1,943 research outputs found
Optimal pricing using online auction experiments: A P\'olya tree approach
We show how a retailer can estimate the optimal price of a new product using
observed transaction prices from online second-price auction experiments. For
this purpose we propose a Bayesian P\'olya tree approach which, given the
limited nature of the data, requires a specially tailored implementation.
Avoiding the need for a priori parametric assumptions, the P\'olya tree
approach allows for flexible inference of the valuation distribution, leading
to more robust estimation of optimal price than competing parametric
approaches. In collaboration with an online jewelry retailer, we illustrate how
our methodology can be combined with managerial prior knowledge to estimate the
profit maximizing price of a new jewelry product.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS503 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Improved minimax predictive densities under Kullback--Leibler loss
Let and be
independent p-dimensional multivariate normal vectors with common unknown mean
. Based on only observing , we consider the problem of obtaining a
predictive density for that is close to as
measured by expected Kullback--Leibler loss. A natural procedure for this
problem is the (formal) Bayes predictive density
under the uniform prior , which is best
invariant and minimax. We show that any Bayes predictive density will be
minimax if it is obtained by a prior yielding a marginal that is superharmonic
or whose square root is superharmonic. This yields wide classes of minimax
procedures that dominate , including Bayes
predictive densities under superharmonic priors. Fundamental similarities and
differences with the parallel theory of estimating a multivariate normal mean
under quadratic loss are described.Comment: Published at http://dx.doi.org/10.1214/009053606000000155 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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Bayesian forecasting of Prepayment Rates for Individual Pools of Mortgages
This paper proposes a novel approach for modeling prepayment rates of individual pools of mortgages. The model incorporates the empirical evidence that prepayment is past dependent via Bayesian methodology. There are many factors that influence the prepayment behavior and for many of them there is no available (or impossible to gather) information. We implement this issue by creating a Bayesian mixture model and construct a Markov Chain Monte Carlo algorithm to estimate the parameters. We assess the model on a data set from the Bloomberg Database. Our results show that the burnout effect is a significant variable for explaining normal prepayment activities. This result does not hold when prepayment is triggered by non-pool dependent events. We show how to use the new model to compute prices for Mortgage Backed Securities. Monte Carlo simulation is the traditional method for obtaining such prices and the proposed model can be easily incorporated within simulation pricing framework. Prices for standard Pass-Throughs are obtained using simulation.State of Texas Advanced Research Program 003658-0763National Science Foundation CMMI-0457558, DMS-0605102Civil, Architectural, and Environmental Engineerin
Simultaneous Variable and Covariance Selection with the Multivariate Spike-and-Slab Lasso
We propose a Bayesian procedure for simultaneous variable and covariance
selection using continuous spike-and-slab priors in multivariate linear
regression models where q possibly correlated responses are regressed onto p
predictors. Rather than relying on a stochastic search through the
high-dimensional model space, we develop an ECM algorithm similar to the EMVS
procedure of Rockova & George (2014) targeting modal estimates of the matrix of
regression coefficients and residual precision matrix. Varying the scale of the
continuous spike densities facilitates dynamic posterior exploration and allows
us to filter out negligible regression coefficients and partial covariances
gradually. Our method is seen to substantially outperform regularization
competitors on simulated data. We demonstrate our method with a re-examination
of data from a recent observational study of the effect of playing high school
football on several later-life cognition, psychological, and socio-economic
outcomes
Admissible predictive density estimation
Let and be independent
-dimensional multivariate normal vectors with common unknown mean .
Based on observing , we consider the problem of estimating the true
predictive density of under expected Kullback--Leibler loss. Our
focus here is the characterization of admissible procedures for this problem.
