[Double_Org]

A bottling company is interested in determining the accuracy with which their equipment is filling bottles of water. One answer would be "95% percent of the bottles contain between 11.9 and 12.1 ounces". A different way of answering the question would be to estimate the actual distribution of water amounts.

The difference here, is that knowing a distribution is often more useful than just knowing the mean, or the variance, or some confidence intervals. Bayesian methods tend to be useful when you want this sort of information which is often the case when you are using it for decision making (or something like game theory).

Another uses case is when you are making decisions requiring multiple pieces of information that don't neatly fit together. A simple example is cancer screening. A rational decision about the proper threshold requires you to combine information about (1) The accuracy of your test, (2) The prevalence of the cancer in the population.

I will also add that the formula presented in the article is the simple case with discrete distributions. The more general version of the formula can also handle continuous distributions.

[mlevental]

lol is this copypasta? i'm quite familiar with all of these toy examples of inference instead of point estimation. i'm talking about fitting models rather than descriptive statistics (or decision theory).

[Double_Org]

Commonly a model is being used primary to make better decisions. Specifically in the context of fitting models, Bayesian methods are really popular for hyperparameter tuning.

I guess my main point is that at least one reason people are using Bayesian methods is because they are dealing with problems that are qualitatively different than more prototypical prediction problems.