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## The formal definition of reference priors (2009)

Venue: | ANN. STATIST |

Citations: | 36 - 2 self |

### Citations

10537 |
The mathematical theory of communication.
- Shannon, Weaver
- 1949
(Show Context)
Citation Context ...{˜p | p},is ∫ κ{˜p | p}= p(θ)log � p(θ) ˜p(θ) dθ, provided the integral (or the sum) is finite. The properties of κ{˜p | p} have been extensively studied; pioneering works include Gibbs [22], Shannon =-=[38]-=-, Good [24, 25], Kullback and Leibler [35], Chernoff [15], Jaynes [29, 30], Kullback [34] and Csiszar [18, 19]. DEFINITION 2 (Logarithmic convergence). A sequence of probability density functions {pi}... |

2143 |
On Information and Sufficiency.
- Kullback, RA
- 1951
(Show Context)
Citation Context ...(θ) dθ, provided the integral (or the sum) is finite. The properties of κ{˜p | p} have been extensively studied; pioneering works include Gibbs [22], Shannon [38], Good [24, 25], Kullback and Leibler =-=[35]-=-, Chernoff [15], Jaynes [29, 30], Kullback [34] and Csiszar [18, 19]. DEFINITION 2 (Logarithmic convergence). A sequence of probability density functions {pi} ∞ i=1 converges logarithmically to a prob... |

1830 |
Statistical decision theory and bayesian analysis. 2nd ed.
- Berger
- 1985
(Show Context)
Citation Context ...sfies numerous nice properties such as being exact frequentist matching (i.e., a Bayesian 100(1 − α)% credible set will also be a frequentist 100(1 − α)% confidence set; cf. equation (6.22) in Berger =-=[2]-=-). This is in stark contrast to the situation with the Fraser, Monette and Ng example. However, the basic fact remains that posteriors from uniform priors on large compact sets do not seem here to be ... |

1753 | Information Theory and Statistics.
- Kullback
- 1978
(Show Context)
Citation Context ...inite. The properties of κ{˜p | p} have been extensively studied; pioneering works include Gibbs [22], Shannon [38], Good [24, 25], Kullback and Leibler [35], Chernoff [15], Jaynes [29, 30], Kullback =-=[34]-=- and Csiszar [18, 19]. DEFINITION 2 (Logarithmic convergence). A sequence of probability density functions {pi} ∞ i=1 converges logarithmically to a probability density p if, and only if, limi→∞ κ(p |... |

1479 | Bayesian theory.
- Bernardo, Smith
- 1994
(Show Context)
Citation Context ...erable body of conceptual and theoretical literature devoted to identifying appropriate procedures for the formulation of objective priors; for relevant pointers see Section 5.6 in Bernardo and Smith =-=[13]-=-, Datta and Mukerjee [20], Bernardo [11], Berger [3], Ghosh, Delampady and Samanta [23] and references therein. Reference analysis, introduced by Bernardo [10] and further developed by Berger and Bern... |

1126 |
Information theory and statistical mechanics.
- Jaynes
- 1957
(Show Context)
Citation Context ...l (or the sum) is finite. The properties of κ{˜p | p} have been extensively studied; pioneering works include Gibbs [22], Shannon [38], Good [24, 25], Kullback and Leibler [35], Chernoff [15], Jaynes =-=[29, 30]-=-, Kullback [34] and Csiszar [18, 19]. DEFINITION 2 (Logarithmic convergence). A sequence of probability density functions {pi} ∞ i=1 converges logarithmically to a probability density p if, and only i... |

489 | A mathematical theory of communication, The Bell System Tech - Shannon - 1948 |

370 |
I-divergence geometry of probability distributions and minimization problems”, The Annals of Probability 3
- Csiszar
- 1975
(Show Context)
Citation Context ...ties of κ{˜p | p} have been extensively studied; pioneering works include Gibbs [22], Shannon [38], Good [24, 25], Kullback and Leibler [35], Chernoff [15], Jaynes [29, 30], Kullback [34] and Csiszar =-=[18, 19]-=-. DEFINITION 2 (Logarithmic convergence). A sequence of probability density functions {pi} ∞ i=1 converges logarithmically to a probability density p if, and only if, limi→∞ κ(p | pi) = 0. 2. Improper... |

