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I often hear of a debate between Bayesian and Frequentist approaches to statistical inference, with both sides (although I've heard much more from the Bayesian side) giving detailed arguments as to why their side is better. What I haven't been convinced of is why this argument is important at all. If I approach a problem from a Bayesian perspective, would this contradict results if I had first approached it from a Frequentist perspective? I haven't really heard of any concrete distinctions.

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  • $\begingroup$ Why should it..? If you can solve a problem using two different approaches, why using both should be worst then using only a single one? Moreover, I wouldn't say that the whole discussion is about who is "better". $\endgroup$
    – Tim
    Aug 30, 2016 at 22:03
  • $\begingroup$ It seems like many methods are similar, with Bayesian = Frequentist + non-uniform prior(s) ? $\endgroup$
    – GeoMatt22
    Aug 30, 2016 at 22:10
  • $\begingroup$ The results can be contradictory though. $\endgroup$
    – amoeba
    Aug 30, 2016 at 22:51
  • $\begingroup$ If you're a philosopher, sure. But most of us aren't. We mostly care about what works. Sometimes bayesian methods are better (e.g when you don't have tons of data) and sometimes frequentists win ( e.g. If you need a simple answer and have a bunch of data). In this respect, I'd like to point out that having lots of data depends heavily on the problem you're trying to solve. $\endgroup$
    – Yair Daon
    Aug 30, 2016 at 22:57
  • $\begingroup$ @YairDaon is it so clear which approach is favored in the case of big vs. small data? One interesting historical note is that it seems like "Frequentist" methods were more the default in the days of "small" data, while today many "big" data approaches lean "Bayesian" (I am thinking engineering data assimilation/robotics applications in particular). $\endgroup$
    – GeoMatt22
    Aug 30, 2016 at 23:29

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I try to show one aspect of similarity and difference in this example:

We have a sample from a population, everybody agrees that Xbar is the best estimate we have for mu.

Frequentist point of view: Xbar is a random variable that its probability density function is concentrated around mu. The constant mu is probably close to my xbar in this range called Confidence Interval.

Bayesian point of view: Mu is an unknown variable for me, conditional to this data (my sample), if I bet that the population mean is close to Xbar within this range called Credible Interval, I expect to win most of the time.

Great statisticians in the forum, please correct my answer if needed. Yours, Amir

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