Upvoted this mostly because I love seeing these debates here on HN about statistical methods. I don't do this kind of work in my day job, unfortunately, but studied it in college and always look back to a certain "fork in the road" moment that, had I adjusted my priors differently (heh), would've very likely led me to an academic life instead of in business.
That definitely could have been made clearer. I looked all over the page for a link to an actual paper and missed it. I never would have guessed it was the link on what appeared to me to be an author's name.
The linked paper is a very nice overview. Of course there problems are known and there are people trying to fix all of the issues (mostly in the relative obscurity of non-overhyped corners of academia), but the concise example-guided description of these problems is great.
Somehow I think the most fundamentally damning critique, and causality shares this problem, is also the most vague. That applied scientists/experimentalists look at the "automation" that these approaches are supposed to enable and say "that's either doing the trivial part of the job or giving you BS answers".
I've always felt bayesian statistics got more attention from researchers than was warranted (including today despite deterministic methods taking over the world) because it has a nice principled "theory of everything" starting point. But then of course you have to approximate the heck out of it to be able to solve it. Often far more than with other methods.
You have to approximate the heck out of the Schrödinger equation as well, otherwise we would be stuck describing the Hydrogen and maybe the Helium atoms and nothing more.
Yes, however as I said in the subsequent sentence, the approximation is "often far more than other methods". For the most obvious example, a point estimate like MAP doesn't need to compute the denominator in Bayes law. That's two (generally easier) terms to approximate rather than three. Those using Bayesian methods point out the value in providing a full distribution, but the necessary additional approximations to get it means the location of its maximum can now actually be less accurate than a simple MAP estimate. But what always bugged me is multivariate problems where the Bayesian paper presumes everything is independent and Gaussian. Great after getting all psyched by that intro talk about the value of getting a distribution, we get the simplest imaginable one, just a mean and variance for each variable.
...the inferential procedure of Bayesian statistics is to assume a prior distribution and a probability model for data and then use probability theory to determine the posterior. But if these steps, or something approximating them, are necessary, if you can’t just look at your data and come up with a subjective posterior distribution, then how is it reasonable to suppose that you could able to come up with an unassailable subjective distribution before seeing the data?
Is the point to end up with an "unassailable subjective distribution"? I believe the power of Bayesian thinking is that you can take a subjective prior, which is necessarily assailable in its subjectivity, and then combine it with data. The result is something that is better than either the subjective prior alone or the likelihood estimator gleaned from data alone.
I haven't read the linked paper yet, but the blog post points a little to what's on my mind.
To borrow a bit from a different paper and anonymous reviewer on one of my papers, inference serves different aims or philosophies. Sometimes it serves more of an estimation function, to increase information about some quantity, or to improve the estimate of that quantity. But sometimes it serves an evaluative, competitive function, in the Popperian sense of affording risky tests of one or more theories or models.
In this latter Popperian aim of inference, priors are to be minimized, which is in many ways the opposite scenario that is assumed with standard subjective Bayesian methods. And even with the former "estimation" inferential aims, there may be situations where you truly have no information or don't feel comfortable assuming it.
What's nice about Bayesian statistics is it still provides a framework for this scenario, in the form of reference priors, in that if nothing else your design and model supporting the parameter(s) to be inferred about implies some kind of assumptions about what you're making inferences about. That in turn can be transformed into a "least informative" prior. So it allows an objective Bayesian framework.
However, in that framework, in many cases you're still often left with uniform priors, which then reduce to frequentist statistics. And in a broader sense, frequentist methods are even further removed from making assumptions in that they completely eliminate the prior from inferential consideration.
There's a tension then, in that in small samples your priors will bias your estimates. If you use least informative priors, you're often doing something akin to frequentist methods anyway. And in large samples the likelihood dominates the posterior so it matters still less.
From a certain perspective, ultimately with Bayesian methods you're making a bet that your priors are accurate enough that the increased bias in estimates will be small enough to be offset by decreased variance due to use of a prior. It's a gamble though, the risks of which will probably vary depending on the costs and benefits of different types of error.
It's nice to see a paper trying to be honest about the problems with Bayesian inference, as it's a bit overhyped at the moment imho.