334 |
Information-type measures of difference of probability distributions and indirect
- Csiszar
- 1967
(Show Context)
Citation Context ...ties of κ{˜p | p} have been extensively studied; pioneering works include Gibbs [22], Shannon [38], Good [24, 25], Kullback and Leibler [35], Chernoff [15], Jaynes [29, 30], Kullback [34] and Csiszar =-=[18, 19]-=-. DEFINITION 2 (Logarithmic convergence). A sequence of probability density functions {pi} ∞ i=1 converges logarithmically to a probability density p if, and only if, limi→∞ κ(p | pi) = 0. 2. Improper... |

319 |
An invariant form for the prior probability in estimation problems
- Jeffreys
- 1946
(Show Context)
Citation Context ...istic arguments were given to justify an explicit expression in terms of the expectation under sampling of the logarithm of the asymptotic posterior density, which reduced to Jeffreys prior (Jeffreys =-=[31, 32]-=-) under asymptotic posterior normality. In multiparameter problems it was argued that one should not maximize the joint Received March 2007; revised December 2007. 1 Supported by NSF Grant DMS-01-0326... |

310 |
Elementary principles of statistical mechanics
- Gibbs
- 1902
(Show Context)
Citation Context ..., denoted by κ{˜p | p},is ∫ κ{˜p | p}= p(θ)log � p(θ) ˜p(θ) dθ, provided the integral (or the sum) is finite. The properties of κ{˜p | p} have been extensively studied; pioneering works include Gibbs =-=[22]-=-, Shannon [38], Good [24, 25], Kullback and Leibler [35], Chernoff [15], Jaynes [29, 30], Kullback [34] and Csiszar [18, 19]. DEFINITION 2 (Logarithmic convergence). A sequence of probability density ... |

248 | Prior Probabilities
- Jaynes
- 1968
(Show Context)
Citation Context ...l (or the sum) is finite. The properties of κ{˜p | p} have been extensively studied; pioneering works include Gibbs [22], Shannon [38], Good [24, 25], Kullback and Leibler [35], Chernoff [15], Jaynes =-=[29, 30]-=-, Kullback [34] and Csiszar [18, 19]. DEFINITION 2 (Logarithmic convergence). A sequence of probability density functions {pi} ∞ i=1 converges logarithmically to a probability density p if, and only i... |

211 |
On a Measure of the Information Provided by an Experiment
- Lindley
- 1956
(Show Context)
Citation Context ...rmality given in Clarke [16] is monotone. 3. Reference priors. 3.1. Definition of reference priors. Key to the definition of reference priors is Shannon expected information (Shannon [38] and Lindley =-=[36]-=-). DEFINITION 6 (Expected information). The information to be expected from one observation from model M ≡{p(x | θ),x ∈ X,θ ∈ �}, when the prior for θ is q(θ),is ∫ ∫ I{q | M}= p(x | θ)q(θ)log dx dθ (3... |

200 |
Reference posterior distributions for Bayesian inference (with discussion
- Bernardo
- 1979
(Show Context)
Citation Context ...rs see Section 5.6 in Bernardo and Smith [13], Datta and Mukerjee [20], Bernardo [11], Berger [3], Ghosh, Delampady and Samanta [23] and references therein. Reference analysis, introduced by Bernardo =-=[10]-=- and further developed by Berger and Bernardo [4–7], and Sun and Berger [42], has been one of the most utilized approaches to developing objective priors; see the references in Bernardo [11]. Referenc... |

170 |
Theory of Probability”, 3rd ed
- Jeffreys
- 1961
(Show Context)
Citation Context ...istic arguments were given to justify an explicit expression in terms of the expectation under sampling of the logarithm of the asymptotic posterior density, which reduced to Jeffreys prior (Jeffreys =-=[31, 32]-=-) under asymptotic posterior normality. In multiparameter problems it was argued that one should not maximize the joint Received March 2007; revised December 2007. 1 Supported by NSF Grant DMS-01-0326... |

