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    $\begingroup$ I would be careful about the interpretation of non-Bayesian approaches being `Bayesian inference in disguise'; there are many cases where approaches are superficially the same (e.g. the same equations show up), but the inferences drawn are substantially different. $\endgroup$ Commented Jan 13, 2020 at 17:41
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    $\begingroup$ Perhaps, though I would still argue that this is not really the issue. One of the key misconceptions which drives this perception is the idea that Bayesian inference just means using MAP estimators instead of the MLE, which is not really accurate. One frequently hears the claim that the Lasso is `just Bayes with Laplace priors'; this is extremely untrue if one is computing posterior means instead of the MAP. $\endgroup$ Commented Jan 13, 2020 at 19:40
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    $\begingroup$ @πr8 sure, I totally agree with you. I'm trying to say, that the non-Bayesian approaches to such problems are often quite strongly related to their Bayesian counterparts, & the distion gets blurry. $\endgroup$ Commented Jan 13, 2020 at 20:47
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    $\begingroup$ @Tim: to hardcore Bayesians, everything that works does so because it's similar to a Bayesian method. I'm not a hardcore Bayesian (i.e., very rarely do I actually use Bayesian methods to analyze data), but I still mostly agree with that idea anyways. $\endgroup$ Commented Jan 14, 2020 at 17:51
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    $\begingroup$ @πr8 I think you're alluding to the fact that Bayesian inference isn't only about using MAP estimators these days, and that that's been a historical focus because it's easier to compute. If I understand your comment then you're saying it is not true that Lasso can be considered Bayesian when something like Hamiltonian Monte Carlo is used to solve for the full posterior - if you then calculate the mean of the posterior rather than the mode (as in MAP)? Does it "become" Bayesian again, if you look at the mode rather than the mean - even under HMC? I've no idea without a lot more thought! $\endgroup$ Commented Jan 15, 2020 at 11:40