An illustration of the risk of borrowing information via a shared likelihood

TL;DR

This paper demonstrates that using shared likelihoods to incorporate supplementary data can sometimes worsen inference if the prior over a nuisance parameter is misspecified, even when the likelihood and main prior are correct.

Contribution

It provides a stylized example showing the potential pitfalls of joint likelihood approaches due to prior misspecification over nuisance parameters.

Findings

Inferences can be degraded by joint likelihood methods.

Misspecification of the nuisance prior affects inference quality.

Correct likelihood and main prior do not guarantee improved inference.

Abstract

A concrete, stylized example illustrates that inferences may be degraded, rather than improved, by incorporating supplementary data via a joint likelihood. In the example, the likelihood is assumed to be correctly specified, as is the prior over the parameter of interest; all that is necessary for the joint modeling approach to suffer is misspecification of the prior over a nuisance parameter.