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This article stands in contrast to the thoughts expressed in a recent interview with George Ellis[1]. The question of Bayesian vs. frequentist is fundamentally

by bjterry 12y ago

This article stands in contrast to the thoughts expressed in a recent interview with George Ellis[1]. The question of Bayesian vs. frequentist is fundamentally a philosophical question, and it's often discussed in these terms in the articles about the debate, including the recent one which caused this post. The philosophy that we accept affects the tools that we use to solve problems, and the way we think about problems, so you can't simply ignore the debate. Ellis' argument even extends to his example, about ψ (the quantum wave function), about which there is extensive philosophical debate. When people identify as supporting Bayesian methods over frequentist methods, that actually changes the way they perform science. It's an argument against bad statistical interpretations that frequently arise in the social sciences and medicine[2] and elsewhere because p-values have become the altar at which all publishing researchers must prostrate themselves. It's not a case of a debate without substance, which would have no observable effects. I can understand the frustrations of the author. Every tool has its use, and frequentist statistical methods are frequently useful. They help you get published, they are broadly supported by software packages, they are often faster and easier to perform, etc., so there are many situations in which they are the right choice. But I feel like in attempting to be agnostic about methods, the author is losing nuance in their argument. Bayesian reasoning really does represent the ground truth (assuming imperfect knowledge), and even when we are using frequentist methods, we are only doing so because the benefits outweigh the costs. 1: http://blogs.scientificamerican.com/cross-check/2014/08/21/quantum-gravity-expert-says-philosophical-superficiality-has-harmed-physics/?-k http://blogs.scientificamerican.com/cross-check/2014/08/21/q... 2: http://www.sciencebasedmedicine.org/prior-probability-the-dirty-little-secret-of-evidence-based-alternative-medicine-2/ http://www.sciencebasedmedicine.org/prior-probability-the-di...

