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buybuydandavis comments on The dangers of zero and one - Less Wrong
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I think Phil's experience suggests a reasonable way to attack these problems.
Do the analysis for a series of epsilons, starting with your measurement delta and working down, and see if it makes any difference to the results.
Also, in one of Jaynes' paper on the marginalization "paradox" he suggested that ignoring the existence of a variable gives you a different result than applying an ignorance prior (showing yet again that the term "ignorance prior" is a stupefying oxymoron).
Re: ignoring the existence of a variable... yes, absolutely. I meant in the more sweeping, less useful sense of "go work on a different problem altogether, or perhaps just have a beer and watch telly."
Re: seeing how results vary for different possible values of an unknown variable... yup, agreed.