arXiv
(422.153 recursos)
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Mostrando recursos 1 - 16 de 16
1.
Getting the Measure of the Flatness Problem - Evrard, Guillaume; Coles, Peter
The problem of estimating cosmological parameters such as $\Omega$ from noisy
or incomplete data is an example of an inverse problem and, as such, generally
requires a probablistic approach. We adopt the Bayesian interpretation of
probability for such problems and stress the connection between probability and
information which this approach makes explicit.
This connection is important even when information is ``minimal'' or, in other
words, when we need to argue from a state of maximum ignorance. We use the
transformation group method of Jaynes to assign minimally--informative prior
probability measure for cosmological parameters in the simple example of a dust
Friedman model, showing that the usual statements of...
2.
Confidence Intervals from One One Observation - Rodriguez, Carlos C.
Robert Machol's surprising result, that from a single observation it is
possible to have finite length confidence intervals for the parameters of
location-scale models, is re-produced and extended. Two previously unpublished
modifications are included. First, Herbert Robbins nonparametric confidence
interval is obtained. Second, I introduce a technique for obtaining confidence
intervals for the scale parameter of finite length in the logarithmic metric.
Keywords: Theory/Foundations , Estimation, Prior Distributions,
Non-parametrics & Semi-parametrics Geometry of Inference,
Confidence Intervals, Location-Scale models
3.
Bayes linear adjustment for variance matrices - Wilkinson, Darren J; Goldstein, Michael
We examine the problem of covariance belief revision using a geometric
approach. We exhibit an inner-product space where covariance matrices live
naturally --- a space of random real symmetric matrices. The inner-product on
this space captures aspects of our beliefs about the relationship between
covariance matrices of interest to us, providing a structure rich enough for us
to adjust beliefs about unknown matrices in the light of data such as sample
covariance matrices, exploiting second-order exchangeability specifications.
4.
Bayes linear covariance matrix adjustment for multivariate dynamic
linear models - Wilkinson, Darren J; Goldstein, Michael
A methodology is developed for the adjustment of the covariance matrices
underlying a multivariate constant time series dynamic linear model. The
covariance matrices are embedded in a distribution-free inner-product space of
matrix objects which facilitates such adjustment. This approach helps to make
the analysis simple, tractable and robust. To illustrate the methods, a simple
model is developed for a time series representing sales of certain brands of a
product from a cash-and-carry depot. The covariance structure underlying the
model is revised, and the benefits of this revision on first order inferences
are then examined.
5.
Minimal information in velocity space - Evrard, Guillaume
Jaynes' transformation group principle is used to derive the objective prior
for the velocity of a non-zero rest-mass particle. In the case of classical
mechanics, invariance under the classical law of addition of velocities, leads
to an improper constant prior over the unbounded velocity space of classical
mechanics. The application of the relativistic law of addition of velocities
leads to a less simple prior. It can however be rewritten as a uniform
volumetric distribution if the relativistic velocity space is given a
non-trivial metric.
6.
Bayesian Variable Selection with Related Predictors - Chipman, Hugh
In data sets with many predictors, algorithms for identifying a good subset
of predictors are often used. Most such algorithms do not account for any
relationships between predictors. For example, stepwise regression might select
a model containing an interaction AB but neither main effect A or B. This paper
develops mathematical representations of this and other relations between
predictors, which may then be incorporated in a model selection procedure. A
Bayesian approach that goes beyond the standard independence prior for variable
selection is adopted, and preference for certain models is interpreted as prior
information. Priors relevant to arbitrary interactions and polynomials, dummy
variables for categorical factors, competing predictors, and...
7.
Bayesian Method of Moments (BMOM) Analysis of Mean and Regression Models - Zellner, Arnold
A Bayesian method of moments/instrumental variable (BMOM/IV) approach is
developed and applied in the analysis of the important mean and multiple
regression models. Given a single set of data, it is shown how to obtain
posterior and predictive moments without the use of likelihood functions, prior
densities and Bayes' Theorem. The posterior and predictive moments, based on a
few relatively weak assumptions, are then used to obtain maximum entropy
densities for parameters, realized error terms and future values of variables.
Posterior means for parameters and realized error terms are shown to be equal
to certain well known estimates and rationalized in terms of quadratic loss
functions. Conditional maxent posterior...
8.
Bayes linear covariance matrix adjustment - Wilkinson, Darren J
In this thesis, a Bayes linear methodology for the adjustment of covariance
matrices is presented and discussed. A geometric framework for quantifying
uncertainties about covariance matrices is set up, and an inner-product for
spaces of random matrices is motivated and constructed. The inner-product on
this space captures aspects of our beliefs about the relationship between
covariance matrices of interest to us, providing a structure rich enough for us
to adjust beliefs about unknown matrices in the light of data such as sample
covariance matrices, exploiting second-order exchangeability and related
specifications to obtain representations allowing analysis.
