Bayesian Hypothesis Testing: a Reference Approach

Bernardo, José M.; Rueda, Rauil

doi:10.1111/j.1751-5823.2002.tb00175.x

Cited by 72 publications

(19 citation statements)

References 27 publications

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“…Not many divergence measures in functional analysis satisfy the desiderata mentioned above, but they are all satisfied by the intrinsic discrepancy, a divergence measure introduced in Bernardo and Rueda (2002), which has both an information theoretical justification, and a simple operational interpretation in terms of average log-density ratios.…”

Section: The Intrinsic Loss Functionmentioning

confidence: 99%

“…The corresponding Bayes point estimators, Bayes credible regions and Bayes test criteria will respectively be referred to as reference intrinsic estimators, credible regions or test criteria. The basic ideas were respectively introduced in Bernardo and Juárez (2003), Bernardo (2005), and Bernardo and Rueda (2002). All inference summaries depend on the data only through the expected reference intrinsic loss, d(θ0 | z), the expectation of intrinsic loss with respect to the appropriate joint reference posterior…”

Section: Integrated Reference Analysismentioning

confidence: 99%

“…Some relevant results in this direction (from the approach proposed here) may be found in Bernardo and Rueda (2002), where simultaneous testing of H0i ≡ {µi = 0}, for i = 1, . .…”

Section: Objective Bayesian Estimation and Hypothesis Testing 57mentioning

confidence: 99%

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Integrated Objective Bayesian Estimation and Hypothesis Testing

Bernardo

2011

Bayesian Statistics 9

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SummaryThe complete final product of Bayesian inference is the posterior distribution of the quantity of interest. Important inference summaries include point estimation, region estimation, and precise hypotheses testing. Those summaries may appropriately be described as the solution to specific decision problems which depend on the particular loss function chosen. The use of a continuous loss function leads to an integrated set of solutions where the same prior distribution may be used throughout. Objective Bayesian methods are those which use a prior distribution which only depends on the assumed model and the quantity of interest. As a consequence, objective Bayesian methods produce results which only depend on the assumed model and the data obtained. The combined use of the intrinsic discrepancy, an invariant information-based loss function, and appropriately defined reference priors, provides an integrated objective Bayesian solution to both estimation and hypothesis testing problems. The ideas are illustrated with a large collection of non-trivial examples.

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Section: The Intrinsic Loss Functionmentioning

confidence: 99%

Section: Integrated Reference Analysismentioning

confidence: 99%

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Integrated Objective Bayesian Estimation and Hypothesis Testing

Bernardo

2011

Bayesian Statistics 9

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“…This results in an actual data set y o (which is conceptually a realization of Y d ), and an actual information gain from the prior to the posterior state via Equation (14). The actual information gain can be smaller or larger than the expected value according to the distribution of D KL in Equation (16).…”

Section: Limits On Mutual Information In Experimental Design For Modementioning

confidence: 99%

Entropy-Based Experimental Design for Optimal Model Discrimination in the Geosciences

Nowak

Guthke

2016

Entropy

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Abstract:Choosing between competing models lies at the heart of scientific work, and is a frequent motivation for experimentation. Optimal experimental design (OD) methods maximize the benefit of experiments towards a specified goal. We advance and demonstrate an OD approach to maximize the information gained towards model selection. We make use of so-called model choice indicators, which are random variables with an expected value equal to Bayesian model weights. Their uncertainty can be measured with Shannon entropy. Since the experimental data are still random variables in the planning phase of an experiment, we use mutual information (the expected reduction in Shannon entropy) to quantify the information gained from a proposed experimental design. For implementation, we use the Preposterior Data Impact Assessor framework (PreDIA), because it is free of the lower-order approximations of mutual information often found in the geosciences. In comparison to other studies in statistics, our framework is not restricted to sequential design or to discrete-valued data, and it can handle measurement errors. As an application example, we optimize an experiment about the transport of contaminants in clay, featuring the problem of choosing between competing isotherms to describe sorption. We compare the results of optimizing towards maximum model discrimination with an alternative OD approach that minimizes the overall predictive uncertainty under model choice uncertainty.

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“…Model comparisons based on coefficients of determination apply to models related by specifications according to either (1) or (2). They are useful to find a parsimonious relative explanatorily powerful model within a set of related models.…”

Section: Use Of a Coefficient Of Determination In Model Comparisonmentioning

confidence: 99%

On association in regression: the coefficient of determination revisited

Linde

Tutz

2008

Statistics

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Universal coefficients of determination are investigated which quantify the strength of the relation between a vector of dependent variables Y and a vector of independent covariates X. They are defined as measures of dependence between Y and X through θ(x), with θ(x) parameterizing the conditional distribution of Y given X = x. If θ(x) involves unknown coefficients γ the definition is conditional on γ, and in practice γ, respectively the coefficient of determination has to be estimated. The estimates of quantities we propose generalize R 2 in classical linear regression and are also related to other definitions previously suggested. Our definitions apply to generalized regression models with arbitrary link functions as well as multivariate and nonparametric regression. The definition and use of the proposed coefficients of determination is illustrated for several regression problems with simulated and real data sets.

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Bayesian Hypothesis Testing: a Reference Approach

Cited by 72 publications

References 27 publications

Integrated Objective Bayesian Estimation and Hypothesis Testing

Integrated Objective Bayesian Estimation and Hypothesis Testing

Entropy-Based Experimental Design for Optimal Model Discrimination in the Geosciences

On association in regression: the coefficient of determination revisited

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