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What Is Bayesian/Frequentist Inference? (2012)

74 points by spekcular 4y ago

clircle 4y ago
This blog is by Larry Wasserman, so i think his advice should be taken seriously. I agree that there are uses of both philosophies, and that statisticians should be pragmatic rather than dogmatic. My issue is that his advice is most useful for statisticians working in the abstract, but it doesn’t really help people working with real data. Scientists and data analysts just want to know how to analyze their data, and this does not help them. I know that stats isn’t a cookbook, but we could put some guard rails down that help practitioners with their problems.
- biomcgary 4y ago
  >Scientists and data analysts just want to know how to analyze their data, and this does not help them. I'm a computational biologist that uses Bayesian and Frequentist approaches depending on what I'm trying to achieve. This article was very helpful for making explicit a distinction I had not recognized before. With most of science (in my field) being done by people with Doctorates of Philosophy, I think it is reasonable to expect them to understand the underlying concepts of the math they are using. But, I'm anachronistic in wanting science to have a bit more natural philosophy rather than just "shut up and calculate" (https://en.wikiquote.org/wiki/Shut_up_and_calculate https://en.wikiquote.org/wiki/Shut_up_and_calculate).
- okennedy 4y ago
  Guardrails for stats are something that I've put a lot of thought towards. The fundamental challenge is that statistics is operating in a world of incomplete information. Statistical measures are almost never monotone with respect to new information, and so any new piece of context might completely invalidate an analysis. Going beyond the abstract requires an intimate knowledge of the domain being analyzed, and the limitations of statistical methods as applied to that domain. "Guardrails for stats" have to be domain- and even dataset-specific.
- kqr 4y ago
  The problem I see with guard rails is that it's very hard to know if you're doing statistics right, due to its nature. Inference is sometimes hard enough on its own (and I sometimes use computational methods in addition to theory just to double-check my results, but that's just the innermost layer. Outside of that you have to define appropriate and efficient samples, which is more difficult. You have to know what population you're actually interested in, which is less obvious than it sounds like, and on top of that you have to pick an experimental/observational method that minimises error and ideally lets you quantify it -- extremely hard in most practical cases. Add to that the fact that the outcome of statistical analysis might often be, "well, we still don't know anything meaningful!" But if you say that, someone else will sound more confident and guess who people will listen to? ---- The way out is not guard rails, it's much better training from earlier ages. This stuff is hard and we are not born with intuition for it. We need lots of practise. I still don't get why there's so much analysis and calculus in our curricula -- those are problems we can solve with numerical (sometimes statistical) methods. We ought to replace at least half off that with more probability and statistical inference and experimental design.
  - tpoacher 4y ago
    Guardrails is actually an apt term. Just because you train people not to peer over the edge doesn't mean there's no need for safeguards. (Death Star laser operating platform, I'm looking at you!). And just because there's safeguards doesn't mean you stop warning people not to peer over the edge. The implicit assumption in your text is that if there's a guardrail, people will be leaning all day. Whereas my experience is that if there isn't, people will try that stuff regardless, and keep falling off the platform. Obligatory: https://www.youtube.com/watch?v=9bSZXucTH4A https://www.youtube.com/watch?v=9bSZXucTH4A
wirrbel 4y ago
Dunno whether I agree to this. I agree that both are acceptable ways to do statistics. However 1. Bayesian stats is an approach that tends to make model assumptions fairly explicity, whereas in frequentist approaches, many assumptions are fairly implicit (Normal distribution of data, etc.) 2. I would consider myself a Bayesianist but I am sceptical about too much mention of esoteric terminology like "Belief". Bayesian probabilities are probabilities following the Kolmogorov axioms, which is also the foundation of Frequentist stats. For decades, Bayesian inference was impractical because we need to resort to sampling methods and (a) computational power was insuffient and (b) we didn't have algorithms like No U-Turn Sampler (NUTS). Both aspects are 'solved', so why is Bayesianism not universally adopted? Of course it still has a reputational disadvantage, but I think more importantly its because * frequentist methods are good enough for purposes of publishing research [] for some problems we really have a hard time assembling bayesian graphs * some inference methods (e..g. Kalman filter) can both be seen as frequentist or Bayesian As a bayesianist I am amazed at how well frequentism can work, even when the 'traditional' way of applying it contradicts the derivations of founding fathers like Fisher/Pearson. It's almost as if we have an evolutionary process at play. [*] That is, if you use p-Values as publication thresholds
- NohatCoder 4y ago
