Philosophy and the practice of Bayesian statistics [pdf]

I never thought I'd see a 31-page paper by Andrew Gelman on the front page of Hacker News. And certainly not a paper coauthored with a well-known frequentist!<p>I was lucky enough to work with Prof. Gelman as his research assistant while I was in school - I can't even being to tell you how prolific and brilliant that man is. His name may not be known very much outside academic circles, but I'd go as far as to say that he's the most important Bayesian statistician since Thomas Bayes.<p>He used to be a contributor to FiveThirtyEight, back before the Times picked it up. I used to explain FiveThirtyEight as 'one of the six blogs Andrew Gelman writes for'. Now, I explain Andrew Gelman as 'a former contributor to Nate Silver's blog'. How times have changed!<p>Gelman's approach to statistics is more wholly Bayesian than most people with a moderate level of statistical training are likely familiar with. It was from Gelman that I learned why I never need to perform an F-test[0]; at the same time, it was from Gelman that I learned some of the potential pitfalls of pure Bayesian reasoning[1] (and how to address them).<p>When people ask me where to get started with statistics, both of the books I recommend are Gelman's: <i>Teaching Statistics: A Bag of Tricks</i> and <i>Data Analysis Using Regression and Multilevel/Hierarchical Models</i>.<p>Both have tremendously off-putting titles, but they're actually incredibly accessible. Gelman is great at many things, but picking sexy titles is not one.<p>If you're interested in understanding the concepts behind this paper, I'd start there.<p>[0] <a href="http://andrewgelman.com/2009/05/18/noooooooooooooo/" rel="nofollow">http://andrewgelman.com/2009/05/18/noooooooooooooo/</a><p>[1] The linked paper provides a good analysis

If you are interested in the <i>practical</i> practice of Bayesian methods (and you love Python), check out our open-source project/book <i>Bayesian Methods for Hackers</i>:<p><a href="https://github.com/CamDavidsonPilon/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers" rel="nofollow">https://github.com/CamDavidsonPilon/Probabilistic-Programmin...</a><p>We aim to empower the non-mathematician with really cool tools and methods to solve otherwise very difficult problems. Plus it's all opensource, and every plot/diagram is reproducible and extendable.

Haven't had time to read the whole article yet, but these two paragraphs from the conclusion (p. 24-25) are excellent:<p><i>"In our hypothetico-deductive view of data analysis, we build a statistical model out of available parts and drive it as far as it can take us, and then a little farther. When the model breaks down, we dissect it and figure out what went wrong. For Bayesian models, the most useful way of figuring out how the model breaks down is through posterior predictive checks, creating simulations of the data and comparing them to the actual data. The comparison can often be done visually; see Gelman et al. (2004, Chapter 6) for a range of examples. Once we have an idea about where the problem lies, we can tinker with the model, or perhaps try a radically new design. Either way, we are using deductive reasoning as a tool to get the most out of a model, and we test the model – it is falsifiable, and when it is consequentially falsified, we alter or abandon it. None of this is especially subjective, or at least no more so than any other kind of scientific inquiry, which likewise requires choices as to the problem to study, the data to use, the models to employ, etc. – but these choices are by no means arbitrary whims, uncontrolled by objective conditions.</i><p><i>"Conversely, a problem with the inductive philosophy of Bayesian statistics – in which science ‘learns’ by updating the probabilities that various competing models are true – is that it assumes that the true model (or, at least, the models among which we will choose or over which we will average) is one of the possibilities being considered. This does not fit our own experiences of learning by finding that a model does not fit and needing to expand beyond the existing class of models to fix the problem."</i><p>And section 4, which discusses issues that arise in Bayesian statistics when working with multiple candidate models, is interesting and agrees with my limited experience, especially 4.3: "Why not just compare the posterior probabilities of different models?"<p>ps (to the submitter), it might be helpful when submitting a 30 page paper to mention what part of the paper you'd like to discuss. It makes it easier to get started.

As someone who didn't study statistics in college, a paper like this is right in that uncomfortable no-man's land between what I understand and what I'm interested in - it seems to tie into several subjects I have layman's interest in.<p>For instance, there is the controversy over how useful models are - are they worthwhile goals we can actually draw conclusions from, or are they simply shortcuts on our way to a more reductionist understanding of a phenomena? Is "emergence" a meaningful concept or an empty concept? Is "systems thinking" a valid concept or just a lack of discipline in the effort to understanding things in a reductionist manner?<p>People here seem to hate Stephen Wolfram but his writing has made some concepts approachable to me that I might not have grasped otherwise - for instance, that computational irreducibility means that even if the world is entirely reductionist, it still doesn't mean that we can deduce the reductionist reality/inputs from an output. And so therefore, models are useful even though they are wrong. This is also a point that Paul Krugman often makes about economic models - people who disregard models on the grounds that they are wrong just don't grasp the value of them, he argues.<p>Most of what I've learned about Bayesianism is what I've read from the first few articles over at lesswrong.com - but I noticed pretty early on that I had a discomfort in using probability as a description of what I <i>believed</i> to be true. It seems the general point of this paper is that Bayesianism is useful for deductive techniques - as a tool in a toolset to support a frequentist view? - but not so much as an expression of a subjectivist philosophy. I appreciated this point:<p>"Beyond the philosophical difficulties, there are technical problems with methods that purport to determine the posterior probability of models, most notably that in models with continuous parameters, aspects of the model that have essentially no effect on posterior inferences within a model can have huge effects on the comparison of posterior probability among models."<p>More generally the paper seems to be making the point that using the Bayesian philosophy to address models is improper in general since the premise of Bayesianism is to update <i>beliefs</i> based off of evidence/data, while we know that belief-in-a-model is pointless since models are wrong. But past that point I got pretty lost.

Frequentist here. This paper makes me hate Bayesians a little less. The reason is because a general thrust of the paper (since I have only had time to give it a once-over) seems to be that just because you are a Bayesian it doesn't mean that you have to get rid of model adequacy checks. Not having model adequacy checks is why I think Bayesians run around with a magic wand saying, "poof! there's an optimal model." After proving a theoretical optimality they never check to see if the real world data supports their arguments. So I'm glad to see a prominent Bayesian saying that you don't have to throw model checking out the window.<p>On a secondary note, I have to lament the use of philosopy in a math paper. I realize that many prominent mathematicians are/were also philosophers and that the two subjects are somehow linked at some level. But really I think that putting philosophy in a math paper is an excuse to use more big words and sound smart. Most of us would like to have a set of formulas to apply and not worry about-- forgive me if I for what I'm about to say-- fuzzy non-science like philosophy and the implications that it might have on our cold hard numbers.<p>Can each 31 page paper that combines math with philosophy come with a 5 page companion paper that leaves out the philosopy and just has the applicable math stuff?

The author's point that the most successful forms of Bayesian statistics accord much better with sophisticated forms of hypothetico-deductivism is reminiscent of the epistemology of normative value(s) which furnish a provisional lens for the analysis of the systemization of statistical transparency.<p>OK, half of that sentence is from an academic bullshit generator. I won't tell you which half.<p>Unfair, I know. That paper is clearly not meant for the general public, but still, learn how to communicate.

I... uh.... How did this get here?