A Case Study in Bayesian Sensitivity: Fish Response to Lake Acidification
Nov 30 1988
The purpose of this paper is to compare the importance of two sensitivity issues in a Bayesian analysis. the first issue is the choice of a probability distribution to represent prior uncertainty. Even assuming that there is a "true" regression function, we are surely unable to describe that distribution with complete accuracy. Instead, for convenience, we often choose a distribution that is close to the true on in the hope that this approximation does not lead us too far astray.
The second issue is the use of a parametric family of regression functions to represent the set of all regression functions that we deem plausible. When we believe the true regression function to be close, in some appropriate sense, to a parametric family of regression functions, then, for convenience, we often restrict attention to the parametric family and ignore the other plausible functions in the hope the this simplification does not lead us too far astray.
Both issue s have been treated separately elsewhere, the first in the Bayesian robustness literature, the second in the Bayesian nonparametric literature. Lavine (1988) provides a unified framework for dealing with them both. the current paper shwos how to implement Lavine's ideas in a problem previously studied by Reckhow (1987, 1988), of a Bayesian analysis of fish response to lake acidification. for Adirondack lakes taht were known previously to have supported brook trout, the probability p that a lake continues to support brook trout was modeled as a logistic function of the lake's pH and calcium. Prior opinions were elicited from an expert. Data were collected on a large number of lakes and a Bayesian analysis performed. For the purpose of this paper we simply the original problem somewhat by ignoring calcium and modeling the probability that a lake still supports brook trout as a function only of the lake's pH.
Our results in this example indicate the the second sensitivity issue can be much more important that the first, and the the usual Bayesian sensitivity analyses, by concentrating only on the first, are overlooking a potentially important source of posterior uncertainty.
Keywords:Bayesian robustness, Bayesian sensitivity, binomial regression, logistic regression