156 |
Probability and the weighing of evidence
- GOOD
- 1950
(Show Context)
Citation Context ... ∫ κ{˜p | p}= p(θ)log � p(θ) ˜p(θ) dθ, provided the integral (or the sum) is finite. The properties of κ{˜p | p} have been extensively studied; pioneering works include Gibbs [22], Shannon [38], Good =-=[24, 25]-=-, Kullback and Leibler [35], Chernoff [15], Jaynes [29, 30], Kullback [34] and Csiszar [18, 19]. DEFINITION 2 (Logarithmic convergence). A sequence of probability density functions {pi} ∞ i=1 converge... |

93 | Objective Bayesian Analysis of Spatially Correlated Data
- Berger, Oliveira, et al.
- 2001
(Show Context)
Citation Context ...med model. If observations are dependent, as in time series or spatial models, the reference prior may well depend on the sample size (see, e.g., Berger and Yang [9] and Berger, de Oliveira and Sansó =-=[8]-=-). THEOREM 5 (Compatibility with sufficient statistics). Consider the model M ={p(x | θ), x ∈ X,θ ∈ �} with sufficient statistic t = t(x) ∈ T , and let Mt ={p(t | θ),t ∈ T ,θ ∈ �} be the corresponding... |

71 |
Jeffrey’s prior is asymptotically least favorable under entropy
- Clarke, Barron
- 1994
(Show Context)
Citation Context ...ernardo [7] gavemoreprecisedefinitions of this sequential reference process, but restricted consideration to continuous multiparameter problems under asymptotic posterior normality. Clarke and Barron =-=[17]-=- established regularity conditions under which joint maximization of the missing information leads to Jeffreys multivariate priors. Ghosal and Samanta [27] and Ghosal [26] provided explicit results fo... |

57 |
Estimating a product of means: Bayesian analysis with reference priors
- BERGER, BERNARDO
- 1990
(Show Context)
Citation Context ...ta and Mukerjee [20], Bernardo [11], Berger [3], Ghosh, Delampady and Samanta [23] and references therein. Reference analysis, introduced by Bernardo [10] and further developed by Berger and Bernardo =-=[4, 5, 6, 7]-=-, and Sun and Berger [42], has been one of the most utilized approaches to developing objective priors; see the references in Bernardo [11]. Reference analysis uses information-theoretical concepts to... |

45 |
Asymptotic inference for mixture models using data-dependent priors
- WASSERMAN
- 2000
(Show Context)
Citation Context ...d an improper posterior in univariate problems. It is also of interest because there is no natural objective prior available for the problem. (There are data-dependent objective priors: see Wasserman =-=[43]-=-.) Theorem 2 can easily be modified to apply to models that can be transformed into a location model. COROLLARY 1. Consider M ≡{p(x | θ),θ ∈ �,x ∈ X}. If there are monotone functions y = y(x) and φ = ... |

37 |
An Introduction to Bayesian Analysis: Theory and Methods
- Ghosh
- 2006
(Show Context)
Citation Context ... procedures for the formulation of objective priors; for relevant pointers see Section 5.6 in Bernardo and Smith [13], Datta and Mukerjee [20], Bernardo [11], Berger [3], Ghosh, Delampady and Samanta =-=[23]-=- and references therein. Reference analysis, introduced by Bernardo [10] and further developed by Berger and Bernardo [4–7], and Sun and Berger [42], has been one of the most utilized approaches to de... |

34 |
On the development of reference priors (with Discussion
- BERGER, BERNARDO
- 1992
(Show Context)
Citation Context ...rs, reference priors. 905906 J. O. BERGER, J. M. BERNARDO AND D. SUN missing information but proceed sequentially, thus avoiding known problems such as marginalization paradoxes. Berger and Bernardo =-=[7]-=- gavemoreprecisedefinitions of this sequential reference process, but restricted consideration to continuous multiparameter problems under asymptotic posterior normality. Clarke and Barron [17] establ... |

32 |
Ordered group reference priors with application to the multinomial problem
- BERGER, BERNARDO
- 1992
(Show Context)
Citation Context ...ta and Mukerjee [20], Bernardo [11], Berger [3], Ghosh, Delampady and Samanta [23] and references therein. Reference analysis, introduced by Bernardo [10] and further developed by Berger and Bernardo =-=[4, 5, 6, 7]-=-, and Sun and Berger [42], has been one of the most utilized approaches to developing objective priors; see the references in Bernardo [11]. Reference analysis uses information-theoretical concepts to... |

