Comment by antics | Hacker News Reader

Comment by antics | Hacker News Reader

There has been a lot of rich discussion about the relative merits of the Bayesian and frequentist perspectives on statistical inference. If you are thinking about applying inference techniques to some problem, then it is well worth your time to make sure that you really, really understand this debate, because picking the correct tool for your job is likely to make your life a lot easier (and, yes, there are both situations where the Bayesian perspective is clearly better and where it is clearly worse).

Unfortunately this post completely and totally ignores the things that will enable you to make this decision. This post is called "Understand the Math Behind it All", but you will learn nothing about math, or really, Bayesian statistics, at all. You will not learn, for example, how to apply Bayesian inference to problems, what it means to do basic Bayesian tasks like "use an expert" or "condition on evidence", or even what a Bayesian statistic is. In fact, Bayes' theorem is never even mentioned. There's just a hand-wavy collection of statements like "The Bayesian approach is to rely on past knowledge and then adjust accordingly". That is so vague that it is not even clear that they are talking about Bayesian analysis. This is the sort of statement that fools people into believing they understand something that they really don't.

If you really want to understand this material, you should watch[1], a talk by Mike Jordan called "Are you a Bayesian or a Frequentist?". It's a bit much for beginners, but if you are willing to look up some of the math, it is entirely digestible, and it is by far the best comparison of the two communities I have found. I say "by far" because it is (1) a more or less complete representation of both communities, (2) it is pretty much an unbiased representation of both communities' strengths and weaknesses, and (3) it is as direct as it can get, meaning that it is not tied up in a lot of external knowledge, and is intent on delivering this message, rather than delivering it in an off-hand way as a method of getting to something else.

[1] http://videolectures.net/mlss09uk_jordan_bfway/

Very much agree.

I'd also prefer not to encourage referring to people as "a Bayesian" or "a frequentist", as though it's a hard philosophical preference, a binary choice one has to make.

I realise there are historical reasons for this, but really guys, they're both tools. Know the pros and cons and pick the right one for the job; neither is uniformly better.

Which framework you use can be a fairly subtle decision sometimes, requiring you to think fairly deeply about what you really want to get out of the analysis, what assumptions you're comfortable making, what kind of interpretation it would be most useful to be able to place on the results. But it's not just an arbitrary choice that you can leave to aesthetic or philosophical preference.

Thanks. Just yesterday I came across this first paragraph in a paper I now see is by Jordan:

"Statistics has both optimistic and pessimistic faces, with the Bayesian perspective often associated with the former and the frequentist perspective with the latter, but with foundational thinkers such as Jim Berger reminding us that statistics is fundamentally a Janus-like creature with two faces." [Janus is a Roman god with two faces]

In my current project (stroke and [since the leap] gesture recognition), I'm using the covariance matrix of a training set and the difference a feature vector from the mean to calculate the mahalanobis distance of the input vector from the training set. I plan to use this same covariance matrix in the Gaussian density estimation formula to generate a probability distribution function (and then use the likelihood function instead of mahalanobis distance). Still trying to mentally connect this to the bigger-picture stuff though.

http://www.cs.berkeley.edu/~jordan/papers/berger-festschrift...

Berger's actually written a seminal book on the topic called "Statistical Decision Theory and Bayesian Analysis"[1]. If you're interested in that area, consider checking it out of your library.

I'm not quite as sharp in the area as I used to be, but feel free to hit me up over email if you have more questions, I can't guarantee that I'll know the answers, but I'm happy to give it a shot.

[1] http://www.amazon.com/Statistical-Decision-Bayesian-Analysis...

Thanks again! I'll do that if I form a coherent question.

Statsexchange is actually a great resource as well if you form that question: http://stats.stackexchange.com/

When are Bayesian methods "clearly worse" than frequentist methods, apart from computationally?

There are times when, even as a Bayesian, one is interested in calibration. Model checking without a specified alternative is an example. Frequentist ideas -- sampling from the model and comparing it to the observed distribution -- can be helpful here. I'm thinking of Rubin (1984): http://www.cs.princeton.edu/courses/archive/fall11/cos597C/r...

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.

It 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.

As 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.

Frequentist 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.

A 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.

The useless answer is that they both do different things, so it depends what of those things you want :)

One aspect of frequentist techniques that perhaps others haven't emphasised so much, is that they tend to give guarantees about expected behaviour which hold uniformly over all possible values of the unknown parameters.

Whereas the Bayesian approach, the guarantees you obtain will only hold in an 'averaged-out' sense over the prior distribution you specify.

If you're a bit paranoid and you want a probabilistic bound on what might happen in the worst case, you might sometimes find the former a little more comforting than the latter.

In particular if you don't have much data, the influence of the choice of prior will be bigger and so the distinction will matter more.

Hope that helps, and that any stats PhDs will correct me if I've over-simplified things here.

This is an attempt to give a pragmatic overview, without the math.

Bayesian methods rely on the concept of "priors". A prior is a known probability about a fact, or event, or whatever it is you are modeling. Priors "seed" the network.

Bayesian networks generally need fewer samples of data to make predictions, and as a result the downside is the sensitivity of the data (aka priors) increases. Whereas frequentist approaches rely on much larger data sets that can handle noise, more effectively.

Think about GMail's spam filter for example (A Bayesian approach), if you train HAM as SPAM, that is going to have devastating effect on the efficacy of your filtering.

Thus practically speaking, if you have tons of data and looking for a signal, use frequentist. If you're building, for example, an expert diagnostic engine (think Sherlock Holmes), requiring few pieces of information to make a prediction, consider using Bayesian.

I'm over-simplifying of course, but that seems to be the gist of it.

I don't think that's right. If you have enough data points, your prior gradually gets more relevant. And Bayesian statistics has the concept of an "ignorance prior", which mathematically represents the position where all possibilities are equally likely. Bayesian statistics also offers the possibility of adding more data after running your experiment, and computing a new answer in a consistent way. Whereas doing this with frequentist statistics completely is completely invalid.

"Ignorance priors" are not the common use case for Bayesian in general, at least in my experience.

Furthermore, who said adding more data is not allowed? Of course it is, and I agree fully, but do struggle to see how that contradicts what was said.

Think about a Bayesian spam filter, you keep training it with more "priors", over time and it gets better.

That's a cool lecture. Looks like it was part of the Cambridge Machine Learning Summer School 2009 (http://videolectures.net/mlss09uk_cambridge/), which has a bunch of cool talks, for example Josh Tenenbaum's "Machine Learning and Cognitive Science" (http://videolectures.net/mlss09uk_tenenbaum_mlcs/)

One of the cooler things about the field of machine learning is that conferences like ICML, and little mini-summer schools like this are very "rich", so talks like this are all over the Internet.

For those who are curious Tenenbaum has recently gotten very famous in the community for Bayesian style work. His students are getting very very prestigious jobs, so a talk like that is worth a look for sure.

That's a truly excellent talk and really even the first slide, titled "Statistical Inference," should be enough to gain a ton of information, so if you're intimidated by the length just give the first slide a try. Michael Jordan is one of my favorite statisticians/machine learning researchers, and if I see he's speaking somewhere I always try to go. I don't go to many stats talks, but his talks are always some of the most mathy I do see, and he's not afraid to dive in to the mathematical mechanics of methods.

Reply on news.ycombinator.com