Your question is more on the semantic side: when can I call a model "Bayesian"?
Drawing conclusions from this excellent paper:
Fienberg, S. E. (2006). When did bayesian inference become "bayesian"?When did bayesian inference become "bayesian"? Bayesian Analysis, 1(1):1-40.
there are 2 answers:
- Your model is first Bayesian if it uses Bayes' rule (that's the "algorithm").
- More broadly, if you infer (hidden) causes from a generative model of your system, then you are Bayesian (that's the "function").
Surprisingly, the "Bayesian models" terminology that is used all over the field only settled down around the 60s. There are many things to learn about machine learning just by looking at its history!