1,590 research outputs found
Approximate Models and Robust Decisions
Decisions based partly or solely on predictions from probabilistic models may
be sensitive to model misspecification. Statisticians are taught from an early
stage that "all models are wrong", but little formal guidance exists on how to
assess the impact of model approximation on decision making, or how to proceed
when optimal actions appear sensitive to model fidelity. This article presents
an overview of recent developments across different disciplines to address
this. We review diagnostic techniques, including graphical approaches and
summary statistics, to help highlight decisions made through minimised expected
loss that are sensitive to model misspecification. We then consider formal
methods for decision making under model misspecification by quantifying
stability of optimal actions to perturbations to the model within a
neighbourhood of model space. This neighbourhood is defined in either one of
two ways. Firstly, in a strong sense via an information (Kullback-Leibler)
divergence around the approximating model. Or using a nonparametric model
extension, again centred at the approximating model, in order to `average out'
over possible misspecifications. This is presented in the context of recent
work in the robust control, macroeconomics and financial mathematics
literature. We adopt a Bayesian approach throughout although the methods are
agnostic to this position
Scalable Bayesian nonparametric regression via a Plackett-Luce model for conditional ranks
We present a novel Bayesian nonparametric regression model for covariates X
and continuous, real response variable Y. The model is parametrized in terms of
marginal distributions for Y and X and a regression function which tunes the
stochastic ordering of the conditional distributions F(y|x). By adopting an
approximate composite likelihood approach, we show that the resulting posterior
inference can be decoupled for the separate components of the model. This
procedure can scale to very large datasets and allows for the use of standard,
existing, software from Bayesian nonparametric density estimation and
Plackett-Luce ranking estimation to be applied. As an illustration, we show an
application of our approach to a US Census dataset, with over 1,300,000 data
points and more than 100 covariates
A General Framework for Updating Belief Distributions
We propose a framework for general Bayesian inference. We argue that a valid
update of a prior belief distribution to a posterior can be made for parameters
which are connected to observations through a loss function rather than the
traditional likelihood function, which is recovered under the special case of
using self information loss. Modern application areas make it is increasingly
challenging for Bayesians to attempt to model the true data generating
mechanism. Moreover, when the object of interest is low dimensional, such as a
mean or median, it is cumbersome to have to achieve this via a complete model
for the whole data distribution. More importantly, there are settings where the
parameter of interest does not directly index a family of density functions and
thus the Bayesian approach to learning about such parameters is currently
regarded as problematic. Our proposed framework uses loss-functions to connect
information in the data to functionals of interest. The updating of beliefs
then follows from a decision theoretic approach involving cumulative loss
functions. Importantly, the procedure coincides with Bayesian updating when a
true likelihood is known, yet provides coherent subjective inference in much
more general settings. Connections to other inference frameworks are
highlighted.Comment: This is the pre-peer reviewed version of the article "A General
Framework for Updating Belief Distributions", which has been accepted for
publication in the Journal of Statistical Society - Series B. This article
may be used for non-commercial purposes in accordance with Wiley Terms and
Conditions for Self-Archivin
A Hierarchical Bayesian Framework for Constructing Sparsity-inducing Priors
Variable selection techniques have become increasingly popular amongst
statisticians due to an increased number of regression and classification
applications involving high-dimensional data where we expect some predictors to
be unimportant. In this context, Bayesian variable selection techniques
involving Markov chain Monte Carlo exploration of the posterior distribution
over models can be prohibitively computationally expensive and so there has
been attention paid to quasi-Bayesian approaches such as maximum a posteriori
(MAP) estimation using priors that induce sparsity in such estimates. We focus
on this latter approach, expanding on the hierarchies proposed to date to
provide a Bayesian interpretation and generalization of state-of-the-art
penalized optimization approaches and providing simultaneously a natural way to
include prior information about parameters within this framework. We give
examples of how to use this hierarchy to compute MAP estimates for linear and
logistic regression as well as sparse precision-matrix estimates in Gaussian
graphical models. In addition, an adaptive group lasso method is derived using
the framework.Comment: Submitted for publication; corrected typo
Scalable Bayesian nonparametric measures for exploring pairwise dependence via Dirichlet Process Mixtures
In this article we propose novel Bayesian nonparametric methods using
Dirichlet Process Mixture (DPM) models for detecting pairwise dependence
between random variables while accounting for uncertainty in the form of the
underlying distributions. A key criteria is that the procedures should scale to
large data sets. In this regard we find that the formal calculation of the
Bayes factor for a dependent-vs.-independent DPM joint probability measure is
not feasible computationally. To address this we present Bayesian diagnostic
measures for characterising evidence against a "null model" of pairwise
independence. In simulation studies, as well as for a real data analysis, we
show that our approach provides a useful tool for the exploratory nonparametric
Bayesian analysis of large multivariate data sets
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