28 |
Reference priors in a variance components problem
- Berger, Bernardo
- 1992
(Show Context)
Citation Context ...ta and Mukerjee [20], Bernardo [11], Berger [3], Ghosh, Delampady and Samanta [23] and references therein. Reference analysis, introduced by Bernardo [10] and further developed by Berger and Bernardo =-=[4, 5, 6, 7]-=-, and Sun and Berger [42], has been one of the most utilized approaches to developing objective priors; see the references in Bernardo [11]. Reference analysis uses information-theoretical concepts to... |

28 | Bayesian hypothesis testing: A reference approach
- Bernardo, Rueda
- 2002
(Show Context)
Citation Context ...ties in dealing with this is the need for a better notion of divergence that is symmetric in its arguments. One possibility is the symmetrized form of the logarithmic divergence in Bernardo and Rueda =-=[12]-=-, but the analysis is considerably more difficult. 2.2. Permissible priors. Based on the previous considerations, we restrict consideration of possibly objective priors to those that satisfy the expec... |

28 |
Probability Matching Priors: Higher Order Asymptotics. Lecture Notes in Statistics.
- Datta, Mukerjee
- 2004
(Show Context)
Citation Context ... and theoretical literature devoted to identifying appropriate procedures for the formulation of objective priors; for relevant pointers see Section 5.6 in Bernardo and Smith [13], Datta and Mukerjee =-=[20]-=-, Bernardo [11], Berger [3], Ghosh, Delampady and Samanta [23] and references therein. Reference analysis, introduced by Bernardo [10] and further developed by Berger and Bernardo [4–7], and Sun and B... |

23 | Reference Analysis
- Bernardo
- 2005
(Show Context)
Citation Context ...l literature devoted to identifying appropriate procedures for the formulation of objective priors; for relevant pointers see Section 5.6 in Bernardo and Smith [13], Datta and Mukerjee [20], Bernardo =-=[11]-=-, Berger [3], Ghosh, Delampady and Samanta [23] and references therein. Reference analysis, introduced by Bernardo [10] and further developed by Berger and Bernardo [4–7], and Sun and Berger [42], has... |

13 |
Yang: Noninformative priors and Bayesian testing for the AR(1) model
- Berger, Y
- 1994
(Show Context)
Citation Context ... x to be a random sample from the assumed model. If observations are dependent, as in time series or spatial models, the reference prior may well depend on the sample size (see, e.g., Berger and Yang =-=[9]-=- and Berger, de Oliveira and Sansó [8]). THEOREM 5 (Compatibility with sufficient statistics). Consider the model M ={p(x | θ), x ∈ X,θ ∈ �} with sufficient statistic t = t(x) ∈ T , and let Mt ={p(t |... |

8 | Asymptotic normality of the posterior in relative entropy
- Clarke
- 1999
(Show Context)
Citation Context ... D. SUN PROOF. The proof of this theorem is given in Appendix C. □ As an aside, the above result suggests that, as the sample size grows, the convergence of the posterior to normality given in Clarke =-=[16]-=- is monotone. 3. Reference priors. 3.1. Definition of reference priors. Key to the definition of reference priors is Shannon expected information (Shannon [38] and Lindley [36]). DEFINITION 6 (Expecte... |

8 |
Reference priors in multiparameter nonregular cases
- Ghosal
- 1997
(Show Context)
Citation Context ...normality. Clarke and Barron [17] established regularity conditions under which joint maximization of the missing information leads to Jeffreys multivariate priors. Ghosal and Samanta [27] and Ghosal =-=[26]-=- provided explicit results for reference priors in some types of nonregular models. This paper has three goals. GOAL 1. Make precise the definition of the reference prior. This has two different aspec... |