> However, in that framework, in many cases you're still often left with uniform priors, which then reduce to frequentist statistics.
Several commenters have made this claim, which seems to be true only for maximum a posteriori (MAP) estimates. Frequentist methods do not construct a distribution over parameters.
That's a good point regarding MAP estimates versus, e.g., EAP estimates, although it does lead to questions about whether one or the other is preferable, and I think good arguments can be made either way.
Yes, it's a bit ironic that the likelihood principle ("all the evidence drawn from data is summarized by the likelihood function") is something that both frequentist and Bayesian statisticians like to claim as their own. Frequentist statistics is basically Bayesian statistics done with a "flat" prior distribution (where "flat" can depend on how the model is parameterized, however). Seen from this POV, it may be somewhat weird to claim that Bayesian stats has "holes" in it.
Yes, the only difference is that in Bayesian stats the biggest hole (priors) is explicit and in your face. In frequentist stat that "flat" prior you mention is implicit and flies under the radar.
> Frequentist statistics is basically Bayesian statistics done with a "flat" prior distribution
I disagree strongly - there’s a huge difference in the kind of inference which is provided by each paradigm. To oversimplify, frequentist statistics constructs decision rules with good properties concerning unknown (but fixed) parameters, while Bayesian statistics uses probability to directly reason about those parameters. Confidence and credible intervals are not the same thing - you’re actually flipping what’s considered fixed and random.
Maximum likelihood is equivalent to Bayesian maximum a-posteriori (MAP) given a uniform prior and the right choice of parameters. However, decision theory tells us, that MAP estimation is rarely, if ever, a good point estimator. The mean or the median of the posterior distribution is usually better, but the optimal choice may be something entirely different.
(Why even consider MAP when it's dumb? Only because it's computationally cheaper than anything else.)
Junk data can’t improve accuracy, that’s the flaw with Bayesian thinking. If I hand you a normal looking coin and ask what’s the odds for heads, tails, or edge to +\- 0.0001%, the absolute minimal trials is going to be based on zero priors.
It’s valuable when you have a combination of useful prior knowledge and insufficient data to review. But, having useful prior knowledge can be difficult to distinguish from useless prior knowledge. Incorrect belief that something is safe means more people are harmed before you update that belief.
This means you can’t say if Bayesian think is a net positive or a net negative abstractly.
I'm confused, in cases where you have truly no prior knowledge, such as your thought experiment, a bayesian can reasonably select a uniform prior. But often in real world statistical experiments, we have priors we can believe in.
You likely have fairly accurate priors about that coin flip I just mentioned. They are just not accurate enough to be useful. The math for designing sample sizes may be unfamiliar, so you can just try running the same simulation a few times. Then try a few priors on each data set and compare.
This is effectively pure math so there is nothing to disagree or get upset about. What’s up for debate is how accurate your priors are in the real world.
Your comments apply to statistics and machine learning as a whole, not just Bayes statistics. At the end of the day all you have is your prior knowledge and what you learn from the data. If both of those are suspect, it does not matter what method you use. I just don't see how this can be considered a problem specific to Bayesian statistics.
That’s a qualitative assessment of a quantitative process. Measuring how each system works in specific situations allows you to compare them, which considering the subject matter seems appropriate.
Are you measuring the process or the inputs? Because in your post the inputs are what you criticized (the prior and the data), not the process itself. Every machine learning algorithm takes these inputs. So I was just saying that it seems unfair to penalize Bayesian statistics for this and not any other method.
Every time I see an article like this, I eagerly read it hoping to find a cogent and coherent criticism of Bayesian stats, and it always ends up being a straw man or a very fair critique of somebody not correctly applying or interpreting an application of Bayes’ theorem.
Bayesian stats is nothing more than a rigorous way to transform beliefs + data into a posterior. Yes, flat priors are not always the best choice. That’s not a criticism of Bayesian stats. It’s a statement about how actually formulating a prior is often times the hardest part of a problem. Is Bayesian stats useful for describing or understanding QM? Idk, again, not really it’s job...
