## International Statistical Review

- Internat. Statist. Rev.
- Volume 72, Number 3 (2004), 307-330.

### A Foundational Justification for a Weighted Likelihood Approach to Inference

#### Abstract

Two types of probability are discussed, one of which is additive whilst the
other is non-additive.
Popular theories that attempt to justify the importance of the additivity of
probability are then critically reviewed.
By making assumptions the two types of probability put forward are utilised to
justify a method of inference which involves betting preferences being revised
in light of the data.
This method of inference can be viewed as a justification for a weighted
likelihood approach to inference where the plausibility of different values of
a parameter *θ* based on the data
$\tilde{x}$
is measured by the quantity
$q\left(\theta \right)=l(\tilde{x},\theta )w\left(\theta \right)$, where
$l(\tilde{x},\theta )$ is the
likelihood function and *w(θ)* is a weight function.
Even though, unlike Bayesian inference, the method has the disadvantageous
property that the measure *q(θ)* is generally non-additive, it is argued
that the method has other properties which may be considered very desirable
and which have the potential to imply that when everything is taken into
account, the method is a serious alternative to the Bayesian approach in many
situations.
The methodology that is developed is applied to both a toy example and a real
example.

#### Article information

**Source**

Internat. Statist. Rev., Volume 72, Number 3 (2004), 307-330.

**Dates**

First available in Project Euclid: 8 December 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.isr/1102516473

**Keywords**

Non-additive probability Weighted likelihood Dutch-book argument Ellsberg paradox

#### Citation

Bowater, Russell J. A Foundational Justification for a Weighted Likelihood Approach to Inference. Internat. Statist. Rev. 72 (2004), no. 3, 307--330. https://projecteuclid.org/euclid.isr/1102516473