If you understand that both are methods with relatively good asymptotic properties (and limitations), and that a prior can be a good or bad idea depending on the case (no free lunch), then you can use them wisely. No need of magical subjective learning to enjoy Bayes.
How useful is Bayesian econometrics?

http://www.stat.columbia.edu/~gelman/research/published/philosophy.pdf
a bayesian and a nonbayesian write a paper on bayesian philosophyOne of the problems with Bayesian philosophy is that each Bayesian has a different one, there is a lot of different interpretations and different schools arguing about how to use the method (informative or less informative priors), subjectivism,...They remind me of the movie the "life of Brian", when two small liberation groups fight to dead even sharing their hate for Romans. Bayesians share in general their rejection of frequentist, but each one disagrees with another.

http://www.stat.columbia.edu/~gelman/research/published/philosophy.pdf
a bayesian and a nonbayesian write a paper on bayesian philosophyOne of the problems with Bayesian philosophy is that each Bayesian has a different one, there is a lot of different interpretations and different schools arguing about how to use the method (informative or less informative priors), subjectivism,...They remind me of the movie the "life of Brian", when two small liberation groups fight to dead even sharing their hate for Romans. Bayesians share in general their rejection of frequentist, but each one disagrees with another.
The time of ideological Baysianism is long over. Nowadays you use what leads to the easier and cleaner computational solution.

http://www.stat.columbia.edu/~gelman/research/published/philosophy.pdf
a bayesian and a nonbayesian write a paper on bayesian philosophyOne of the problems with Bayesian philosophy is that each Bayesian has a different one, there is a lot of different interpretations and different schools arguing about how to use the method (informative or less informative priors), subjectivism,...They remind me of the movie the "life of Brian", when two small liberation groups fight to dead even sharing their hate for Romans. Bayesians share in general their rejection of frequentist, but each one disagrees with another.
The time of ideological Baysianism is long over. Nowadays you use what leads to the easier and cleaner computational solution.
exactly. the whole point of the bayesian position is that all of your cards are on the table. Priors are explicit.

ML completely bypasses bayesian, specially for time series. That would mean bayesian is useless because of the ML hype.
Very. With the current ML hype, bayesian is the future of econometrics.
Loopy Belief propagation, Expectation Propagation, Variational Bayes and other approximate inference methods are all bayesian?

you asked a good question, but at a wrong place. usefulness of an econometrics method is now determined by industry, not us
Currently doing PhD but haven't explored Bayesian stuff much. There's a course delivered next Fall and I'm wondering whether it has much application in theoretical/applied econometrics.

I'm wondering whether it has much application in theoretical/applied econometrics.
Traditionally the Bayesian approach has been the escape route for when your time series sample is too small estimate your VAR models the old frequentist way. Quite frankly there aren't many applications beyond that.
I'm willing to change my mind if someone can point me to a recent top5 publication using Bayesian metrics (but not Bayesian VAR).Two people nogooded but nobody gave an example. What gives?

Currently doing PhD but haven't explored Bayesian stuff much. There's a course delivered next Fall and I'm wondering whether it has much application in theoretical/applied econometrics.
Really useful for time series analysis.
any good references to bayesian time series?

Really? What can bayes add to TS that unit root testing, Granger Causality, and cointegration cannot answer?
Depends. Useful for macro (time series + estimation of DSGE models) but other fields are not much into bayesian stuff.
Quick survey: are you from a thirdworld country ?
Have you not heard of timevarying parameters models, Markovswitching multivariate models, and the like ? Do you think maximizing the likelihood function of such a model is an easy task ?

Under regularity conditions, if the parameters have a finite dimension then the Bayesian posteriors asymptotically shrink to the maximum likelihood estimator, and therefore to the parameter of interest. Moreover, any centrality measure of the posterior (conditional mean, or mode) has the same asymptotic distribution as maximum likelihood. In practice, there is absolutely no advantage in one over another, they work the same. Any other statement is just cheap talk of illiterate people.
Useful view from asymptotia. most applications in the real world have finite samples and it’s easy for this equivalence not to hold, especially with many parameters even if n is large

There's a lot of applications in IO/marketing. You could look at people like Peter Rossi and JP Dube for example who have published several papers in good econ journals. Hierarchical Bayesian models are very useful to understand the distribution of consumer heterogeneity.
I generally use Bayesian methods (industry) because I often work with medium sized datasets consisting of some small number of observation for many units, and I need to generate unitlevel forecasts. My methods wouldn't fly in top econ journals, but they make much better predictions than frequentist/ML methods in practice. So if you may end up in industry, it's a very useful skill, and a relatively rare one.
Also if you open up any recent volume of JASA, you'll see Bayes is very popular among statisticians and you can get a good idea of what they are working on. There's a lot of interesting work in nonparametrics in particular.