An introduction to Bayesian reference analysis: inference on the ratio of multinomial parameters

Bernardo, José M.; Ramón, José

doi:10.1111/1467-9884.00118

Cited by 53 publications

(50 citation statements)

References 165 publications

(167 reference statements)

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“…In the binomial case, this prior is the beta with α = β = 0.5. Bernardo and Ramón (1998) presented an informative survey article about Bernardo's reference analysis approach (Bernardo 1979), which optimizes a limiting entropy distance criterion. This attempts to derive non-subjective posterior distributions that satisfy certain natural criteria such as invariance, consistent frequentist performance (e.g., large-sample coverage probability of confidence intervals close to the nominal level), and admissibility.…”

Section: Prior Distributions For a Binomial Parametermentioning

confidence: 99%

Bayesian inference for categorical data analysis

Agresti

Hitchcock

2005

JISS

132

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This article surveys Bayesian methods for categorical data analysis, with primary emphasis on contingency table analysis. Early innovations were proposed by Good (1953Good ( , 1956Good ( , 1965 for smoothing proportions in contingency tables and by Lindley (1964) for inference about odds ratios. These approaches primarily used conjugate beta and Dirichlet priors. Altham (1969Altham ( , 1971 presented Bayesian analogs of small-sample frequentist tests for 2×2 tables using such priors. An alternative approach using normal priors for logits received considerable attention in the 1970s by Leonard and others (e.g., Leonard 1972). Adopted usually in a hierarchical form, the logit-normal approach allows greater flexibility and scope for generalization. The 1970s also saw considerable interest in loglinear modeling. The advent of modern computational methods since the mid-1980s has led to a growing literature on fully Bayesian analyses with models for categorical data, with main emphasis on generalized linear models such as logistic regression for binary and multi-category response variables.

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Section: Prior Distributions For a Binomial Parametermentioning

confidence: 99%

Bayesian inference for categorical data analysis

Agresti

Hitchcock

2005

JISS

132

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“…This seems to imply that both Masterton's and Williamson's versions of Objective Baysianism do not fit well in any of Bandyopadhyay's et al [2] 4 categories of Objective Bayesianism. Neither position is technically 'strongly' objective as they both allow that the inference process may fail to deliver a unique probability (though they are arguably strongly objective in spirit), but nor is either position 'moderately' objective because they do not envisage scientific inference as a problem of deciding between competing theories [3]. Certainly, neither position is a version of a Carnapian [4] style of logical/'necessary' objective Bayesianism.…”

Section: Placing Masterton's and Williamson's Objective Bayesianism Imentioning

confidence: 99%

Invariant Equivocation

Landes

Masterton

2016

Erkenn

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Objective Bayesians hold that degrees of belief ought to be chosen in the set of probability functions calibrated with one's evidence. The particular choice of degrees of belief is via some objective, i.e., not agentdependent, inference process that, in general, selects the most equivocal probabilities from among those compatible with one's evidence. Maximising entropy is what drives these inference processes in recent works by Williamson and Masterton though they disagree as to what should have its entropy maximised. With regard to the probability function one should adopt as one's belief function, Williamson advocates selecting the probability function with greatest entropy compatible with one's evidence while Masterton advocates selecting the expected probability function relative to the density function with greatest entropy compatible with one's evidence. In this paper we discuss the significant relative strengths of these two positions. In particular, Masterton's original proposal is further developed and investigated to reveal its significant properties; including its equivalence to the centre of mass inference process and its ability to accommodate higher order evidence.Keywords: Maximum Entropy, Objective Bayesianism, Inference Process, Centre of Mass, Language Invariance 1 Introduction "How should one form graded beliefs?" is a question that has long fascinated philosophers. The answer to this question is highly relevant throughout science, law, operational research and policy-making. Intuitively, it is obvious that one's evidence ought to matter when forming beliefs, whether graded or binary. The best way of caching out this ubiquitous intuition is, however, a matter of lively disagreement. The main philosophical protagonists in this debate are: the subjective Bayesians, who can be further subdivided into the radical (de Finetti This paper is concerned with objective Bayesianism. Objective Bayesians, like their subjective brethren, hold that one's degrees of belief ought to be consistent with one's evidence. However, in cases where the evidential constraints are satisfied by more than one probability function the objective Bayesians differ from the subjective Bayesians in holding that the choice of a particular probability function from those consistent with the evidential constraints ought to made 1 in an objective-i.e., agent-independent-manner. Typically, such a choice is made by applying an inference process that picks from among the probability functions consistent with the evidential constraints that function which is, in some sense, maximally equivocal. 1 Objective Bayesianism is (still!) a minority view with a relatively small but dedicated group of advocates. Even in this relatively small group there are disagreements. One such recent disagreement is that between Williamson [27] and Masterton [16] on how to understand maximal equivocation.We shall herein first rehearse Williamson's account in Section 2.1 and 2.2. Then we improve on Masterton's approach by reformulating it, Section 2...

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“…As a result, the reference prior is indeed the semi-reference prior in case of a = 1 2 and b = 0. Another way to derive the reference prior for the partial immunity model is to use the theorem on reference prior under factorization originally established [12] (see also [13]). This theorem is based on the fact that for any typically regular model with two parameters µ and φ and any suitable prior distribution, the joint posterior distribution on µ and φ is asymptotically close to a normal distribution with covariance matrix I −1 ( µ, φ) where ( µ, φ) is the maximum likelihood estimation and I is the Fisher information matrix of the model (see [15]: Section 5.3.).…”

Section: Semi-informative Prior -mentioning

confidence: 99%

A Bayesian framework for the ratio of two Poisson rates in the context of vaccine efficacy trials

Laurent

Legrand

2012

ESAIM: PS

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Abstract. In many applications, we assume that two random observations x and y are generated according to independent Poisson distributions P(λS) and P(µT ) and we are interested in performing statistical inference on the ratio φ = λ/µ of the two incidence rates. In vaccine efficacy trials, x and y are typically the numbers of cases in the vaccine and the control groups respectively, φ is called the relative risk and the statistical model is called 'partial immunity model'. In this paper we start by defining a natural semi-conjugate family of prior distributions for this model, allowing straightforward computation of the posterior inference. Following theory on reference priors, we define the reference prior for the partial immunity model when φ is the parameter of interest. We also define a family of reference priors with partial information on µ while remaining uninformative about φ. We notice that these priors belong to the semi-conjugate family. We then demonstrate using numerical examples that Bayesian credible intervals for φ enjoy attractive frequentist properties when using reference priors, a typical property of reference priors.

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An introduction to Bayesian reference analysis: inference on the ratio of multinomial parameters

Cited by 53 publications

References 165 publications

Bayesian inference for categorical data analysis

Bayesian inference for categorical data analysis

Invariant Equivocation

A Bayesian framework for the ratio of two Poisson rates in the context of vaccine efficacy trials

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