8 |
Non-smooth sailing or triangular distributions revisited after some 50 years. The Statistician 48, 179–187 part 2
- Johnson, Kotz
- 1999
(Show Context)
Citation Context ...e interval (0, 1). The nonsymmetric standard triangular distribution on (0, 1), { 2x/θ, for 0 <x≤ θ, p(x | θ)= 0 <θ<1, 2(1 − x)/(1 − θ), for θ<x<1, was first studied by Ayyangar [1]. Johnson and Kotz =-=[33]-=- revisited nonsymmetric triangular distributions in the context of modeling prices. The triangular density has a unique mode at θ and satisfies Pr[x ≤ θ]=θ, a property that can be used to obtain an es... |

7 | What is the use of a distribution? Multivariate Analysis 2 - Good - 1969 |

5 |
Expansion of Bayes risk for entropy loss and reference prior in nonregular cases
- Ghosal, Samanta
- 1997
(Show Context)
Citation Context ...totic posterior normality. Clarke and Barron [17] established regularity conditions under which joint maximization of the missing information leads to Jeffreys multivariate priors. Ghosal and Samanta =-=[27]-=- and Ghosal [26] provided explicit results for reference priors in some types of nonregular models. This paper has three goals. GOAL 1. Make precise the definition of the reference prior. This has two... |

4 |
Coherent inference from improper priors and from finitely additive priors
- Heath, Sudderth
- 1989
(Show Context)
Citation Context ...ifications revolve around showing that π(θ | x) is a suitable limit of posteriors obtained from proper priors. A variety of versions of such arguments exist; cf. Stone [40, 41] and Heath and Sudderth =-=[28]-=-. Here, we consider approximations based on restricting the prior to an increasing sequence of compact sets and using logarithmic convergence to define the limiting process. The main motivation is, as... |

4 |
Necessary and sufficient condition for convergence in probability to invariant posterior distributions
- Stone
- 1970
(Show Context)
Citation Context ...ensity. The most convincing justifications revolve around showing that π(θ | x) is a suitable limit of posteriors obtained from proper priors. A variety of versions of such arguments exist; cf. Stone =-=[40, 41]-=- and Heath and Sudderth [28]. Here, we consider approximations based on restricting the prior to an increasing sequence of compact sets and using logarithmic convergence to define the limiting process... |

3 |
The case for objective Bayesian analysis (with discussion). Bayesian Analysis
- Berger
- 2006
(Show Context)
Citation Context ...devoted to identifying appropriate procedures for the formulation of objective priors; for relevant pointers see Section 5.6 in Bernardo and Smith [13], Datta and Mukerjee [20], Bernardo [11], Berger =-=[3]-=-, Ghosh, Delampady and Samanta [23] and references therein. Reference analysis, introduced by Bernardo [10] and further developed by Berger and Bernardo [4–7], and Sun and Berger [42], has been one of... |

3 |
Numbers and functions
- Moll
- 2012
(Show Context)
Citation Context ... = 1, a1(θ) = θ and a2(θ) = θ 2 . Then, b1 = 2θ − 1andb2 = (2θ − 1)/(2θ).Itiseasytoshowthatb −1 2 = b−1 1 + 1. For the digamma function920 J. O. BERGER, J. M. BERNARDO AND D. SUN (see Boros and Moll =-=[14]-=-), ψ(z + 1) = ψ(z)+ 1/z, forz>0, so that ψ(1/b1) = ψ(1/b2) − b1. The reference prior (4.6) thus becomes 2θ − 1 π(θ) = θ(θ − 1) exp { [ ( ) 1 1 b1 + b1ψ − b 2 ( )]} 1 1 − b2ψ (4.7) = ∝ 2θ − 1 θ(θ − 1) ... |

3 |
Large-sample theory: Parametric
- Chernoff
- 1956
(Show Context)
Citation Context ...d the integral (or the sum) is finite. The properties of κ{˜p | p} have been extensively studied; pioneering works include Gibbs [22], Shannon [38], Good [24, 25], Kullback and Leibler [35], Chernoff =-=[15]-=-, Jaynes [29, 30], Kullback [34] and Csiszar [18, 19]. DEFINITION 2 (Logarithmic convergence). A sequence of probability density functions {pi} ∞ i=1 converges logarithmically to a probability density... |

