"p-value" as a term already has a definition -- one that imposes a frequentist perspective -- so you'd have to come up with a new term to define a Bayesian analogue of that concept.
I fail to understand why the terminology matters. If you need to make a decision then there will be a threshold in the probability that will determine it. It doesn’t matter if you call it p values or anything else.
I'm curious how much professional statistical experience you have? In my experience probabilities alone are very rarely used and distributions of beliefs are combined with distributions of value so you can choose optimal risk/reward scenarios.
Bayesian decision making typically involves combining a cost function with your posterior distribution.
It's worth pointing out that this is typically not possible in most frequentist frameworks since they work in double negative style assertions and arbitrary threshold rather than modeling the problem itself.
While you can map Bayesian approaches to frequentists tests, this is usually a misunderstanding of Bayesian methods coming from a purely frequentist background. Bayesian analysis is fundamentally more flexible since it just the application of the sum and product rules (you don't even need Bayes' theorem since it can be trivially derived from these) as well as corresponding cost/value functions.
I'm not understanding your confusion. Different concepts require different words to distinguish them in conversation. p-value is a name, not merely a descriptor, of one possible way to define a threshold. Other ways of defining thresholds get different names so that everyone knows what's being discussed.
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