An invariant form for the prior probability in estimation problems

By H. Jeffreys. 1946

It is shown that a certain differential form depending on the values of the parameters in a law of chance is invariant for all transformations of the parameters when the law is differentiable with regard to all parameters. For laws containing a location and a scale parameter a form with a somewhat restricted type of invariance is found even when the law is not everywhere differentiable with regard to the parameters. This form has the properties required to give a general rule for stating the prior probability in a large class of estimation problems.

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Citations in the corpus

Lifelong Generative Modeling (2017, background): “This generalization derives from the fact that the FIM can be cast as a KL divergence between the posterior and an epsilon perturbation of the same random variable [51].”
Aleatoric and Epistemic Uncertainty in Machine Learning: A Tutorial Introduction (2019, background): “See work on “non-informative” priors, however (Jeffreys, 1946; Bernardo, 1979).”
Bayesian coalescent inference of past population dynamics from molecular sequences. (2005, background): “…exponential distribution with a mean equal to the previous population size: hj;Expðhj 1Þ; 2 j m: ð5Þ In addition, we introduce a scale-invariant prior (Jeffreys 1946) on the first element fh1ðh1Þ}1=h1 to signify that at N orthern A rizona U niversity on January 23, 2015 http://m…”
On f-Divergences: Integral Representations, Local Behavior, and Inequalities (2018, unknown): “2) Jeffrey’s divergence [35]: J(P‖Q) := D(P‖Q) +D(Q‖P ) (19) = Df (P‖Q) (20)”
Is Bayes Posterior just Quick and Dirty Confidence (2011, methodology): “Many extensions of the Bayes mathematical prior have been proposed trying to achieve the favorable behavior of the original Bayes, for example, Jeﬀreys (1939, 1946) and Bernardo (1979).”
A Didactic Presentation of Snijders’s lz* Index of Person Fit With Emphasis on Response Model Selection and Ability Estimation (2012, unknown): “…ð34Þ hence r0ðyÞ ¼ Pn i¼1 riðyÞ Pi 00ðyÞ 2 Pn i¼1 riðyÞ Pi 0ðyÞ : ð35Þ Note eventually that Hoijtink and Boomsma (1995, p. 57) have established that in the case of the 1PL model, the function h(y) used in Equation 30 is nothing else than the so-called Jeffreys’ prior density (Jeffreys, 1939, 1946).”
Adaptive computation and machine learning (1998, background): “parameterization of the likelihood (Jeffreys 1946; Jaynes 1968).”
Optimal estimation of a large-dimensional covariance matrix under Stein’s loss (2018, unknown): “It is equal to the Jeffreys [21] divergence between the multivariate normal distribution with zero mean and covariance matrix n and the multivariate normal distribution with zero mean and covariance matrix S̃n, rescaled by the factor 1/p.”
Joint additive Kullback–Leibler residual minimization and regularization for linear inverse problems (2007, background): “The work reported in [11] appears to have been the first to point out that the symmetric form (4) had special statistical properties from a Bayesian perspective.”
Measurement Error Models (2000, methodology): “The symmetric version of (l), J(f,g) = K ( f : g) + K(g : f), was used by Jeffreys (1946) as a measure of divergence between two distributions, but not as an information function.”
Weighted exponential distribution: properties and different methods of estimation (2015, background): “Therefore, it is not unreasonable to assume independent gamma priors on the shape and scale parameters for a two-parameter lifetime distribution, because gamma distributions are very flexible and Jeffreys’ (non-informative) prior, introduced by Jeffreys’,[17] is a special case of this.”