Jeffreys' a priori distribution

As Jeffreys' a-priori distribution is referred to in the in Bayesian statistics , an a-priori distribution (a probability distribution, which is assumed to be independent of measurement data as a given). It is named after Sir Harold Jeffreys and distinguishes itself from other a priori distributions in that it is invariant to a reparametrization of the model parameters. Because of this invariance, the Jeffreys' a priori distribution is also called non-informative.

The probability density function of the Jeffreys' a priori distribution is proportional to the square root of the determinant of the Fisher information : ${\displaystyle \wp \left({\vec {\theta }}\right)}$

${\displaystyle \wp \left({\vec {\theta }}\right)\propto {\sqrt {\det {\mathcal {I}}\left({\vec {\theta }}\right)}}.\,}$

motivation

The problem arises from Bayes' theorem, in which the a posteriori distribution is given as proportional to the product of the likelihood and the a priori distribution , where represents the data, and model parameters (such as the mean and standard deviation of a normal distribution). Typically, the likelihood, that is, the probability distribution of the data, can be determined if a model parameter value is given. However, in order to obtain a probability distribution of the model parameters given the measurement data D, a data-independent a priori probability distribution is also necessary. ${\displaystyle \Pr(M\mid D)}$ ${\displaystyle \Pr(D\mid M)}$ ${\displaystyle \Pr(M)}$ ${\displaystyle D}$ ${\displaystyle M}$ ${\displaystyle M}$

For a long time, the orthodoxy of statistics rejected Bayesian statistics because the choice of a suitable a priori distribution seemed too arbitrary. What was particularly annoying was that the method for choosing an a priori distribution led to other distributions, depending on how the models were used were parameterized. Jeffreys took up this point of criticism and made it a boundary condition in the search for a method for choosing a priori distribution functions.

approach

Given a monotonic transformation which assigns an alternative model parameter to each model parameter , the invariance condition required is that the probability functions of the a priori distributions are the same for - pairs. ${\displaystyle \phi =h(\theta )}$ ${\displaystyle \theta }$ ${\displaystyle \phi }$ ${\displaystyle \theta }$ ${\displaystyle \phi }$

${\displaystyle \Pr \left(\Theta <\theta \right)=\Pr \left(\Phi <h(\theta )\right)}$

Since it is a monotonic transformation, a connection between the probability density functions of the two parameter spaces can be established. The method sought to find an a priori probability function must therefore satisfy the following relationship: ${\displaystyle h(\cdot )}$

${\displaystyle \wp (\theta )=\wp (\phi )\vert h'(\theta )\vert }$

It can be shown that the fishermen have a direct connection with information for the parameterization with and with ${\displaystyle \theta }$ ${\displaystyle \phi }$

${\displaystyle I_{\theta }(\theta )=(h'(\theta ))^{2}\cdot I_{\phi }(\phi =h(\theta )),}$

Thus, by choosing an a priori distribution, it can actually be concluded that invariant a priori distributions can be found from the Fischer information under reparameterization ${\displaystyle \wp \left(\eta \right)\propto {\sqrt {I_{\eta }(\eta )}}}$

${\displaystyle \wp \left(\theta \right)\propto \vert h'(\theta )\vert \wp \left(\phi \right),}$

and if the a priori distributions can be normalized, they are also genuinely the same.

Individual evidence

1. ↑ Torsten Becker, et al .: Stochastic risk modeling and statistical methods. Springer Spectrum, 2016. p. 330.
2. ↑ Jaynes, ET: Prior Probabilities . In: IEEE Trans. On Systems Science and Cybernetics . 4, No. 3, 1968, pp. 227-241. doi : 10.1109 / TSSC.1968.300117 .