Roughly speaking, we can divide Bayesianism into two, maybe three or more, separate but related meanings:
1. Adherence to a form of Bayesian epistemology. You think that knowledge comes in degrees of belief, and the correct way to update your beliefs on seeing new information is to use Bayes theorem. It's usually done informally.
2. Adherence to Bayesian statistics. You believe that frequentist inference is invalid and that frequentist measures of an estimator's quality should not be used. Instead, you prefer to use precisely defined priors and likelihoods, derive their posteriors, and report a quantity based solely on that. Moreover, you would often espouse some form of Bayesian decision theory - i.e., you have a loss function in addition to your prior and likelihood, and report (or act on) the optimal decision according to your framework. All of this is usually done formally.
Your comments about Dirichlet don't make sense. Are you thinking about the Dirichlet distribution? If so, it is more widely used in Bayesian statistics than frequentist statistics, as it is the conjugate prior to the multinomial distribution. Regarding your comments about the SAS institute, I can say this: Most of the members of this forum are deeply interested in deep learning. Is deep learning Bayesian? No. Not even Bayesian deep learning is properly Bayesian. Does that matter to you, as a Bayesian epistemologist? No, as deep learning has little to nothing to do with epistemology. Does it matter to you, as a Bayesian statistician? No, as deep learning is not about inference or decision theory, which is what Bayesian statisticians care about (for the most part).
By the way, Bayes theorem isn't a "statistical technique", it's just a theorem. Used by all statisticians without a second thought. It's when you use it to do inference you become a Bayesian statistician.
Roughly speaking, we can divide Bayesianism into two, maybe three or more, separate but related meanings:
1. Adherence to a form of Bayesian epistemology. You think that knowledge comes in degrees of belief, and the correct way to update your beliefs on seeing new information is to use Bayes theorem. It's usually done informally.
2. Adherence to Bayesian statistics. You believe that frequentist inference is invalid and that frequentist measures of an estimator's quality should not be used. Instead, you prefer to use precisely defined priors and likelihoods, derive their posteriors, and report a quantity based solely on that. Moreover, you would often espouse some form of Bayesian decision theory - i.e., you have a loss function in addition to your prior and likelihood, and report (or act on) the optimal decision according to your framework. All of this is usually done formally.
Your comments about Dirichlet don't make sense. Are you thinking about the Dirichlet distribution? If so, it is more widely used in Bayesian statistics than frequentist statistics, as it is the conjugate prior to the multinomial distribution. Regarding your comments about the SAS institute, I can say this: Most of the members of this forum are deeply interested in deep learning. Is deep learning Bayesian? No. Not even Bayesian deep learning is properly Bayesian. Does that matter to you, as a Bayesian epistemologist? No, as deep learning has little to nothing to do with epistemology. Does it matter to you, as a Bayesian statistician? No, as deep learning is not about inference or decision theory, which is what Bayesian statisticians care about (for the most part).
By the way, Bayes theorem isn't a "statistical technique", it's just a theorem. Used by all statisticians without a second thought. It's when you use it to do inference you become a Bayesian statistician.