Parameter Restrictions for the Sake of Identification: Is there Utility in Asserting that Perhaps a Restriction Holds?

TL;DR

This paper explores the use of mixture priors in Bayesian models to handle untestable assumptions that may or may not hold, aiming to improve understanding of partial identification in statistical inference.

Contribution

It introduces a novel perspective on incorporating uncertain assumptions via mixture priors within Bayesian analysis, addressing a gap in the literature on model averaging with partially identified models.

Findings

Mixture priors can represent 'maybe' assumptions in Bayesian models.

Bayesian model averaging can be applied across identified and partially identified models.

The approach offers a way to handle untestable assumptions in statistical inference.

Abstract

Statistical modeling can involve a tension between assumptions and statistical identification. The law of the observable data may not uniquely determine the value of a target parameter without invoking a key assumption, and, while plausible, this assumption may not be obviously true in the scientific context at hand. Moreover, there are many instances of key assumptions which are untestable, hence we cannot rely on the data to resolve the question of whether the target is legitimately identified. Working in the Bayesian paradigm, we consider the grey zone of situations where a key assumption, in the form of a parameter space restriction, is scientifically reasonable but not incontrovertible for the problem being tackled. Specifically, we investigate statistical properties that ensue if we structure a prior distribution to assert that `maybe' or `perhaps' the assumption holds. Technically this simply devolves to using a mixture prior distribution putting just some prior weight on the assumption, or one of several assumptions, holding. However, while the construct is straightforward, there is very little literature discussing situations where Bayesian model averaging is employed across a mix of fully identified and partially identified models.