We show that the class of all generalized Bayes rules is a complete class, and
that the easily interpretable conditions of Brown and Hwang [Statistical
Decision Theory and Related Topics (1982) III 205--230] are sufficient for a
formal Bayes rule to be admissible.Comment: Published in at http://dx.doi.org/10.1214/07-AOS506 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Dilution Priors: Compensating for Model Space Redundancy
For the general Bayesian model uncertainty framework, the focus of this paper is on the development of model space priors which can compensate for redundancy between model classes, the so-called dilution priors proposed in George (1999). Several distinct approaches for dilution prior construction are suggested. One is based on tessellation determined neighborhoods, another on collinearity adjustments, and a third on pairwise distances between models
Variable selection for BART: An application to gene regulation
We consider the task of discovering gene regulatory networks, which are
defined as sets of genes and the corresponding transcription factors which
regulate their expression levels. This can be viewed as a variable selection
problem, potentially with high dimensionality. Variable selection is especially
challenging in high-dimensional settings, where it is difficult to detect
subtle individual effects and interactions between predictors. Bayesian
Additive Regression Trees [BART, Ann. Appl. Stat. 4 (2010) 266-298] provides a
novel nonparametric alternative to parametric regression approaches, such as
the lasso or stepwise regression, especially when the number of relevant
predictors is sparse relative to the total number of available predictors and
the fundamental relationships are nonlinear. We develop a principled
permutation-based inferential approach for determining when the effect of a
selected predictor is likely to be real. Going further, we adapt the BART
procedure to incorporate informed prior information about variable importance.
We present simulations demonstrating that our method compares favorably to
existing parametric and nonparametric procedures in a variety of data settings.
To demonstrate the potential of our approach in a biological context, we apply
it to the task of inferring the gene regulatory network in yeast (Saccharomyces
cerevisiae). We find that our BART-based procedure is best able to recover the
subset of covariates with the largest signal compared to other variable
selection methods. The methods developed in this work are readily available in
the R package bartMachine.Comment: Published in at http://dx.doi.org/10.1214/14-AOAS755 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The Median Probability Model and Correlated Variables
The median probability model (MPM) Barbieri and Berger (2004) is defined as
the model consisting of those variables whose marginal posterior probability of
inclusion is at least 0.5. The MPM rule yields the best single model for
prediction in orthogonal and nested correlated designs. This result was
originally conceived under a specific class of priors, such as the point mass
mixtures of non-informative and g-type priors. The MPM rule, however, has
become so very popular that it is now being deployed for a wider variety of
priors and under correlated designs, where the properties of MPM are not yet
completely understood. The main thrust of this work is to shed light on
properties of MPM in these contexts by (a) characterizing situations when MPM
is still safe under correlated designs, (b) providing significant
generalizations of MPM to a broader class of priors (such as continuous
spike-and-slab priors). We also provide new supporting evidence for the
suitability of g-priors, as opposed to independent product priors, using new
predictive matching arguments. Furthermore, we emphasize the importance of
prior model probabilities and highlight the merits of non-uniform prior
probability assignments using the notion of model aggregates
Fully Bayes factors with a generalized g-prior
For the normal linear model variable selection problem, we propose selection
criteria based on a fully Bayes formulation with a generalization of Zellner's
-prior which allows for . A special case of the prior formulation is
seen to yield tractable closed forms for marginal densities and Bayes factors
which reveal new model evaluation characteristics of potential interest.Comment: Published in at http://dx.doi.org/10.1214/11-AOS917 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Fast Bayesian Factor Analysis via Automatic Rotations to Sparsity
Rotational post hoc transformations have traditionally played a key role in enhancing the interpretability of factor analysis. Regularization methods also serve to achieve this goal by prioritizing sparse loading matrices. In this work, we bridge these two paradigms with a unifying Bayesian framework. Our approach deploys intermediate factor rotations throughout the learning process, greatly enhancing the effectiveness of sparsity inducing priors. These automatic rotations to sparsity are embedded within a PXL-EM algorithm, a Bayesian variant of parameter-expanded EM for posterior mode detection. By iterating between soft-thresholding of small factor loadings and transformations of the factor basis, we obtain (a) dramatic accelerations, (b) robustness against poor initializations, and (c) better oriented sparse solutions. To avoid the prespecification of the factor cardinality, we extend the loading matrix to have infinitely many columns with the Indian buffet process (IBP) prior. The factor dimensionality is learned from the posterior, which is shown to concentrate on sparse matrices. Our deployment of PXL-EM performs a dynamic posterior exploration, outputting a solution path indexed by a sequence of spike-and-slab priors. For accurate recovery of the factor loadings, we deploy the spike-and-slab LASSO prior, a two-component refinement of the Laplace prior. A companion criterion, motivated as an integral lower bound, is provided to effectively select the best recovery. The potential of the proposed procedure is demonstrated on both simulated and real high-dimensional data, which would render posterior simulation impractical. Supplementary materials for this article are available online
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