keithwinstein 12y ago
Don't think that's quite what the author was getting at. A "frequentist" who makes a decision based on a p-value or confidence interval, and a "Bayesian" who makes a decision based on a posterior or predictive probability distribution, can both be viewed as making a decision according to a procedure that minimizes some statistic of a loss function. Both of these broad families of methods are diverse, but to generalize broadly, decisions informed by "frequentist" tools are often concerned with the worst-case value of the loss function within some universe, and decisions based on "Bayesian" methods can be viewed as caring about the expected value of the loss function given some prior. Both of those families (and many others) are "the ground truth," in that they both make statements that can be proved as theorems of mathematics. A "frequentist" confidence interval will always achieve its guaranteed coverage no matter what, if the likelihood function is true. A "Bayesian" credibility interval will include the true value of the parameters at exactly the advertised rate, when averaged over each possible value of the parameters weighted according to the prior, and assuming the likelihood function is true. The author says, and I agree, that the thing worth arguing over is the "norm we use to choose our optimal procedure" and the nature of the loss function. Whether you care about controlling the worst case or the expected value or something else depends on what matters. (The author points out that in an adversarial situation where your strategy is known to your opponent, minimizing the worst case may be advisable...) Some days, we care about the worst-case performance of QuickSort, and some days we care about its average-case performance (given an assumption that all input orderings are uniformly likely). It's ok to care about different things depending on the application; we don't have to split into warring tribes over it. More here: http://blog.keithw.org/2013/02/q-what-is-difference-between-bayesian.html http://blog.keithw.org/2013/02/q-what-is-difference-between-... http://jsteinhardt.wordpress.com/2014/02/10/a-fervent-defense-of-frequentist-statistics/ http://jsteinhardt.wordpress.com/2014/02/10/a-fervent-defens... http://cs.stanford.edu/~jsteinhardt/stats-essay.pdf http://cs.stanford.edu/~jsteinhardt/stats-essay.pdf
- Matumio 12y ago
  Do Bayesian methods imply a choice of loss function? I think it's a framework for observational statistics, not for policy making or decision theory. You can use it as a tool during policy or decision making, where you should indeed have a debate about picking a loss function that corresponds to your goals.
  - mjw 12y ago
    If you're reporting point or interval estimates (rather than the entire posterior) then you are implicitly or explicitly optimising some kind of loss function. Also, worth a reminder that continuous estimation problems can be viewed as decision problems too, it's not just about discrete decision-making. I don't think it's great to take the view that, because I'm not making a decision based on this estimate myself, I don't need to worry about which loss function I'm implicitly optimising for when choosing an estimator. Someone else may need to.
- jules 12y ago
  Bayesian statistics gives you a posterior distribution. What you do with that distribution is up to you. If you want to find the decision that minimizes the maximum loss instead of the expected loss, that fits perfectly well within the Bayesian framework. The posterior gives you all the information that you need to make a decision, whatever your loss function. Frequentist statistics on the other hand gives you no such thing. You can't make decisions based on frequentist statistics. Frequentist statistics is all about reasoning about things that did not happen. Things that did not happen are irrelevant for making decisions in a situation where that thing did happen. So while frequentist statistics is mathematically correct, it's also strictly speaking useless in practice. It's only useful insofar as it gives us heuristics for decision making when the Bayesian approach is impractical. Here's an example. Suppose there are two types of berries: edible and poisonous. We devise a statistical procedure where you measure some properties of the berry (lets say we look at the color), and the procedure should help you decide whether to eat that berry. Now in frequentist statistics, you'll get a procedure that gives you the correct answer with probability at least p regardless of what your measurement was. Suppose that p=90% and we observe that the color is blue, and the procedure says: this berry is edible. Can we now eat the berry? No! This says absolutely nothing about the probability of the blue berry being poisonous or not. For example suppose 95% of berries are edible and red, and 5% of the berries are poisonous and blue. Then a valid procedure would be one that says that all berries are edible. It's correct >90% of the time, so yay! But if the berry we are holding in our hands is blue, it's incorrect 100% of the time. The fact that the procedure would have given us the right answer if the berry we found was red is irrelevant for making the decision of whether to eat the berry in the situation where the berry we found was blue. Things that did not happen are irrelevant for making decisions in a situation where that thing did happen! tl;dr: frequentist vs bayesian is not about worst case vs average case. It's about P(measurement | true value) vs P(true value | measurement). The former is irrelevant for making decisions, the latter is exactly what you want.
scotty79 12y ago
If you are restricting your tools on the grounds of philosophy you subscribe to, you are doing it wrong. Ditch the philosophy, stick to math, try everything and go with what results in models of predictive utility. That's what actual scientists do.
- analog31 12y ago
  ... and test the prediction.
  - TeMPOraL 12y ago
    That is implied in model being of predictive utility.
- judk 12y ago
  How do you decide which models actually have predictive utility, under real world incomplete information? Back to square One.
  - TeMPOraL 12y ago
    How do you decide if the water in your bathtub is warm? Stick your finger into it and see for yourself. Similarly here, you use your models to predict something, check if they predicted it right, and either try to fix them or ditch them completely.
- eruditely 12y ago
  Incorrect. The philosophers know 'math'. The half-mathematicians that think they can escape philosophy should read.... well anything. Almost all grand-standing that people have about 'philosophy' is some weird fictional idea that people have in their mind. Notice how all the recipients of the Lakatos awards and how they are people like Judea Pearl. Here's a quote from here (Terrence Tao, definitely a mathematician). http://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theory/ http://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-revi... "probability theory is only “allowed” to study concepts and perform operations which are preserved with respect to extension of the underlying sample space." Any attempt to side-step these issues leads to people wandering off into the middle of nowhere. Misuse of mathematical tools to areas where is does not apply and all sorts of other major and minor errors that could have been avoided. There is no computer science without Godel, Frege, Russell, and Turing.
wirrbel 12y ago
As a bayesianist I would like to defend Frequentism here. p-value hacking IS NOT "Frequentist", it is malpractice has its roots in lack of education (copy-paste-analyses), publication pressure and using the p-value as a strong metric for judgement of results (which it was never intended to be). Frequentism is a toolset of best practices and experiences about statistics, a KISS approach to statistics so to speak. Bayesianism is a beautiful approach to probabilities and statistics, yet often turns out to be really complex when performing calculations and it is also not fool-proof and it also requires experience to be applied correctly.

by bjterry 12y ago

keithwinstein 12y ago

Don't think that's quite what the author was getting at. A "frequentist" who makes a decision based on a p-value or confidence interval, and a "Bayesian" who makes a decision based on a posterior or predictive probability distribution, can both be viewed as making a decision according to a procedure that minimizes some statistic of a loss function. Both of these broad families of methods are diverse, but to generalize broadly, decisions informed by "frequentist" tools are often concerned with the worst-case value of the loss function within some universe, and decisions based on "Bayesian" methods can be viewed as caring about the expected value of the loss function given some prior. Both of those families (and many others) are "the ground truth," in that they both make statements that can be proved as theorems of mathematics. A "frequentist" confidence interval will always achieve its guaranteed coverage no matter what, if the likelihood function is true. A "Bayesian" credibility interval will include the true value of the parameters at exactly the advertised rate, when averaged over each possible value of the parameters weighted according to the prior, and assuming the likelihood function is true. The author says, and I agree, that the thing worth arguing over is the "norm we use to choose our optimal procedure" and the nature of the loss function. Whether you care about controlling the worst case or the expected value or something else depends on what matters. (The author points out that in an adversarial situation where your strategy is known to your opponent, minimizing the worst case may be advisable...) Some days, we care about the worst-case performance of QuickSort, and some days we care about its average-case performance (given an assumption that all input orderings are uniformly likely). It's ok to care about different things depending on the application; we don't have to split into warring tribes over it. More here: http://blog.keithw.org/2013/02/q-what-is-difference-between-bayesian.html http://blog.keithw.org/2013/02/q-what-is-difference-between-... http://jsteinhardt.wordpress.com/2014/02/10/a-fervent-defense-of-frequentist-statistics/ http://jsteinhardt.wordpress.com/2014/02/10/a-fervent-defens... http://cs.stanford.edu/~jsteinhardt/stats-essay.pdf http://cs.stanford.edu/~jsteinhardt/stats-essay.pdf

