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Are there really people other than philosophers who are actually "Bayesians" or "Frequentists"? Frequentist statistics are just a special case of Bayesian st
by eemax 12y ago
Are there really people other than philosophers who are actually "Bayesians" or "Frequentists"?
Frequentist statistics are just a special case of Bayesian statistics with certain implicit, built-in priors - if you pick these priors using Bayesian statistics you'll get the same answers.
The frequentest toolbox is just a collection of these useful special cases of Bayesian statistics with priors that usually make sense in practice. The advantage is this greatly simplifies the statistical analysis for many problems. Sometimes these methods fail though, when the priors they rely on implicitly are far off from the actual, and so a Bayesian analysis is needed.
Of course, sometimes it's difficult or impossible to use Bayesian methods.
It's analogous to classical vs. quantum/relativistic physics. For many cases, you can get the right answer to a problem using classical physics. But under certain conditions, classical physics breaks down, and you must apply quantum or relativistic physics to get a meaningful answer. On the other hand, for many problems it would be silly or impossible to use quantum or relativistic physics because classical is perfectly good.
So you might get into argument about whether a specific case can be adequately handled by frequentest statistical methods, or whether a bayesian analysis can/must be applied.
The philosophical debate about the different approaches to the nature of probability is just that - a philosophical one, and one that has no real bearing on the usefulness and correctness of bayesian or frequentist statistics in practice.
- keithwinstein 12y agoI'm afraid you're mistaken -- frequentist statistics are not a special case of Bayesian statistics, and there is generally no "implicit, built-in prior" that allows you to get frequentist worst-case guarantees from Bayesian techniques. Consider the case of a confidence interval on a binary proportion given a finite number of samples. A frequentist method will produce an interval that includes the true value of the proportion with at least x% probability, even in the worst case, for any proportion between 0 and 1. (E.g. the Blyth-Still-Casella method or the Clopper-Pearson method.) An x% credible interval will include the true value exactly x% of the time, averaged over all values of the proportion weighted according to the prior. This will not provide the same guarantee (much less the same interval extent!), no matter what prior is used. Which is not to say the credible interval is bad or inappropriate. It's just not the same kind of tool. It is optimizing a different penalty function of non-inclusion. (Another example where the "Bayesian" technique is not quite as conservative as necessary to achieve a frequentist-style guarantee, even with a uniform prior: http://www.quora.com/I-have-burned-200-disks-and-I-want-to-make-sure-that-they-are-all-in-perfect-working-order-What-is-the-smallest-size-sample-I-could-test-in-order-to-be-relatively-confident-that-98-of-all-the-disks-are-fine-burned-correctly?share=1 http://www.quora.com/I-have-burned-200-disks-and-I-want-to-m...)
- andrea_s 12y agoStatistics researchers in academia definitely express a preference between frequentist and Bayesian inference. So no, it's not limited to philosophers - and I'm pretty sure all but the most forgiving frequentists would not agree with the concept "frequentist = Bayesian with flat priors". Remember that you cannot really formulate a flat infinite (and infinitely thin) distribution in mathematical terms: you need to resort to limit calculation. While the concept "Bayesian + flat prior = frequentist" is useful to explain a high level connection between the two worlds, there is a lot more to the topic - and in my opinion, it's something that can hardly be scratched without a formal education in the field.
- judk 12y agoWhy is "flat prior" here? A requentist method assumes a certain prior, for example Normal
- kgwgk 12y agoMaybe you mean that a frequentist method assumes a certain model, for example normal. A frequentist method will sometimes give similar results to a Bayesian analysis that uses a non-informative prior. For example, if we want to estimate a value from repeated measurements with Gaussian noise the frequentist result is equivalent to the Bayesian result if a "flat" (improper) prior is used. [http://en.wikipedia.org/wiki/Jeffreys_prior#Gaussian_distribution_with_mean_parameter http://en.wikipedia.org/wiki/Jeffreys_prior#Gaussian_distrib...]
- mjw 12y agoThe other problem with the "Bayes with flat prior = frequentist maximum likelihood" idea is that, even if you ignore the issues with improper priors, the concept of a "flat prior" is inherently dependent on arbitrary choices in the way a model is parameterised. It's not possible for a prior to be "flat" with respect to all re-parameterisations of a continuous parameter in a model. E.g. a flat prior for the variance isn't flat for its inverse (precision) or its square root (the std. dev.), and the choice of which of these alternative parameterisations you use to express the unknown quantity in the model is arbitrary. In the frequentist case it doesn't affect the result of the inference; in the Bayesian case it matters which of the parameterisations you choose your prior to be flat with respect to.
- jules 12y agoThis is true in some cases: maximum likelihood estimation often corresponds to MAP estimation with a certain prior (though it can be argued whether MAP estimation is truly a Bayesian concept, or just a way to transplant MLE to a Bayesian setting by using a very weird loss function). But many frequentist concepts have no Bayesian counterpart such that if you choose a particular prior you get the frequentist concept.