  Scientific publishing has largely gone off the rails, thanks in no small part to the frequentist p-value obsession. It is not good enough, people just use it anyway. I think most people want to avoid the dance of picking a prior, that is why frequentism is still so widespread.
lalaithion 4y ago
This falls apart in higher dimensions, but in the example given in the article the two answers only differ because they have different priors. If you repeat the bayesian analysis using the prior \theta ~ N(0, x), and let x go to infinity, then you approach the frequentist answer. In my opinion, https://stats.stackexchange.com/questions/2272/whats-the-difference-between-a-confidence-interval-and-a-credible-interval https://stats.stackexchange.com/questions/2272/whats-the-dif... is a better explanation of the difference between confidence and credible intervals. Editing to add more commentary to my link: If you've read the link, one of the principal objections of the frequentist is "What if the jar is type B? Then your interval will be wrong 80% of the time, and only correct 20% of the time!" This is because, if you look at the original numbers, jar B has all types of cookies, and therefore any single draw from jar B can "look like" any other jar, and because other jars have more concentrated cookie types, they are "more likely" answers for each potential sample. This issue also comes up with the frequentist analysis! If you look at the confidence intervals, they all say "This jar could be jar B". Instead of being bad at detecting jar B, they are good at always considering jar B no matter the evidence – but it's the same uncertainty. The bayesian version of this objection is "when you pull a cookie with 3 chips your interval is only correct 41% of the time". This is because if we get jar A, we'll probably draw a 2-chip cookie, so it's outside of our confidence interval. But note that we probably don't really care about the confidence interval or credibility interval. It's basically a hack to take a probabilistic problem and turn our answer into black and white. To say "these hypotheses are valid and these are invalid". But this is statistics! If you just take the Bayesian approach, and throw out the need to create an arbitrary interval, you can just stop at the table titled P(Jar|Chips). That's all the information you need. If you draw a N-chip cookie, you can use that table to update your P(Chips_2) for a second draw, and you'll get a concrete probabilistic answer. Yes, you have to assume a prior. But frequentist statistics literally can't answer this question! Without a prior, there's no way to turn P(Chips | Jar) into a P(Jar | Chips) to update on, so you can't track your evidence to get better predictions. You just sit there saying "well, my interval meets the criterion even in the worst case".
- spekcular 4y ago
  Honestly, I think Wasserman does a better job. The cookie interval example gets the fact that frequentists require uniform coverage properties with respect to the unknown parameter right. But the "when you pull a cookie with 3 chips your interval is only correct 41% of the time" thing isn't really an essential Bayesian vs. Frequentist issue. As Wasserman notes, coverage is a minimal requirement for something being a confidence interval; we usually also construct them to avoid obvious deficiencies. All this objection shows is that the particular procedure in the example might not be the best one; it's not an argument that can be applied to all CIs in general. (And it's clear, if you play with the numbers, that the example can be improved.) Also, regarding "But note that we probably don't really care about the confidence interval or credibility interval." Many times we do - giving a point estimate and an associated quantification of its uncertainty is one of the most basic statistical tasks. Further, it's somewhat misleading to critique frequentists by saying they don't give probabilities for P(Jar | Chips), because in the frequentist setup the jar is a fixed and unknown parameter, not stochastic. For the two-cookie setting, it's trivial to generalize the construction in the M.SE post, so saying "frquentists can't track evidence to get better predictions" is simply wrong.
dekhn 4y ago
Whilst successful in my career and user of probability, statistics, and inference on a regular basis, I simply cannot understand what's being discussed here. I don't even want to understand it. Just like quantum, half the argument seems to be the a mismatch between mental models and actual reality.
- ordu 4y ago
  > half the argument seems to be the a mismatch between mental models and actual reality. Which half seems to be a mismatch to you? A bayesian half or a frequentist one?
  - dekhn 4y ago
    Every time I've tried to understand the entire argument it just raises more questions to me. For example as I was first introduced to it, frequentists simple count frequencies observed in nature and then compute stats on them, and then build inferential models using those stats without assuming any complex underlying distribution. While Bayesians count frequencies, apply a prior correction (say, adding a pseudocount of one for every unobserved possible event, or any other way of assuming the generative process has a distribution that we've previously estimated), some stats,then build models from that. however, after I was told that, I've seen several other arguments that quickly dive into: the distribution of the underlying events (I've heard that frequentists assume one type while bayesian assume another). Other folks just sort of give the example of the base rate fallacy. Throughout all of this I've realized: I don't understand stats at all. I came to the scientific world with a view much more like physics: there is a microscopic event system (a particle simulation, or whatever) that we are observing, but due to limitations, we can only make macroscopic observations, which represent biased aggregations of the underlying microscopic event system. We can figure out those biases and use the aggregate data to build predictive models of the underlying systems- without ever really knowing the true details of the microscopic model. From what I can tell, everything about what physicists do to model the world mentally is more Bayesian than Frequentist, if I understand what the hell people mean when they argue about it. However, as I said, every time I look at the arguments, I realize I don't understand stats, while I understand the physics approach which seems to be fairly obvious.
