Philosophical justification for possible but "implausible" (probability = 0) events in Bayesian Inference? It violates some intuition in me that no matter what particular outcome has occurred of a variable described by a continuous distribution, it must have had probability 0.
In Frequentist inference, I feel ok (but not great) by recognizing that the asymptotic value of the number of occurrences of any outcome divided by n (number of trials) is 0 as n approaches infinity.
In Bayesian, I don't feel satisfied (yet) with just saying that you can (correctly) have had 0 expectation of certainty for an outcome that has indeed happened.
I'm just hoping to find a perspective to understand this better.
Thanks!
Edit: Due to some confusion in the comments, I just want to emphasize that this is not a question about why point events must always have probability 0 in a (pure) continuous distribution. That is a strictly mathematical consequence because, for a given probability space, there cannot exist an uncountable number of disjoint events with probability greater than 0.
I'm just having trouble reconciling how we can interpret that with a Bayesian approach.
 A: Putting aside the philosophical waste-of-time that is the contemplation of whether or not the universe is continuous, the important thing is that there is nothing that is measurable as continuous. Any observation method has intrinsic errors; any observation method has finite resolution.
Therefore, any apparently continuous measurand must in reality be described in terms of an interval that is larger than a single point. And, for a continuous pdf, the pdf over the interval will integrate to more than zero, as long the pdf is positive over at least some sub-interval of it.
If the posterior pdf is zero over the whole interval that represents the measured value, then either the prior is wrong, or the likelihood is wrong (or they're both wrong).

And, to go a little further, and only half tongue-in-cheek, if you have observed something inexplicable, it's much more likely to be the result of something you thought to be impossible, rather than something you knew to be improbable. (with a nod to Douglas Adams's creation Dirk Gently, who was, to my knowledge, the first to express that idea, though not in those words)
