You can squish it into Bayes by considering it a uniform prior on the real number line that you never update, but you’re not really doing things “in the spirit” of Bayes, then.
IMO, null hypothesis testing is way overused. We should be quantifying the comparison of the null versus alternative hypotheses. People already compare them. Might as well bring some math into it.
If your hypothesis is that A is greater than B, then you're test boils down to "The probability that A is greater than B", which is arrived at through parameter estimation via Bayes' Theorem.
You can consider the distribution of estimates for a parameter to represent a space of possible hypothesis for the true value of that parameter and their absolutely or relative likelihood based on the information available.
The also has the pleasant consequence that hypothesis testing using Bayesian methods nearly always is closer to the actual question someone wants to answer rather than replying with confusing statements such as "the probability of rejecting the null hypothesis", which contains an implicit double negative.
but even in this situation you would need a threshold for decision making, i.e. a p-value.
Bob bob bob'bob bob bob bob bob, bob bob bob bob bob "Bob".
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.
In the most common situations this typically means the expected value well exceeds the expected cost and you're comfortable with the risk if you're wrong.
A prior could be any previous knowledge that you have on the subject, or anything about the world.
Let me give you a silly example. Imagine I do an experiment, on if person A can see the future, and predict coin flips.
Now, imagine that I do this experiment, and lo and behold it passes the 5% threshold P value!
Remember what a P value is. It is simply the percent chance that the results that you got were by chance.
So, if I do that experiment, and it actually gets at the "P value" of 5%, which means that there is a 5% chance of it happening by chance, do you know believe that Person A can see the future?
I wouldn't. I would need a much lower P value to start believing that.
Thats all a prior is. It is using the fact that "Person A can see the future!" is an extraordinary claim. And for that, we require extraordinary evidence. Much better evidence than a mere "5% that it happened by chance". P
P-values are fundamentally Frequentist because this framework see that observed data as random, whereas Bayesian statistics believe our observations are the only thing that is actually known, and all the other parameters are what is random in our experiments. That is: the data is the only part of an experiment that is real, everything else that we can't directly observe is something we must hypothesize about.