Your intuitions are what give you your axioms and Bayesian priors, the starting point of deduction and analysis, as well as your values and top-level goals.
You can't justify any belief at all without axioms/priors, or make any decisions about what to do without values/goals.
Intuition is the thing that gives you those axioms and values; it's really the "only game in town" for generating them.
There really isn't any Bayesian "prior" for us. We exist as agents interacting with an environment qua data stream. Every single moment brings new flows of "data" and as such there isnt a sense of having a prior and posterior since this milliseconds prior is last milliseconds posterior.
> Your intuitions are what give you your axioms and Bayesian priors
No, I would say it's your perceptions and memories (of past perceptions).