Rissanen's universal prior for integers

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Description

It returns (up to normalizing constant) the mass assigned to each positive integer or a vector of integers by Rissanen's universal prior for positive integers

Usage

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Arguments

n

a vector of positive integers

Details

Rissanen's universal prior on positive integers is one of the default options for eliciting a noninformative prior distribution on the unknown population size N. It is a proper prior with tails of the order between 1/N and 1/N^2

Value

The mass assigned to each positive integer in the input vector of integers n by Rissanen's universal prior for positive integers

Q(n)=2^{-\log^*(n)} \qquad n>0

where \log^*(x)= \log(x)+\log( \log (x)) + \log( \log (\log (x))).... where the sum involves only the non-negative terms. Notice that masses are not normalized hence they do not add to one but to a finite positive real constant

c = ∑_{n=1}^∞ Q(n)

Author(s)

Danilo Alunni Fegatelli and Luca Tardella

References

Rissanen, J. (1983) A universal prior for integers and estimation by minimum description length. Ann. Statist. 11, no. 2, 416-431

Examples

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# Notice that masses are not normalized hence they do not add to one but to a finite 
# positive real constant c

rissanen(1:5)