariant priors, and (2) to the infinite-dimensional setting,
yielding families of priors on spaces of distribution functions. Our results
can be used to describe conditions under which a particular Dirichlet Process
posterior arises from an optimal Bayesian analysis, in the sense that
invariances in the prior and likelihood lead to one and only one posterior
distribution
Robust Inference of Trees
This paper is concerned with the reliable inference of optimal
tree-approximations to the dependency structure of an unknown distribution
generating data. The traditional approach to the problem measures the
dependency strength between random variables by the index called mutual
information. In this paper reliability is achieved by Walley's imprecise
Dirichlet model, which generalizes Bayesian learning with Dirichlet priors.
Adopting the imprecise Dirichlet model results in posterior interval
expectation for mutual information, and in a set of plausible trees consistent
with the data. Reliable inference about the actual tree is achieved by focusing
on the substructure common to all the plausible trees. We develop an exact
algorithm that infers the substructure in time O(m^4), m being the number of
random variables. The new algorithm is applied to a set of data sampled from a
known distribution. The method is shown to reliably infer edges of the actual
tree even when the data are very scarce, unlike the traditional approach.
Finally, we provide lower and upper credibility limits for mutual information
under the imprecise Dirichlet model. These enable the previous developments to
be extended to a full inferential method for trees.Comment: 26 pages, 7 figure
Atmospheric Retrieval for Super-Earths: Uniquely Constraining the Atmospheric Composition with Transmission Spectroscopy
We present a retrieval method based on Bayesian analysis to infer the
atmospheric compositions and surface or cloud-top pressures from transmission
spectra of exoplanets with general compositions. In this study, we identify
what can unambiguously be determined about the atmospheres of exoplanets from
their transmission spectra by applying the retrieval method to synthetic
observations of the super-Earth GJ 1214b. Our approach to infer constraints on
atmospheric parameters is to compute their joint and marginal posterior
probability distributions using the MCMC technique in a parallel tempering
scheme. A new atmospheric parameterization is introduced that is applicable to
general atmospheres in which the main constituent is not known a priori and
clouds may be present. Our main finding is that a unique constraint of the
mixing ratios of the absorbers and up to two spectrally inactive gases (such as
N2 and primordial H2+He) is possible if the observations are sufficient to
quantify both (1) the broadband transit depths in at least one absorption
feature for each absorber and (2) the slope and strength of the molecular
Rayleigh scattering signature. The surface or cloud-top pressure can be
quantified if a surface or cloud deck is present. The mean molecular mass can
be constrained from the Rayleigh slope or the shapes of absorption features,
thus enabling to distinguish between cloudy hydrogen-rich atmospheres and high
mean molecular mass atmospheres. We conclude, however, that without the
signature of Rayleigh scattering--even with robustly detected infrared
absorption features--there is no reliable way to tell if the absorber is the
main constituent of the atmosphere or just a minor species with a mixing ratio
of <0.1%. The retrieval method leads us to a conceptual picture of which
details in transmission spectra are essential for unique characterizations of
well-mixed atmospheres.Comment: 23 pages, 13 figures, accepted at ApJ, submitted to ApJ on Nov 4,
201
A Gaussian process framework for modelling instrumental systematics: application to transmission spectroscopy
Transmission spectroscopy, which consists of measuring the
wavelength-dependent absorption of starlight by a planet's atmosphere during a
transit, is a powerful probe of atmospheric composition. However, the expected
signal is typically orders of magnitude smaller than instrumental systematics,
and the results are crucially dependent on the treatment of the latter. In this
paper, we propose a new method to infer transit parameters in the presence of
systematic noise using Gaussian processes, a technique widely used in the
machine learning community for Bayesian regression and classification problems.
Our method makes use of auxiliary information about the state of the
instrument, but does so in a non-parametric manner, without imposing a specific
dependence of the systematics on the instrumental parameters, and naturally
allows for the correlated nature of the noise. We give an example application
of the method to archival NICMOS transmission spectroscopy of the hot Jupiter
HD 189733, which goes some way towards reconciling the controversy surrounding
this dataset in the literature. Finally, we provide an appendix giving a
general introduction to Gaussian processes for regression, in order to
encourage their application to a wider range of problems.Comment: 6 figures, 1 table, accepted for publication in MNRA
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