A prior is not a program that tells you what to do with the data. A prior is a set of hypotheses with a number assigned to each. When data comes in, we compute the likelihoods of the data given each hypothesis on the list, and use these numbers to obtain a posterior over the same hypotheses. There’s no general way to have a “none of the above” (NOTA) hypothesis in your prior, because you can’t compute the likelihood of the data given NOTA.
Another equivalent way to think about it: because of the marginalization step (dividing everything by the sum of all likelihoods), Bayesian updating doesn’t use the total likelihood of the data given all current hypotheses—only the relative likelihoods of one hypothesis compared to another. This isn’t easy to fix because “total likelihood” is a meaningless number that doesn’t indicate anything—it could easily be 1000 in a setup with an incorrect prior or 0.001 in a setup with a correct prior.
I’m not sure exactly what can qualify as a prior.
Is “Anomalies may be clues about a need to make deep changes in other priors” a possible prior?
A prior is not a program that tells you what to do with the data. A prior is a set of hypotheses with a number assigned to each. When data comes in, we compute the likelihoods of the data given each hypothesis on the list, and use these numbers to obtain a posterior over the same hypotheses. There’s no general way to have a “none of the above” (NOTA) hypothesis in your prior, because you can’t compute the likelihood of the data given NOTA.
Another equivalent way to think about it: because of the marginalization step (dividing everything by the sum of all likelihoods), Bayesian updating doesn’t use the total likelihood of the data given all current hypotheses—only the relative likelihoods of one hypothesis compared to another. This isn’t easy to fix because “total likelihood” is a meaningless number that doesn’t indicate anything—it could easily be 1000 in a setup with an incorrect prior or 0.001 in a setup with a correct prior.
People have beliefs about how various sorts of behavior will work out, though I think it’s rare to have probabilities attached.