I believe one major advantage of Bayesian inference is the intuitiveness of interpretation. This is my primary interest. However, it's not completely clear to me when it's OK to make such an interpretation.
I make the potentially false assumption that fitting a probability model in the frequentist way is virtually the same as fitting the same model with a flat prior in a Bayesian way. Please nuance or correct that as interest number one (1).
And my main interest (2) is if my assumption is true (and if a flat prior happens to be the best prior I could possibly come up with), does the posterior of the model fitted in a frequentist way, but sampled say with an MCMC sampler, allow for me to make a Bayesian type of interpretation? e.g., the probability that an individual described by some particular configuration of $X$ will have an income ($Y$) greater than $100K is 76%.
I've read that what makes an analysis Bayesian is that it involves prior information. Is that really the essence? I've also read that you can't make such interpretations as my example about income from frequentist results. Does sampling from the posterior with MCMC methods move me away from frequentist methods sufficiently to make a Bayesian interpretation?
I greatly appreciate your direction. Thank you.