I have been thinking about Ben's post that introduced me to the fundamental notion of possibility measure. This post is highly similar.

I can assign possibility measures somewhat arbitrarily within what the axioms allow, but I'll confess I don't find that satisfying from a modelling perspective. In some cases I could see myself assigning classical possibility measures based on domain knowledge (e.g. allowed/disallowed states of a game or impossibility of violations of certain laws of physics).

With probabilities we might use data to get empirical relative frequencies a direct estimates, or to fit distributions with maximum likelihood estimation, or to update priors.

Analogously, are there methods for "fitting" possibility measures or computing them data?


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I think the simple answer here is that we usually implicitly form a view on a space of possible events when we form a probability model. Typically we will demarcate the model to allow a set of outcomes that forms the sample space, and anything outside the sample space is considered impossible. In the event that we observe an "impossible" event we would then need to revise our model accordingly, or we might consider it okay to use the same model but treat this as an approximation that underestimates the probability of a purportedly (but not actually) impossible event. I don't agree that we can assign the possibility space arbitrarily --- if we do that we will often get terrible models that assert highly probable things to be impossible. However, we might indulge in some wishful simplification where we rule out possible but highly unlikely (and perhaps uninteresting) outcomes as "impossible" within the model.

A simple example of this is when we form a Bernoulli model for coin-flipping. In this situation we usually consider the possible outcomes to be heads and tails. We don't usually incorporate the possibility that the coin comes to rest on its edge, or that the coin-flip is interrupted by the building blowing up, etc. This is an approximation, and it is unrealistic insofar as it is possible that the coin might come to rest on its edge, or that the building might blow up, etc. However, this is not an arbitrary demarcation. Rather, we form the view that the heads and tails outcomes are the things that almost always occur, and we are uninterested in the other possible outcomes. (If the coin comes to rest on its edge then we might just repeat the coin-flip and say that outcome doesn't count. If the building blows up then we typically have bigger problems than accurate statistical modelling of coin flips.)


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