Use Bayesian stats, not with an air of suspicion, but a respect for the fact that it will give you the results implied by your data and prior, under your model assumptions. Nothing more, nothing less. By the way, what is the alternative if you find yourself in a situation where your result depends strongly on your prior and you aren’t really sure how to choose your prior? Wave your hands and find an ad hoc frequentist approach? How about just admitting to yourself that your data isn’t enough to make up for the fact that you can’t really quantify your true prior belief?
If you read this and disagree, I sincerely implore you to comment — I just don’t understand the “debate” aspect of Bayesian stats. Some people misunderstand what Bayesian stats is sometimes (very understandable) but I have yet to see a legitimate philosophical or mathematical critique of the Bayesian approach that really made any sense to me. I would like to know if I am wrong though...
That’s awesome, and maybe I misunderstand his argument (very possible, in fact my main reason for posting this comment). Am I not understanding the arguments being made here? I have not been following these developments regarding using Bayesian stats in QM, it’s just that if Bayesian stats doesn’t help us understand QM it doesn’t mean there’s a “hole” because Bayesian stats makes no claim whatsoever that it has any applicability to QM. Am I just not understanding?
I very much agree. It’s well known that probability theory is insufficient to explain the evolution of a QM system (i.e. the Schrödinger equation cannot be expressed in terms of probability). This is an interesting fact about QM, but in no way constitutes a hole in probability theory or Bayesian statistics.
> That’s not a criticism of Bayesian stats. It’s a statement about how actually formulating a prior is often times the hardest part of a problem.
Bayesian statatistics doesn't give a satisfying answer/method for choosing priors. Choosing priors is part of statistics. Therefore Bayesian statistics has a "hole" in it.
> Bayesian statistics doesn’t give a satisfying answer/method for choosing priors
I absolutely agree with you. Priors are the (very difficult) job of the practitioner.
> choosing priors is a part of statistics
Ah so I think I agree, maybe would nitpick and say priors are part of statistics and CHOOSING priors is part of modeling maybe? Not totally comfortable with that characterization either it’s just that “priors” and “choosing priors” are two different things.
> therefore Bayesian statistics has a “hole” in it.
Completely disagree; this is my main understanding of the argument against Bayesian statistics, I just don’t understand how this follows from the last statement. Priors are hard. But they are explicit! And a necessary ingredient of the posterior. It’s not like we can say “oh priors are hard, so let’s not do Bayesian stats but instead let’s do X” what is X?
E. T. Jaynes gave some very good answers (symmetry groups and maximum entropy principle). You should read "Probability Theory: The Logic Of Science". In it, you will learn that the supposed "hole" has been closed pretty nicely.
Now that I've taken some time to actually READ the article referenced in this post, I have a couple more things to say; basically: I agree with virtually everything that they're saying in the paper (except that I still don't understand why Bayesian stats failing in QM says absolutely anything about the validity of Bayesian stats), and I think I set up my own straw man here a bit. I thought this was another "Bayesian stats is bad because priors" post, which in a way it SORT of is, but their arguments are a bit more on the side of "you should take 'Bayesian' analyses with a large grain of salt sometimes because...well...priors" <- and THAT I completely agree with.
The only beef I have with these types of articles are when they seem to imply that because priors sometimes have an outsized effect on the results, we should not use Bayesian statistics/analyses. But really, there is no other option. If you have a statistical question, you're stuck with having to answer for your priors. That being said, what IS good advice is to not simply believe every 'Bayesian' conclusion for exactly that reason -- that result is only as good as the (1) prior, (2) model assumptions, and (3) the data. (2) and (3) are easy to fight about, but (1) can be hard to evaluate, and easily can lead all of us astray even when the prior sounds reasonable (flat priors).
I'm turned off by the invocation of quantum mechanics here. Most applications of Bayesian statistics have nothing to do with QM, so "it's not compatible with QM" doesn't seem like a strong argument.
QM is the number-one favorite topic of crank scientists and pseudoscience. When using it in the discussion of a seemingly-unrelated topic, authors should take extra care to motivate why QM is relevant.
(Of course probability theory and QM are not unrelated topics, but probability exists independently of QM.)
While reading the blog post/abstract I had the same thought, but the full paper makes it clear that the author intends something different:
> The second challenge that the uncertainty principle poses for Bayesian statistics is that [...] we routinely treat the act of measurement as a direct application of conditional probability.
