35,587 research outputs found
A Comparative Review of Dimension Reduction Methods in Approximate Bayesian Computation
Approximate Bayesian computation (ABC) methods make use of comparisons
between simulated and observed summary statistics to overcome the problem of
computationally intractable likelihood functions. As the practical
implementation of ABC requires computations based on vectors of summary
statistics, rather than full data sets, a central question is how to derive
low-dimensional summary statistics from the observed data with minimal loss of
information. In this article we provide a comprehensive review and comparison
of the performance of the principal methods of dimension reduction proposed in
the ABC literature. The methods are split into three nonmutually exclusive
classes consisting of best subset selection methods, projection techniques and
regularization. In addition, we introduce two new methods of dimension
reduction. The first is a best subset selection method based on Akaike and
Bayesian information criteria, and the second uses ridge regression as a
regularization procedure. We illustrate the performance of these dimension
reduction techniques through the analysis of three challenging models and data
sets.Comment: Published in at http://dx.doi.org/10.1214/12-STS406 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A universal approximate cross-validation criterion and its asymptotic distribution
A general framework is that the estimators of a distribution are obtained by
minimizing a function (the estimating function) and they are assessed through
another function (the assessment function). The estimating and assessment
functions generally estimate risks. A classical case is that both functions
estimate an information risk (specifically cross entropy); in that case Akaike
information criterion (AIC) is relevant. In more general cases, the assessment
risk can be estimated by leave-one-out crossvalidation. Since leave-one-out
crossvalidation is computationally very demanding, an approximation formula can
be very useful. A universal approximate crossvalidation criterion (UACV) for
the leave-one-out crossvalidation is given. This criterion can be adapted to
different types of estimators, including penalized likelihood and maximum a
posteriori estimators, and of assessment risk functions, including information
risk functions and continuous rank probability score (CRPS). This formula
reduces to Takeuchi information criterion (TIC) when cross entropy is the risk
for both estimation and assessment. The asymptotic distribution of UACV and of
a difference of UACV is given. UACV can be used for comparing estimators of the
distributions of ordered categorical data derived from threshold models and
models based on continuous approximations. A simulation study and an analysis
of real psychometric data are presented.Comment: 23 pages, 2 figure
Minimum Rates of Approximate Sufficient Statistics
Given a sufficient statistic for a parametric family of distributions, one
can estimate the parameter without access to the data. However, the memory or
code size for storing the sufficient statistic may nonetheless still be
prohibitive. Indeed, for independent samples drawn from a -nomial
distribution with degrees of freedom, the length of the code scales as
. In many applications, we may not have a useful notion of
sufficient statistics (e.g., when the parametric family is not an exponential
family) and we also may not need to reconstruct the generating distribution
exactly. By adopting a Shannon-theoretic approach in which we allow a small
error in estimating the generating distribution, we construct various {\em
approximate sufficient statistics} and show that the code length can be reduced
to . We consider errors measured according to the
relative entropy and variational distance criteria. For the code constructions,
we leverage Rissanen's minimum description length principle, which yields a
non-vanishing error measured according to the relative entropy. For the
converse parts, we use Clarke and Barron's formula for the relative entropy of
a parametrized distribution and the corresponding mixture distribution.
However, this method only yields a weak converse for the variational distance.
We develop new techniques to achieve vanishing errors and we also prove strong
converses. The latter means that even if the code is allowed to have a
non-vanishing error, its length must still be at least .Comment: To appear in the IEEE Transactions on Information Theor
Information measures and classicality in quantum mechanics
We study information measures in quantu mechanics, with particular emphasis
on providing a quantification of the notions of classicality and
predictability. Our primary tool is the Shannon - Wehrl entropy I. We give a
precise criterion for phase space classicality and argue that in view of this
a) I provides a measure of the degree of deviation from classicality for closed
system b) I - S (S the von Neumann entropy) plays the same role in open systems
We examine particular examples in non-relativistic quantum mechanics. Finally,
(this being one of our main motivations) we comment on field classicalisation
on early universe cosmology.Comment: 35 pages, LATE
Optimal Recovery of Local Truth
Probability mass curves the data space with horizons. Let f be a multivariate
probability density function with continuous second order partial derivatives.
Consider the problem of estimating the true value of f(z) > 0 at a single point
z, from n independent observations. It is shown that, the fastest possible
estimators (like the k-nearest neighbor and kernel) have minimum asymptotic
mean square errors when the space of observations is thought as conformally
curved. The optimal metric is shown to be generated by the Hessian of f in the
regions where the Hessian is definite. Thus, the peaks and valleys of f are
surrounded by singular horizons when the Hessian changes signature from
Riemannian to pseudo-Riemannian. Adaptive estimators based on the optimal
variable metric show considerable theoretical and practical improvements over
traditional methods. The formulas simplify dramatically when the dimension of
the data space is 4. The similarities with General Relativity are striking but
possibly illusory at this point. However, these results suggest that
nonparametric density estimation may have something new to say about current
physical theory.Comment: To appear in Proceedings of Maximum Entropy and Bayesian Methods
1999. Check also: http://omega.albany.edu:8008
Nonsubjective priors via predictive relative entropy regret
We explore the construction of nonsubjective prior distributions in Bayesian
statistics via a posterior predictive relative entropy regret criterion. We
carry out a minimax analysis based on a derived asymptotic predictive loss
function and show that this approach to prior construction has a number of
attractive features. The approach here differs from previous work that uses
either prior or posterior relative entropy regret in that we consider
predictive performance in relation to alternative nondegenerate prior
distributions. The theory is illustrated with an analysis of some specific
examples.Comment: Published at http://dx.doi.org/10.1214/009053605000000804 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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