Bayesian Probability of Zero?

Question

I've been reading a few different philosophical papers/books which have mentioned a "Bayesian belief". Within these texts I've been basically inferring that within the Bayesian theorem, there's something that says there's always a non-zero chance of something happening. Is there ever a time that the Bayesian belief would assign a probability of exactly 0% that something will not happen? After reading the wiki, I'm getting the sense that my inference was wrong, so now I'm confused on what the authors were attempting to convey. As an example, in a book by Nick Bostrom called Superintelligence, he writes:

Unless the AI's motivation system is of a special kind, or there are additional element in its final goal that penalize strategies that have excessively wide-ranging impacts on the world, there is no reason for the AI to cease activity upon achieving its goal. On the contrary: if the AI is a sensible Bayesian agent, it would never assign exactly zero probability to the hypothesis that it has no yet achieved its goal - this, after all, being an empirical hypothesis against which the Ai can have only uncertain perceptual evidence.

If within the Bayesian theorem, there is something that says there's a zero chance of something happening, in what circumstance would that be (in layman's terms ) and also in what circumstances is there never a zero probability?

Answer 1

Cromwell's rule as advocated by famous Bayesian statistician Lindley states that we should never assign zero probability to events, unless they are physically impossible. As soon as you assign zero probability, you cannot be convinced anymore by the data, no matter how strong the actual evidence.

Answer 2

Bayes theorem is one of the many important concepts in the field of AI. And, for the varied application that it is used for, imagine of all the scenarios to which it has been and will be applied, what percentage of them ensure that a particular event will not occur? It rarely or never is the case, and it would be wrong to assume a probability of 0 for some event to occur — unless we are absolutely sure of it, and no more such related events are to occur, and the experiment is over.

Suppose in an NLP task, where the aim is to correctly classify an email into spam and not spam. We base our model w.r.t. the words that we have seen in the training example. But it is absolutely not certain that the training set emails have all the words in the English vocabulary. And, while your training set had an email with the word "unfaithful" or "deceitful", the test set may throw you "amatorculist". Should a probability of 0 be assigned to the occurrence of this word? Well, it would be wrong to do so! Thus, it would be a wrong assumption about the hypothesis that the "amatorculist" will never occur. Most of the cases that you will find fall in the same line.

And for the cases of AI, a probability of zero is seldom ever 0, as most of the time it is data dependent, and to have seen all of the data even for testing is never the case.