Statistical Decision Theory and Bayesian Analysis

Berger, James O.

doi:10.1007/978-1-4757-4286-2

Cited by 5,845 publications

(4,399 citation statements)

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“…This is formally related to the use of conjugate [gamma] priors for scale parameters like f (λ i ) (cf. Berger, 1985), when they are noninformative. Both imply a flat prior on the log-precision, which means its derivatives with respect to ln f (λ i ) = λ i vanish (because it has no maximum).…”

Section: A1 Hyper-parameterising Covariancesmentioning

confidence: 99%

Variational free energy and the Laplace approximation

et al. 2007

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This note derives the variational free energy under the Laplace approximation, with a focus on accounting for additional model complexity induced by increasing the number of model parameters. This is relevant when using the free energy as an approximation to the log-evidence in Bayesian model averaging and selection. By setting restricted maximum likelihood (ReML) in the larger context of variational learning and expectation maximisation (EM), we show how the ReML objective function can be adjusted to provide an approximation to the log-evidence for a particular model. This means ReML can be used for model selection, specifically to select or compare models with different covariance components. This is useful in the context of hierarchical models because it enables a principled selection of priors that, under simple hyperpriors, can be used for automatic model selection and relevance determination (ARD). Deriving the ReML objective function, from basic variational principles, discloses the simple relationships among Variational Bayes, EM and ReML. Furthermore, we show that EM is formally identical to a full variational treatment when the precisions are linear in the hyperparameters. Finally, we also consider, briefly, dynamic models and how these inform the regularisation of free energy ascent schemes, like EM and ReML.

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Section: A1 Hyper-parameterising Covariancesmentioning

confidence: 99%

Variational free energy and the Laplace approximation

et al. 2007

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“…Since the calculation of marginal likelihoods using a mixture of g-priors involves only a one dimensional integral, this approach provides an attractive computational solution that made the original g-priors popular while insuring robustness to misspecification of g (see Zellner (1986) and Fernandez, Ley and Steel (2001) to mention a few). To guard against mispecifying the distributions of the priors, many suggest considering classes of priors (see Berger (1985)). …”

Section: The General Setupmentioning

confidence: 99%

“…The robust Bayesian approach relies upon a class of prior distributions and selects an appropriate prior in a data dependent fashion. An interesting class of prior distributions suggested by Berger (1983Berger ( , 1985 is the ε-contamination class, which combines the elicited prior for the parameters, termed as base prior, with a possible contamination class of prior distributions and implements Type II maximum likelihood (ML-II) procedure for the selection of prior distribution for the parameters. The primary advantage of using such a contamination class of prior distributions is that the resulting estimator obtained by using ML-II procedure performs well even if the true prior distribution is away from the elicited base prior distribution.…”

Section: Introductionmentioning

confidence: 99%

Robust linear static panel data models usingε-contamination

Baltagi¹,

Bresson²,

Chaturvedi³

et al. 2018

Journal of Econometrics

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“…However, it will provide more conservative estimates of significance than both likelihood-based approaches and more traditional significance tests [57]. The Bayes factor will naturally choose smaller models over more complex ones if the quality of fit is comparable and hence provide a control on the size of our trees [3].…”

Section: Bayes Factors-mentioning

confidence: 99%

Bayesian Weibull tree models for survival analysis of clinico-genomic data

Clarke

West

2008

Statistical Methodology

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An important goal of research involving gene expression data for outcome prediction is to establish the ability of genomic data to define clinically relevant risk factors. Recent studies have demonstrated that microarray data can successfully cluster patients into low-and high-risk categories. However, the need exists for models which examine how genomic predictors interact with existing clinical factors and provide personalized outcome predictions. We have developed clinico-genomic tree models for survival outcomes which use recursive partitioning to subdivide the current data set into homogeneous subgroups of patients, each with a specific Weibull survival distribution. These trees can provide personalized predictive distributions of the probability of survival for individuals of interest. Our strategy is to fit multiple models; within each model we adopt a prior on the Weibull scale parameter and update this prior via Empirical Bayes whenever the sample is split at a given node. The decision to split is based on a Bayes factor criterion. The resulting trees are weighted according to their relative likelihood values and predictions are made by averaging over models. In a pilot study of survival in advanced stage ovarian cancer we demonstrate that clinical and genomic data are complementary sources of information relevant to survival, and we use the exploratory nature of the trees to identify potential genomic biomarkers worthy of further study.

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Statistical Decision Theory and Bayesian Analysis

Cited by 5,845 publications

References 0 publications

Variational free energy and the Laplace approximation

Variational free energy and the Laplace approximation

Robust linear static panel data models usingε-contamination

Bayesian Weibull tree models for survival analysis of clinico-genomic data

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