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I'm an experimental physicist so please pardon me if my thinking about this is too concrete. Let's say I am taking a measurement over and over and trying to determine the "real" value of something, but my measurement has some inherent noise in it. Let's say I have taken 100 measurements so far.

From what I understand, in Bayesian statistics I would determine the probability that my thing has value X based on the 'priors' - in this case the distribution I have of my measurements so far, and I would update my prediction based on any further measurements I would make with some math.

In frequentist statistics, I would just update my experimentally measured distribution with each new measurement by adding it in to the set of measurements.

How are these two things different? The Bayesian version seems like an unnecessarily cumbersome way to just add a new measurement to my distribution. What am I missing here?

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  • $\begingroup$ With sufficient data, the prior becomes unimportant (at least in reasonable cases). But one thing to keep in mind is that it's not always possible/feasible to collect a lot of data. There are fields in which studies are published with 10-20 noisy data points. Having a well-defined prior is valuable there because you're less likely to be misled by small amounts of random variation in the study. $\endgroup$ – mkt Feb 24 at 16:13
  • $\begingroup$ I see, thank you. it seems like having a prior which is influenced by one's personal biases may be more dangerous than staying agnostic... $\endgroup$ – SabrinaChoice Feb 24 at 23:49
  • $\begingroup$ No, it isn't, because (1) the prior makes you define and defend your subjective beliefs, and (2) one's biases can come through in many ways other in a frequentist study. It may be different in many areas of physics, where hypotheses are constrained by well-defined, mathematical theory and systems can be isolated into smaller and smaller units for experimentation. In many sciences, this is not true, and hypotheses can seem arbitrary or developed ad hoc while studies are full of noisy confounders (see the replication crisis in social psychology for examples of how this can go wrong). $\endgroup$ – mkt Feb 25 at 7:38
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    $\begingroup$ @mkt thanks for your comment. Can you expand on how the prior leads one to defend the subjective belief? On another note, I agree very much with your point (2)! $\endgroup$ – SabrinaChoice Mar 3 at 1:04
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    $\begingroup$ @ mkt Thank you, that helps clear up some small misunderstandings I had. $\endgroup$ – SabrinaChoice Mar 16 at 3:20
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You are right that in your scenario, where you have lots of previous measurements and want to combine them with new measurements, there are not a lot of differences between the two approaches. I.e. if the prior is flat, then the posterior is just the likelihood.

Differences primarily occur, if you had prior beliefs before your very first measurements (if some things cannot be measured or you only get a few noisy measurements, then getting information from the literature or elicited from experts could be hugely helpful), if you use some rather crude/dubious (but in a sense valid) frequentist approaches (e.g. 95% CIs constructed as the empty set 5% of the time and the whole parameter space 95% of the time) rather than say likelihood based CIs, depending how you summarize information (e.g. maximum likelihood/ maximum a-posteriori, posterior mean or median etc. - these might be differently affected by different parametrizations), due to difficulties in getting appropriate standard errors/CIs using frequentist methods on sparse data situations (often much easier on the Bayesian framework) and due to very different inference tools that are typically used (needing to e.g. maximize a log-likelihood & get the observed Fischer information vs. sampling from a posterior - sometimes one is much easier than the other).

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