Furthermore it states that this problem might also arise for other applications of Bayesian statistics:
> If classical probability theory needs to be generalizedto apply to quantum mechanics, then it makes us wonder if it should be generalized for applicationsin political science, economics, psychometrics, astronomy, and so forth. It’s not clear if there are any practical uses to this idea in statistics, outside of quantum physics. For example, would it make sense to use “two-slit-type” models in psychometrics, to capture the idea that asking one question affects the response to others?
One of the comments at the site finally cracked a little bit of light for me on Bell's theorem. To quote from the comment: "What his theorem says is that the world can not simultaneously be local and have hidden variables. His own position was that it seemed exceedingly likely that the world was non-local and that there were hidden variables (such as where is the photon at any given time)."
I don't know why, but I kept imagining non-local scenarios and thinking, "This seems like it should be OK, so I don't understand what's going on". Having it spelled out is tremendously helpful. I still don't understand what's going on, but at least I don't feel completely crazy ;-)
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[ 2.8 ms ] story [ 92.6 ms ] threadSomehow I think the most fundamentally damning critique, and causality shares this problem, is also the most vague. That applied scientists/experimentalists look at the "automation" that these approaches are supposed to enable and say "that's either doing the trivial part of the job or giving you BS answers".
Is the point to end up with an "unassailable subjective distribution"? I believe the power of Bayesian thinking is that you can take a subjective prior, which is necessarily assailable in its subjectivity, and then combine it with data. The result is something that is better than either the subjective prior alone or the likelihood estimator gleaned from data alone.
To borrow a bit from a different paper and anonymous reviewer on one of my papers, inference serves different aims or philosophies. Sometimes it serves more of an estimation function, to increase information about some quantity, or to improve the estimate of that quantity. But sometimes it serves an evaluative, competitive function, in the Popperian sense of affording risky tests of one or more theories or models.
In this latter Popperian aim of inference, priors are to be minimized, which is in many ways the opposite scenario that is assumed with standard subjective Bayesian methods. And even with the former "estimation" inferential aims, there may be situations where you truly have no information or don't feel comfortable assuming it.
What's nice about Bayesian statistics is it still provides a framework for this scenario, in the form of reference priors, in that if nothing else your design and model supporting the parameter(s) to be inferred about implies some kind of assumptions about what you're making inferences about. That in turn can be transformed into a "least informative" prior. So it allows an objective Bayesian framework.
However, in that framework, in many cases you're still often left with uniform priors, which then reduce to frequentist statistics. And in a broader sense, frequentist methods are even further removed from making assumptions in that they completely eliminate the prior from inferential consideration.
There's a tension then, in that in small samples your priors will bias your estimates. If you use least informative priors, you're often doing something akin to frequentist methods anyway. And in large samples the likelihood dominates the posterior so it matters still less.
From a certain perspective, ultimately with Bayesian methods you're making a bet that your priors are accurate enough that the increased bias in estimates will be small enough to be offset by decreased variance due to use of a prior. It's a gamble though, the risks of which will probably vary depending on the costs and benefits of different types of error.
It's nice to see a paper trying to be honest about the problems with Bayesian inference, as it's a bit overhyped at the moment imho.
Several commenters have made this claim, which seems to be true only for maximum a posteriori (MAP) estimates. Frequentist methods do not construct a distribution over parameters.
I disagree strongly - there’s a huge difference in the kind of inference which is provided by each paradigm. To oversimplify, frequentist statistics constructs decision rules with good properties concerning unknown (but fixed) parameters, while Bayesian statistics uses probability to directly reason about those parameters. Confidence and credible intervals are not the same thing - you’re actually flipping what’s considered fixed and random.
(Why even consider MAP when it's dumb? Only because it's computationally cheaper than anything else.)
It’s valuable when you have a combination of useful prior knowledge and insufficient data to review. But, having useful prior knowledge can be difficult to distinguish from useless prior knowledge. Incorrect belief that something is safe means more people are harmed before you update that belief.
This means you can’t say if Bayesian think is a net positive or a net negative abstractly.
This is effectively pure math so there is nothing to disagree or get upset about. What’s up for debate is how accurate your priors are in the real world.
