3 ms·
My Bayesian theory is a bit rusty, but here we go. Say we have data X, and some non-finite dimensional index into the family of functions that describe the the
by antics 14y ago
My Bayesian theory is a bit rusty, but here we go.
Say we have data X, and some non-finite dimensional index into the family of functions that describe the the data, called \theta.
The Bayesian perspective classically holds \theta constant and optimizes the expected loss, conditioned on the data X. The frequentist perspective, on the other hand, classically optimizes \theta, that is, it picks the best \theta over the data X, unconditionally.
This has two impacts. First, all things equal frequentist statistics will tend to be more stable, and more calibrated, but less coherent. It is commonly said that frequentist statistics will "isolate" one from poor decision making, and all things equal, that will be true.
Specific, clear wins for frequentists are bootstrapping procedures (e.g., Efron's bootstrap, the b of n bootstrap, Jordan's own scalable "bag of bootstraps" from NIPS 2011), which are methods for building what are called "quantifiers" for "estimators". In short, this means that if you have some estimator (e.g., a classifier, or a mean, or whatever), you want to be able to quantify the certainty of your estimator -- so if you've only seen 5 examples, you want to express that you're less certain about this. This is clearly a frequentist application, not a Bayesian application, and in general, it points to the fact that pure frequentist tools not only have a place in inference, but they fill a niche that Bayesian tools necessarily will not, and in some cases, cannot, fill.
- equark 14y agoIt sounds like you're just saying that if you want to know the frequentist properties of an estimator you have to be frequentist. That's a tautology. The harder question is whether there are any decisions you'd prefer to make using a non-Bayesian procedure. That's basically a tautology in the other direction though.
- antics 14y agoAs I said, my Bayesian theory is rusty, but there are no "frequentist properties" of an estimator. Frequentist inference is inference -- it doesn't make guarantees about the underlying thing it's approximating, it provides guarantees about its approximation. The key here is that Bayesian and frequentist procedures provide different sorts of guarantees. Frequentists optimize for \theta, the possible set of things that could describe all of the data X, while Bayesians will assume a single describing function \theta (this might come from an "expert") and simply optimize the expectation conditioned on the data. Neither is "wrong" but in the case of the bootstrap, the result is calibrated in a way that Bayesian inference simply never will be (if it were, it would be frequentist). EDIT: As per your second question, actually I think it's not more interesting. A classifier is a type of estimator, so all of the general frequentist guarantees actually still apply to decisionmaking.
- equark 14y agoFrequentist statistics is about determining the repeated sampling properties of a procedure/statistic/estimator. It's about evaluation not estimation. "Optimizing \theta" or whatever you're envisioning is just one possible procedure you might be interested in evaluating. You can use the repeated sampling properties of your procedure to do frequentist inference or to evaluate other properties like unbiasedness, consistency, risk, etc. Typically the goal is to find procedures that have "good" frequentist (repeated sampling) properties. Most Bayesian-inclined statisticians would tend to argue that many frequentist properties are not important to applied data analysis or optimal decision making.
- taliesinb 14y agoA possible example given in the lecture is figuring out if two sets of numeric data were sampled i.i.d from the same distribution (being hand-wavy about what this precisely means). I don't see how you can sensibly approach this kind of problem in a Bayesian way. From a frequentist perspective you're sort of spoiled for choice about how to approach this problem.