Andrew Gelman has some interesting comments on non-informative priors this morning. Rather than thinking of the prior as a static thing, think of it as a way to prime the pump.
… a non-informative prior is a placeholder: you can use the non-informative prior to get the analysis started, then if your posterior distribution is less informative than you would like, or if it does not make sense, you can go back and add prior information. …
At first this may sound like tweaking your analysis until you get the conclusion you want. It’s like the old joke about consultants: the client asks what 2+2 equals and the consultant counters by asking the client what he wants it to equal. But that’s not what Andrew is recommending.
A prior distribution cannot strictly be non-informative, but there are common intuitive notions of what it means to be non-informative. It may be helpful to substitute “convenient” or “innocuous” for “non-informative.” My take on Andrew’s advice is something like this.
Start with a prior distribution that’s easy to use and that nobody is going to give you grief for using. Maybe the prior doesn’t make much difference. But if your convenient/innocuous prior leads to too vague a conclusion, go back and use a more realistic prior, one that requires more effort or risks more criticism.
It’s odd that realistic priors can be more controversial than unrealistic priors, but that’s been my experience. It’s OK to be unrealistic as long as you’re conventional.
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