Matumio 12y ago
Do Bayesian methods imply a choice of loss function? I think it's a framework for observational statistics, not for policy making or decision theory. You can use it as a tool during policy or decision making, where you should indeed have a debate about picking a loss function that corresponds to your goals.
- mjw 12y ago
  If you're reporting point or interval estimates (rather than the entire posterior) then you are implicitly or explicitly optimising some kind of loss function. Also, worth a reminder that continuous estimation problems can be viewed as decision problems too, it's not just about discrete decision-making. I don't think it's great to take the view that, because I'm not making a decision based on this estimate myself, I don't need to worry about which loss function I'm implicitly optimising for when choosing an estimator. Someone else may need to.
jules 12y ago
Bayesian statistics gives you a posterior distribution. What you do with that distribution is up to you. If you want to find the decision that minimizes the maximum loss instead of the expected loss, that fits perfectly well within the Bayesian framework. The posterior gives you all the information that you need to make a decision, whatever your loss function. Frequentist statistics on the other hand gives you no such thing. You can't make decisions based on frequentist statistics. Frequentist statistics is all about reasoning about things that did not happen. Things that did not happen are irrelevant for making decisions in a situation where that thing did happen. So while frequentist statistics is mathematically correct, it's also strictly speaking useless in practice. It's only useful insofar as it gives us heuristics for decision making when the Bayesian approach is impractical. Here's an example. Suppose there are two types of berries: edible and poisonous. We devise a statistical procedure where you measure some properties of the berry (lets say we look at the color), and the procedure should help you decide whether to eat that berry. Now in frequentist statistics, you'll get a procedure that gives you the correct answer with probability at least p regardless of what your measurement was. Suppose that p=90% and we observe that the color is blue, and the procedure says: this berry is edible. Can we now eat the berry? No! This says absolutely nothing about the probability of the blue berry being poisonous or not. For example suppose 95% of berries are edible and red, and 5% of the berries are poisonous and blue. Then a valid procedure would be one that says that all berries are edible. It's correct >90% of the time, so yay! But if the berry we are holding in our hands is blue, it's incorrect 100% of the time. The fact that the procedure would have given us the right answer if the berry we found was red is irrelevant for making the decision of whether to eat the berry in the situation where the berry we found was blue. Things that did not happen are irrelevant for making decisions in a situation where that thing did happen! tl;dr: frequentist vs bayesian is not about worst case vs average case. It's about P(measurement | true value) vs P(true value | measurement). The former is irrelevant for making decisions, the latter is exactly what you want.

scotty79 12y ago

If you are restricting your tools on the grounds of philosophy you subscribe to, you are doing it wrong. Ditch the philosophy, stick to math, try everything and go with what results in models of predictive utility. That's what actual scientists do.

analog31 12y ago
... and test the prediction.
- TeMPOraL 12y ago
  That is implied in model being of predictive utility.
judk 12y ago
How do you decide which models actually have predictive utility, under real world incomplete information? Back to square One.
- TeMPOraL 12y ago
  How do you decide if the water in your bathtub is warm? Stick your finger into it and see for yourself. Similarly here, you use your models to predict something, check if they predicted it right, and either try to fix them or ditch them completely.
eruditely 12y ago
Incorrect. The philosophers know 'math'. The half-mathematicians that think they can escape philosophy should read.... well anything. Almost all grand-standing that people have about 'philosophy' is some weird fictional idea that people have in their mind. Notice how all the recipients of the Lakatos awards and how they are people like Judea Pearl. Here's a quote from here (Terrence Tao, definitely a mathematician). http://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theory/ http://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-revi... "probability theory is only “allowed” to study concepts and perform operations which are preserved with respect to extension of the underlying sample space." Any attempt to side-step these issues leads to people wandering off into the middle of nowhere. Misuse of mathematical tools to areas where is does not apply and all sorts of other major and minor errors that could have been avoided. There is no computer science without Godel, Frege, Russell, and Turing.

wirrbel 12y ago

As a bayesianist I would like to defend Frequentism here. p-value hacking IS NOT "Frequentist", it is malpractice has its roots in lack of education (copy-paste-analyses), publication pressure and using the p-value as a strong metric for judgement of results (which it was never intended to be). Frequentism is a toolset of best practices and experiences about statistics, a KISS approach to statistics so to speak. Bayesianism is a beautiful approach to probabilities and statistics, yet often turns out to be really complex when performing calculations and it is also not fool-proof and it also requires experience to be applied correctly.