Bayesian stats is nothing more than a rigorous way to transform beliefs + data into a posterior. Yes, flat priors are not always the best choice. That’s not a criticism of Bayesian stats. It’s a statement about how actually formulating a prior is often times the hardest part of a problem. Is Bayesian stats useful for describing or understanding QM? Idk, again, not really it’s job...
Use Bayesian stats, not with an air of suspicion, but a respect for the fact that it will give you the results implied by your data and prior, under your model assumptions. Nothing more, nothing less. By the way, what is the alternative if you find yourself in a situation where your result depends strongly on your prior and you aren’t really sure how to choose your prior? Wave your hands and find an ad hoc frequentist approach? How about just admitting to yourself that your data isn’t enough to make up for the fact that you can’t really quantify your true prior belief?
If you read this and disagree, I sincerely implore you to comment — I just don’t understand the “debate” aspect of Bayesian stats. Some people misunderstand what Bayesian stats is sometimes (very understandable) but I have yet to see a legitimate philosophical or mathematical critique of the Bayesian approach that really made any sense to me. I would like to know if I am wrong though...
I can't tell from your comment if you're aware that Andrew is the first author of the leading textbook on Bayesian methods. http://www.stat.columbia.edu/~gelman/book/
Bayesian statatistics doesn't give a satisfying answer/method for choosing priors. Choosing priors is part of statistics. Therefore Bayesian statistics has a "hole" in it.
I absolutely agree with you. Priors are the (very difficult) job of the practitioner.
> choosing priors is a part of statistics
Ah so I think I agree, maybe would nitpick and say priors are part of statistics and CHOOSING priors is part of modeling maybe? Not totally comfortable with that characterization either it’s just that “priors” and “choosing priors” are two different things.
> therefore Bayesian statistics has a “hole” in it.
Completely disagree; this is my main understanding of the argument against Bayesian statistics, I just don’t understand how this follows from the last statement. Priors are hard. But they are explicit! And a necessary ingredient of the posterior. It’s not like we can say “oh priors are hard, so let’s not do Bayesian stats but instead let’s do X” what is X?
Now that I've taken some time to actually READ the article referenced in this post, I have a couple more things to say; basically: I agree with virtually everything that they're saying in the paper (except that I still don't understand why Bayesian stats failing in QM says absolutely anything about the validity of Bayesian stats), and I think I set up my own straw man here a bit. I thought this was another "Bayesian stats is bad because priors" post, which in a way it SORT of is, but their arguments are a bit more on the side of "you should take 'Bayesian' analyses with a large grain of salt sometimes because...well...priors" <- and THAT I completely agree with.
The only beef I have with these types of articles are when they seem to imply that because priors sometimes have an outsized effect on the results, we should not use Bayesian statistics/analyses. But really, there is no other option. If you have a statistical question, you're stuck with having to answer for your priors. That being said, what IS good advice is to not simply believe every 'Bayesian' conclusion for exactly that reason -- that result is only as good as the (1) prior, (2) model assumptions, and (3) the data. (2) and (3) are easy to fight about, but (1) can be hard to evaluate, and easily can lead all of us astray even when the prior sounds reasonable (flat priors).
QM is the number-one favorite topic of crank scientists and pseudoscience. When using it in the discussion of a seemingly-unrelated topic, authors should take extra care to motivate why QM is relevant.
(Of course probability theory and QM are not unrelated topics, but probability exists independently of QM.)
> The second challenge that the uncertainty principle poses for Bayesian statistics is that [...] we routinely treat the act of measurement as a direct application of conditional probability.
Furthermore it states that this problem might also arise for other applications of Bayesian statistics:
> If classical probability theory needs to be generalizedto apply to quantum mechanics, then it makes us wonder if it should be generalized for applicationsin political science, economics, psychometrics, astronomy, and so forth. It’s not clear if there are any practical uses to this idea in statistics, outside of quantum physics. For example, would it make sense to use “two-slit-type” models in psychometrics, to capture the idea that asking one question affects the response to others?
I don't know why, but I kept imagining non-local scenarios and thinking, "This seems like it should be OK, so I don't understand what's going on". Having it spelled out is tremendously helpful. I still don't understand what's going on, but at least I don't feel completely crazy ;-)