2 |
Right Haar measures for convergence in probability to invariant posterior distributions
- Stone
- 1965
(Show Context)
Citation Context ...ensity. The most convincing justifications revolve around showing that π(θ | x) is a suitable limit of posteriors obtained from proper priors. A variety of versions of such arguments exist; cf. Stone =-=[40, 41]-=- and Heath and Sudderth [28]. Here, we consider approximations based on restricting the prior to an increasing sequence of compact sets and using logarithmic convergence to define the limiting process... |

2 |
Reference priors under partial information
- Sun, Berger
- 1998
(Show Context)
Citation Context ...ardo [11], Berger [3], Ghosh, Delampady and Samanta [23] and references therein. Reference analysis, introduced by Bernardo [10] and further developed by Berger and Bernardo [4–7], and Sun and Berger =-=[42]-=-, has been one of the most utilized approaches to developing objective priors; see the references in Bernardo [11]. Reference analysis uses information-theoretical concepts to make precise the idea of... |

1 |
The triangular distribution
- Ayyangar
- 1941
(Show Context)
Citation Context ...random variables on the interval (0, 1). The nonsymmetric standard triangular distribution on (0, 1), { 2x/θ, for 0 <x≤ θ, p(x | θ)= 0 <θ<1, 2(1 − x)/(1 − θ), for θ<x<1, was first studied by Ayyangar =-=[1]-=-. Johnson and Kotz [33] revisited nonsymmetric triangular distributions in the context of modeling prices. The triangular density has a unique mode at θ and satisfies Pr[x ≤ θ]=θ, a property that can ... |

1 |
Marginalization, likelihood and structural models
- Fraser, Monette, et al.
- 1985
(Show Context)
Citation Context ...virtually any improper prior in Bayes theorem. As illustrated below, however, logarithmic convergence of the approximating posteriors is not necessarily good enough. EXAMPLE 1 (Fraser, Monette and Ng =-=[21]-=-). discrete data and parameter space, � □ Consider the model, with both M = { p(x | θ)= 1/3, x∈{[θ/2], 2θ,2θ + 1}, θ∈{1, 2,...} } ,910 J. O. BERGER, J. M. BERNARDO AND D. SUN where [u] denotes the in... |

1 |
What is the use of a distribution
- Good
- 1969
(Show Context)
Citation Context ... ∫ κ{˜p | p}= p(θ)log � p(θ) ˜p(θ) dθ, provided the integral (or the sum) is finite. The properties of κ{˜p | p} have been extensively studied; pioneering works include Gibbs [22], Shannon [38], Good =-=[24, 25]-=-, Kullback and Leibler [35], Chernoff [15], Jaynes [29, 30], Kullback [34] and Csiszar [18, 19]. DEFINITION 2 (Logarithmic convergence). A sequence of probability density functions {pi} ∞ i=1 converge... |

1 |
Statistical analysis if one-dimentional distributions
- Schmidt
- 1934
(Show Context)
Citation Context ...ely difficult to determine analytically. EXAMPLE 11 (Triangular distribution). The use of a symmetric triangular distribution on (0, 1) can be traced back to the 18th century to Simpson [39]. Schmidt =-=[37]-=- noticed that this pdf is the density of the mean of two i.i.d. uniform random variables on the interval (0, 1). The nonsymmetric standard triangular distribution on (0, 1), { 2x/θ, for 0 <x≤ θ, p(x |... |

1 |
A letter to the right honourable George Earls of Maclesfield. President of the Royal Society, on the advantage of taking the mean of a number of observations in practical astronomy
- Simpson
(Show Context)
Citation Context ...s to be extremely difficult to determine analytically. EXAMPLE 11 (Triangular distribution). The use of a symmetric triangular distribution on (0, 1) can be traced back to the 18th century to Simpson =-=[39]-=-. Schmidt [37] noticed that this pdf is the density of the mean of two i.i.d. uniform random variables on the interval (0, 1). The nonsymmetric standard triangular distribution on (0, 1), { 2x/θ, for ... |

1 | The Psi Function, 10.11 - Boros, Moll - 2004 |

1 | Expansion of Bayes risk for entropy loss and imsart-aos ver. 2007/01/24 file: mainBW.tex date: December 17 - Ghosal, Samanta - 1997 |