I am not sure whether this is the answer you’re looking for, but I think it’s true and could be de-confusing, and others have given the standard/practical answer already.
You can try running a program which computes Bayesian updates to determine what happens when this program is passed as input an ‘observation’ to which it assigns probability 0. Two possible outcomes (of many, dependent on the exact program) that come to mind:
The program returns a ‘cannot divide by 0’ error upon attempting to compute the observation’s update.
The program updates on the observation in a way which rules out the entirety of its probability-space, as it was all premised on the non-0 possibilities. The next time the program tries to update on a new observation, it fails to find priors about that observation.
Bayes’ theorem is an algorithm which is used because it happens to help predict the world, rather than something with metaphysical status.
We could also imagine very-different (mathematical)-worlds where prediction is not needed/useful, or, maybe, where the world is so differently-structured that Bayes’ theorem is not predictive.
I am not sure whether this is the answer you’re looking for, but I think it’s true and could be de-confusing, and others have given the standard/practical answer already.
You can try running a program which computes Bayesian updates to determine what happens when this program is passed as input an ‘observation’ to which it assigns probability 0. Two possible outcomes (of many, dependent on the exact program) that come to mind:
The program returns a ‘cannot divide by 0’ error upon attempting to compute the observation’s update.
The program updates on the observation in a way which rules out the entirety of its probability-space, as it was all premised on the non-0 possibilities. The next time the program tries to update on a new observation, it fails to find priors about that observation.
Bayes’ theorem is an algorithm which is used because it happens to help predict the world, rather than something with metaphysical status.
We could also imagine very-different (mathematical)-worlds where prediction is not needed/useful, or, maybe, where the world is so differently-structured that Bayes’ theorem is not predictive.