In a textbook Probability Theory: The Logic of Science written by E. T. Jaynes and others, on page 13 it reads that:
For many years, there has been controversy over ‘frequentist’ versus ‘Bayesian’ methods of inference, in which the writer has been an outspoken partisan on the Bayesian side.
From this answer, if I am not wrong, I conjecture that any task which suits a frequentist approach can be solved also in a Bayesian way, and vice versa.
I later learned that the rise of MCMC algorithms in the 80's lead to an exponential rise in the use of Bayes in methods as complex models built through hierarchical distributions suddenly were tractable(reference: Mixtures of Conjugate Priors and MCMC).
Then my question is that if the Bayesian approach is better than the frequentist (1. Jaynes taught me that in that book; 2. for instance more parameters in models like the hidden Markov model can learn more information and then be better to reach the reality, the posterior distribution than the Markov model; 3) more reasons from this answer to this question: What is the importance of probabilistic machine learning?) and it has been tractable, then why frequentist methods still dominate the machine learning field?
In the answers to this question I learned why frequentist is better practically. But I wonder if Bayesian can be as practical as frequentist approaches? How can it be?