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Posterior Predictive Assessment of Model Fitness Via Realized Discrepancies
 Statistica Sinica
, 1996
"... Abstract: This paper considers Bayesian counterparts of the classical tests for goodness of fit and their use in judging the fit of a single Bayesian model to the observed data. We focus on posterior predictive assessment, in a framework that also includes conditioning on auxiliary statistics. The B ..."
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Cited by 166 (28 self)
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Abstract: This paper considers Bayesian counterparts of the classical tests for goodness of fit and their use in judging the fit of a single Bayesian model to the observed data. We focus on posterior predictive assessment, in a framework that also includes conditioning on auxiliary statistics. The Bayesian formulation facilitates the construction and calculation of a meaningful reference distribution not only for any (classical) statistic, but also for any parameterdependent “statistic ” or discrepancy. The latter allows us to propose the realized discrepancy assessment of model fitness, which directly measures the true discrepancy between data and the posited model, for any aspect of the model which we want to explore. The computation required for the realized discrepancy assessment is a straightforward byproduct of the posterior simulation used for the original Bayesian analysis. We illustrate with three applied examples. The first example, which serves mainly to motivate the work, illustrates the difficulty of classical tests in assessing the fitness of a Poisson model to a positron emission tomography image that is constrained to be nonnegative. The second and third examples illustrate the details of the posterior predictive approach in two problems: estimation in a model with inequality constraints on the parameters, and estimation in a mixture model. In all three examples, standard test statistics (either a χ 2 or a likelihood ratio) are not pivotal: the difficulty is not just how to compute the reference distribution for the test, but that in the classical framework no such distribution exists, independent of the unknown model parameters. Key words and phrases: Bayesian pvalue, χ 2 test, discrepancy, graphical assessment, mixture model, model criticism, posterior predictive pvalue, prior predictive
Choice of Basis for Laplace Approximation
 Machine Learning
, 1998
"... Maximum a posterJori optimization of parameters and the Laplace approximation for the marginal likelihood are both basisdependent methods. This note compares two choices of basis for models parameterized by probabilities, showing that it is possible to improve on the traditional choice, the prob ..."
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Cited by 23 (1 self)
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Maximum a posterJori optimization of parameters and the Laplace approximation for the marginal likelihood are both basisdependent methods. This note compares two choices of basis for models parameterized by probabilities, showing that it is possible to improve on the traditional choice, the probability simplex, by transforming to the softmax' basis.
Method of Moments Using Monte Carlo Simulation
 Journal of Computational and Graphical Statistics
, 1995
"... We present a computational approach to the method of moments using Monte Carlo simulation. Simple algebraic identities are used so that all computations can be performed directly using simulation draws and computation of the derivative of the loglikelihood. We present a simple implementation using ..."
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Cited by 2 (1 self)
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We present a computational approach to the method of moments using Monte Carlo simulation. Simple algebraic identities are used so that all computations can be performed directly using simulation draws and computation of the derivative of the loglikelihood. We present a simple implementation using the NewtonRaphson algorithm, with the understanding that other optimization methods may be used in more complicated problems. The method can be applied to families of distributions with unknown normalizing constants and can be extended to leastsquares fitting in the case that the number of moments observed exceeds the number of parameters in the model. The method can be further generalized to allow "moments" that are any function of data and parameters, including as a special case maximum likelihood for models with unknown normalizing constants or missing data. In addition to being used for estimation, our method may be useful for setting the parameters of a Bayes prior distribution by spe...
Large Scale Multinomial Inferences and Its Applications in Genome Wide Association Studies
"... Abstract Statistical analysis of multinomial counts with a large number K of categories and a small number n of sample size is challenging to both frequentist and Bayesian methods and requires thinking about statistical inference at a very fundamental level. Following the framework of DempsterShafe ..."
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Abstract Statistical analysis of multinomial counts with a large number K of categories and a small number n of sample size is challenging to both frequentist and Bayesian methods and requires thinking about statistical inference at a very fundamental level. Following the framework of DempsterShafer theory of belief functions, a probabilistic inferential model is proposed for this “large K and small n” problem. The inferential model produces a probability triplet (p,q,r) for an assertion conditional on observed data. The probabilities p and q are for and against the truth of the assertion, whereas r = 1 − p − q is the remaining probability called the probability of “don’t know”. The new inference method is applied in a genomewide association study with very high dimensional count data, to identify association between genetic variants to the disease Rheumatoid Arthritis. 1
BAYESIAN ORDER RESTRICTED METHODS WITH BIOMEDICAL APPLICATIONS
, 2004
"... This dissertation focuses on Bayesian order restricted inference, with interest in applying new methodology to biomedical examples. The first section considers samples of curves restricted to follow a particular shape. For example, progesterone levels in healthy women increase during the menstrual c ..."
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This dissertation focuses on Bayesian order restricted inference, with interest in applying new methodology to biomedical examples. The first section considers samples of curves restricted to follow a particular shape. For example, progesterone levels in healthy women increase during the menstrual cycle to a random peak with decreases thereafter. Reproductive epidemiologists are interested in studying the distribution of the peak and the trajectory for women in different groups. Motivated by this application, we propose a simple approach for restricting each woman’s mean trajectory to follow an umbrella shape. An unconstrained hierarchical Bayesian model is used to characterize the data, and draws from the posterior distribution obtained using a Gibbs sampler are then mapped to the constrained space. Inferences are based on the resulting posterior distribution for the peak and individual woman trajectories. Methods are applied to a study comparing progesterone trajectories for conception and nonconception cycles. The second section addresses studies that collect event time data in which it is often appropriate to assume nondecreasing hazards across dose groups, though dose
Z. Teresa A’marA Management Strategy Evaluation of the harvest policies of the North Pacific Fishery Management Council used for the fishery for
, 2009
"... This is to certify that I have examined this copy of a doctoral dissertation by ..."
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This is to certify that I have examined this copy of a doctoral dissertation by
International Journal of Approximate Reasoning •• • (••••) •••–••• Contents lists available at SciVerse ScienceDirect International Journal of Approximate Reasoning
"... www.elsevier.com/locate/ijar Large scale two sample multinomial inferences and its ..."
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www.elsevier.com/locate/ijar Large scale two sample multinomial inferences and its