I like the emphasis on a type distinction between likelihoods and probabilities, thanks for articulating it!
You seem to ponder a type distinction between prior and posterior probabilities (and ask for English language terminology which might align with that). I can think of a few word-pairings which might be relevant.
Credibility (‘credible’/‘incredible’)
Could be useful for talking about posterior, since it nicely aligns with the concept of a credible interval/region on a Bayesian parameter after evidence.
After gathering evidence, it becomes credible that...
...strongly contradicts our results, and as such we consider it incredible...
Plausibility (‘plausible’/‘implausible’)
Not sure! To me it could connote a prior sort of estimate
It seems implausible on the face of it that the chef killed him. But let’s consult the evidence.′
The following are plausible hypotheses:...
But perhaps unhelpfully I think it could also connote the relationship between a model and an evidence, which I think would correspond to likelihood.
Ah, that’s a more plausible explanation of what we’re seeing!
Completely implausible: the chef would have had to pass the housekeeper in the narrow hallway without her noticing...
Aleatoric and epistemic uncertainty
There’s also a probably-meaningful type distinction between aleatoric uncertainty (aka statistical uncertainty) and epistemic uncertainty, where aleatoric uncertainty refers to things which are ‘truly’ random (at the level of abstraction we are considering them), even should we know the ‘true underlying distribution’ (like rolling dice), and epistemic uncertainty refers to aspects of the domain which may in reality be fixed and determined, but which we don’t know (like the weighting of a die).
I find it helpful to try to distinguish these, though in the real world the line is not necessarily clear-cut and it might be a matter of level of abstraction. For example it might in principle be possible to compute the exact dynamics of a rolling die in a particular circumstance, reducing aleatoric uncertainty to epistemic uncertainty about its exact weighting and starting position/velocity etc. The same could be said about many chaotic systems (like weather).
Another possibility which occurs to me is to use “elegance” for prior probability. This is a common way to refer to the subjective simplicity of a hypothesis in science, so it fits pretty well!
We might then insist that “probability” should always incorporate everything we know to the best of our ability, and “elegance” be used to refer to prior probability.
This seems really weird in certain cases, though, such as “I know my swimming led to my getting bit by a shark, but you can’t blame me; it wasn’t very elegant!”
I like the emphasis on a type distinction between likelihoods and probabilities, thanks for articulating it!
You seem to ponder a type distinction between prior and posterior probabilities (and ask for English language terminology which might align with that). I can think of a few word-pairings which might be relevant.
Credibility (‘credible’/‘incredible’)
Could be useful for talking about posterior, since it nicely aligns with the concept of a credible interval/region on a Bayesian parameter after evidence.
Plausibility (‘plausible’/‘implausible’)
Not sure! To me it could connote a prior sort of estimate
But perhaps unhelpfully I think it could also connote the relationship between a model and an evidence, which I think would correspond to likelihood.
Aleatoric and epistemic uncertainty
There’s also a probably-meaningful type distinction between aleatoric uncertainty (aka statistical uncertainty) and epistemic uncertainty, where aleatoric uncertainty refers to things which are ‘truly’ random (at the level of abstraction we are considering them), even should we know the ‘true underlying distribution’ (like rolling dice), and epistemic uncertainty refers to aspects of the domain which may in reality be fixed and determined, but which we don’t know (like the weighting of a die).
I find it helpful to try to distinguish these, though in the real world the line is not necessarily clear-cut and it might be a matter of level of abstraction. For example it might in principle be possible to compute the exact dynamics of a rolling die in a particular circumstance, reducing aleatoric uncertainty to epistemic uncertainty about its exact weighting and starting position/velocity etc. The same could be said about many chaotic systems (like weather).
Thanks for making some suggestions!
Another possibility which occurs to me is to use “elegance” for prior probability. This is a common way to refer to the subjective simplicity of a hypothesis in science, so it fits pretty well!
We might then insist that “probability” should always incorporate everything we know to the best of our ability, and “elegance” be used to refer to prior probability.
This seems really weird in certain cases, though, such as “I know my swimming led to my getting bit by a shark, but you can’t blame me; it wasn’t very elegant!”