604 research outputs found
Regression approaches for Approximate Bayesian Computation
This book chapter introduces regression approaches and regression adjustment
for Approximate Bayesian Computation (ABC). Regression adjustment adjusts
parameter values after rejection sampling in order to account for the imperfect
match between simulations and observations. Imperfect match between simulations
and observations can be more pronounced when there are many summary statistics,
a phenomenon coined as the curse of dimensionality. Because of this imperfect
match, credibility intervals obtained with regression approaches can be
inflated compared to true credibility intervals. The chapter presents the main
concepts underlying regression adjustment. A theorem that compares theoretical
properties of posterior distributions obtained with and without regression
adjustment is presented. Last, a practical application of regression adjustment
in population genetics shows that regression adjustment shrinks posterior
distributions compared to rejection approaches, which is a solution to avoid
inflated credibility intervals.Comment: Book chapter, published in Handbook of Approximate Bayesian
Computation 201
Sharp Oracle Inequalities for Aggregation of Affine Estimators
We consider the problem of combining a (possibly uncountably infinite) set of
affine estimators in non-parametric regression model with heteroscedastic
Gaussian noise. Focusing on the exponentially weighted aggregate, we prove a
PAC-Bayesian type inequality that leads to sharp oracle inequalities in
discrete but also in continuous settings. The framework is general enough to
cover the combinations of various procedures such as least square regression,
kernel ridge regression, shrinking estimators and many other estimators used in
the literature on statistical inverse problems. As a consequence, we show that
the proposed aggregate provides an adaptive estimator in the exact minimax
sense without neither discretizing the range of tuning parameters nor splitting
the set of observations. We also illustrate numerically the good performance
achieved by the exponentially weighted aggregate
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
Modelling the spatial distribution of DEM Error
Assessment of a DEM’s quality is usually undertaken by deriving a measure of DEM accuracy – how close the DEM’s elevation values are to the true elevation. Measures such as Root Mean Squared Error and standard deviation of the error are frequently used. These measures summarise elevation errors in a DEM as a single value. A more detailed description of DEM accuracy would allow better understanding of DEM quality and the consequent uncertainty associated with using DEMs in analytical applications. The research presented addresses the limitations of using a single root mean squared error (RMSE) value to represent the uncertainty associated with a DEM by developing a new technique for creating a spatially distributed model of DEM quality – an accuracy surface. The technique is based on the hypothesis that the distribution and scale of elevation error within a DEM are at least partly related to morphometric characteristics of the terrain. The technique involves generating a set of terrain parameters to characterise terrain morphometry and developing regression models to define the relationship between DEM error and morphometric character. The regression models form the basis for creating standard deviation surfaces to represent DEM accuracy. The hypothesis is shown to be true and reliable accuracy surfaces are successfully created. These accuracy surfaces provide more detailed information about DEM accuracy than a single global estimate of RMSE
Conditional Transformation Models
The ultimate goal of regression analysis is to obtain information about the
conditional distribution of a response given a set of explanatory variables.
This goal is, however, seldom achieved because most established regression
models only estimate the conditional mean as a function of the explanatory
variables and assume that higher moments are not affected by the regressors.
The underlying reason for such a restriction is the assumption of additivity of
signal and noise. We propose to relax this common assumption in the framework
of transformation models. The novel class of semiparametric regression models
proposed herein allows transformation functions to depend on explanatory
variables. These transformation functions are estimated by regularised
optimisation of scoring rules for probabilistic forecasts, e.g. the continuous
ranked probability score. The corresponding estimated conditional distribution
functions are consistent. Conditional transformation models are potentially
useful for describing possible heteroscedasticity, comparing spatially varying
distributions, identifying extreme events, deriving prediction intervals and
selecting variables beyond mean regression effects. An empirical investigation
based on a heteroscedastic varying coefficient simulation model demonstrates
that semiparametric estimation of conditional distribution functions can be
more beneficial than kernel-based non-parametric approaches or parametric
generalised additive models for location, scale and shape
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