Bayesian Inference Is Unaffected by Selection: Fact or Fiction?

The problem considered is that of making inferences about the value of a parameter vector θ based on the value of an observable random vector y that is subject to selection of the form y∈S (for a known subset S). According to conventional wisdom, a Bayesian approach (unlike a frequentist approach) requires no adjustment for selection, which is generally regarded as counterintuitive and even paradoxical. An alternative considered herein consists (when taking a Bayesian approach in the face of selection) of basing the inferences for the value of θ on the posterior distribution derived from the conditional (on y∈S ) joint distribution of y and θ . That leads to an adjustment in the likelihood function that is reinterpretable as an adjustment to the prior distribution and ultimately leads to a different posterior distribution. And it serves to make the inferences specific to settings that are subject to selection of the same kind as the setting that gave rise to the data. Moreover, even in the absence of any real selection, this approach can be used to make the inferences specific to a meaningful subset of y-values.