Adjustment is associated with orthogonal projection, and illustrated with
examples of adjustments for some...
9.
Toward general solutions to time-series problems: Notes on obstacles and
noise - Gideoni, Iftah
Computational difficulties in the general application of Bretthorsts
formalism to time-series problems, posed by the large number of possible models
and the use of models with nonorthogonal base-functions are discussed. The
specific problem under consideration is a Bayesian procedure for model
selection, parameter estimation, and classification, that was applied to the
search for the {\it {In Vivo}} $T_2$ decay rate distributions in brain tissues.
Through the estimation of the meta-parameter $\sigma$ in the process, we also
gain a better understanding of the meaning and estimation of "noise" in the
frame-work of probability theory as logic.
10.
Bayes linear variance adjustment for time series - Wilkinson, Darren J
This paper exhibits quadratic products of linear combinations of observables
which identify the covariance structure underlying the univariate locally
linear time series dynamic linear model. The first- and second-order moments
for the joint distribution over these observables are given, allowing Bayes
linear learning for the underlying covariance structure for the time series
model. An example is given which illustrates the methodology and highlights the
practical implications of the theory.
11.
Local computation of influence propagation through Bayes linear belief
networks - Wilkinson, Darren J
In recent years there has been interest in the theory of local computation
over probabilistic Bayesian graphical models. In this paper, local computation
over Bayes linear belief networks is shown to be amenable to a similar
approach. However, the linear structure offers many simplifications and
advantages relative to more complex models, and these are examined with
reference to some illustrative examples.
12.
A Geometric Formulation of Occam's Razor for Inference of Parametric
Distributions - Balasubramanian, Vijay
I define a natural measure of the complexity of a parametric distribution
relative to a given true distribution called the {\it razor} of a model family.
The Minimum Description Length principle (MDL) and Bayesian inference are shown
to give empirical approximations of the razor via an analysis that
significantly extends existing results on the asymptotics of Bayesian model
selection. I treat parametric families as manifolds embedded in the space of
distributions and derive a canonical metric and a measure on the parameter
manifold by appealing to the classical theory of hypothesis testing. I find
that the Fisher information is the natural measure of distance, and give a
novel justification...
13.
Statistical Inference, Occam's Razor and Statistical Mechanics on The
Space of Probability Distributions - Balasubramanian, Vijay
The task of parametric model selection is cast in terms of a statistical
mechanics on the space of probability distributions. Using the techniques of
low-temperature expansions, we arrive at a systematic series for the Bayesian
posterior probability of a model family that significantly extends known
results in the literature. In particular, we arrive at a precise understanding
of how Occam's Razor, the principle that simpler models should be preferred
until the data justifies more complex models, is automatically embodied by
probability theory. These results require a measure on the space of model
parameters and we derive and discuss an interpretation of Jeffreys' prior
distribution as a uniform prior over...
14.
Suppressing Random Walks in Markov Chain Monte Carlo Using Ordered
Overrelaxation - Neal, R. M.
Markov chain Monte Carlo methods such as Gibbs sampling and simple forms of
the Metropolis algorithm typically move about the distribution being sampled
via a random walk. For the complex, high-dimensional distributions commonly
encountered in Bayesian inference and statistical physics, the distance moved
in each iteration of these algorithms will usually be small, because it is
difficult or impossible to transform the problem to eliminate dependencies
between variables. The inefficiency inherent in taking such small steps is
greatly exacerbated when the algorithm operates via a random walk, as in such a
case moving to a point n steps away will typically take around n^2 iterations.
Such random walks can...
15.
Probability and Measurement Uncertainty in Physics - a Bayesian Primer - D'Agostini, G.
Bayesian statistics is based on the subjective definition of probability as
{\it ``degree of belief''} and on Bayes' theorem, the basic tool for assigning
probabilities to hypotheses combining {\it a priori} judgements and
experimental information. This was the original point of view of Bayes,
Bernoulli, Gauss, Laplace, etc. and contrasts with later ``conventional''
(pseudo-)definitions of probabilities, which implicitly presuppose the concept
of probability. These notes show that the Bayesian approach is the natural one
for data analysis in the most general sense, and for assigning uncertainties to
the results of physical measurements - while at the same time resolving
philosophical aspects of the problems. The approach, although little known...
16.
A Theory of Measurement Uncertainty Based on Conditional Probability - D'Agostini, G.
A theory of measurement uncertainty is presented, which, since it is based
exclusively on the Bayesian approach and on the subjective concept of
conditional probability, is applicable in the most general cases.
The recent International Organization for Standardization (ISO)
recommendation on measurement uncertainty is reobtained as the limit case in
which linearization is meaningful and one is interested only in the best
estimates of the quantities and in their variances.
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