THE FORMAL DEFINITION OF REFERENCE PRIORS

By James O. Berger, J. Bernardo, Dongchu Sun. 2009

444 citations recorded by the source corpus.

Reference analysis produces objective Bayesian inference, in the sense that inferential statements depend only on the assumed model and the available data, and the prior distribution used to make an inference is least informative in a certain information-theoretic sense. Reference priors have been rigorously defined in specific contexts and heuristically defined in general, but a rigorous general definition has been lacking. We produce a rigorous general definition here and then show how an explicit expression for the reference prior can be obtained under very weak regularity conditions. The explicit expression can be used to derive new reference priors both analytically and numerically.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/739327ea91c9932681416afd7f11ca08a87e8679

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Citations in the corpus

On Some Principles of Statistical Inference (2015, background): “The extensive development of reference priors and other forms of priors meant to be uninformative with respect to the parameters clearly indicates that such priors must be targeted to the parameters of interest (Berger & Bernardo, 1992; Berger et al., 2009; Fraser et al., 2010).”
ANOVA-Like Differential Expression (ALDEx) Analysis for Mixed Population RNA-Seq (2013, methodology): “This is the only method of inferring frequencies from counts that is guaranteed to be invariant to reparameterization, have consistent sampling properties, and be immune to marginalization paradoxes [28,29].”
Bayesian Workflow (2020, background): “Traditionally in Bayesian statistics we speak of noninformative or fully informative priors, but neither of these generally exist: a uniform prior contains some information, as it depends on the parameterization of the model; a reference prior depends on an assumed asymptotic regime for collecting n”
Generalized fiducial inference via discretization (2013, methodology): “8) was slightly shorter than the 95% intervals based on the reference prior solution of Berger et al. (2009) and the proposal of Fraser et al. (2010b). The details of the simulation study are available from the author upon request.”
Penalising Model Component Complexity: A Principled, Practical Approach to Constructing Priors (2014, background): “On the other end of the hunt for the holy grail, “objective” priors (Bernardo, 1979; Berger, 2006; Berger et al., 2009a; Ghosh, 2011) are data-dependent and are not uniformly accepted among Bayesians on philosophical grounds; see for example discussion contributions to Berger (2006) and Goldstein…”
Beyond subjective and objective in statistics (2015, background): “Various principles for doing this have been proposed (maximum entropy, Jaynes, 2003; maximum missing Shannon information, Berger et al., 2009; and a set of desirable properties, Bayarri et al., 2012).”
Marginal Inferential Models: Prior-Free Probabilistic Inference on Interest Parameters (2013, background): “2011), and Bayesian inference with default, reference, or data-dependent priors (Berger 2006; Berger et al. 2009; Fraser 2011; Fraser et al. 2010).”
Generalized Fiducial Inference: A Review and New Results (2016, methodology): “Examples of recent breakthroughs related to reference prior and model selection are Bayarri et al. (2012), Berger (1992), Berger and Sun (2008), Berger, Bernardo, and Sun (2009), and Berger, Bernardo, and Sun (2012).”
Fiducial prediction intervals (2012, methodology): “We point out that in this case the fiducial distribution is the same as a Bayesian posterior computed using the reference prior ðabÞ 1 of Berger et al. (2009). Prediction intervals for the Weibull distribution can be constructed from realizations of the fiducial distribution of ða,bÞ in the same way”
Scale-Dependent Priors for Variance Parameters in Structured Additive Distributional Regression (2016, background): “Another branch of the literature encouraged the choice of ‘objective’ priors (Bernardo, 1979; Berger, 2006; Ghosh, 2011) or Jeffreys priors (Berger et al., 2009).”
A note on p-values interpreted as plausibilities (2012, background): “2011), Dempster–Shafer theory (Dempster 2008; Shafer 1976, 2011), generalized p-values and confidence intervals (Chiang 2001; Tsui and Weerahandi 1989; Weerahandi 1993), and Bayesian inference with default, reference, and/or data-dependent priors (Berger 2006; Berger et al. 2009; Fraser 2011; Fraser”
Constructing Priors that Penalize the Complexity of Gaussian Random Fields (2015, background): “…and are built on the fundamental principle of being the least informative priors, in an information-theoretical sense, for Bayesian inference (Berger, Bernardo, and Sun 2009), and GRFs are often embedded in Bayesian hierarchical models that are too complex for deriving the reference priors.”