dang 4y ago
Discussed at the time: What Is Bayesian/Frequentist Inference? - https://news.ycombinator.com/item?id=4800449 https://news.ycombinator.com/item?id=4800449 - Nov 2012 (27 comments)
sega_sai 4y ago
In the end I believe the Bayesian inference is more straightforward to implement and understand if you can afford computationally sampling of the posterior. So I think at least in physics there is a shift towards Bayesian approaches.
enaaem 4y ago
The difference between Bayesian and Frequentist is in the interpretation of randomness. In Bayesian statistics 'randomness' is not a property of nature but a description of our knowledge. What's randomness in a coin toss? If we had all the information we could perfectly predict the result of a toss. But if we know nothing then at most we can say is that both outcomes are equally probable. Another example, if you had no idea who the next presidential winner will be between two candidates, than saying it's 50-50 is an accurate description of your knowledge. If anyone is more interested I would refer to you to [1]. Here, probability theory is interpreted as an extension of logic. Very interesting stuff. [1] http://www.med.mcgill.ca/epidemiology/hanley/bios601/GaussianModel/JaynesProbabilityTheory.pdf http://www.med.mcgill.ca/epidemiology/hanley/bios601/Gaussia...
- angrais 4y ago
  >> What's randomness in a coin toss? If we had all the information we could perfectly predict the result of a toss. But if we know nothing then at most we can say is that both outcomes are equally probable. Perhaps I'm being dense or overall don't understand, but how is this possible? What is "all the information"? Isn't it at most likely they the outcome is 50/50?
  - idontpost 4y ago
    Each coin-toss is a deterministic physical process governed by laws of motion. If we had perfect information about the motion of all components in the system (hand, coin, air, floor, etc.), then we could, in principle, perfectly predict the outcome of every toss. Each individual toss would have a 100% probability of its predicted outcome. Since we typically lack any of that information, we are stuck with the 50 / 50 prior distribution.
    - davrosthedalek 4y ago
      This is very debatable if you throw QM into the mix. From all what we know, we cannot predict everything with 100% success rate -- QM cannot be explained by a hidden variables model.
tpoacher 4y ago
Fully agree with every single word in this article. Particularly the bit about "identity statistics". Also, regarding failure of notation: I've been arguing for a while that the notation we use for probability is highly problematic and effectively eggregious abuse of notation. And not just the whole "belief vs frequency of hypotheticals", but even the simple fact of what it represents. Does p(x) denote an event? The whole distribution? The distribution over a particular space? An apologist might rightly point out that p(x) is actually shorthand for p(x=X), where x is the event and X is the distribution. But even this screams confusion. For me the ideal notation would have been something that makes it explicit that the probability describes a set of 'sampling events' from a 'pool', e.g. Prob_{x}(X), where x is the set of events, and X is the nature of the distribution (i.e. a function which returns frequencies/beliefs over a domain). And probability density functions would be denoted as actual derivatives, e.g. d/dx P_x (Cum.Normal)

74 points by spekcular 4y ago

This blog is by Larry Wasserman, so i think his advice should be taken seriously. I agree that there are uses of both philosophies, and that statisticians should be pragmatic rather than dogmatic. My issue is that his advice is most useful for statisticians working in the abstract, but it doesn’t really help people working with real data. Scientists and data analysts just want to know how to analyze their data, and this does not help them. I know that stats isn’t a cookbook, but we could put some guard rails down that help practitioners with their problems.

biomcgary 4y ago
>Scientists and data analysts just want to know how to analyze their data, and this does not help them. I'm a computational biologist that uses Bayesian and Frequentist approaches depending on what I'm trying to achieve. This article was very helpful for making explicit a distinction I had not recognized before. With most of science (in my field) being done by people with Doctorates of Philosophy, I think it is reasonable to expect them to understand the underlying concepts of the math they are using. But, I'm anachronistic in wanting science to have a bit more natural philosophy rather than just "shut up and calculate" (https://en.wikiquote.org/wiki/Shut_up_and_calculate https://en.wikiquote.org/wiki/Shut_up_and_calculate).
okennedy 4y ago
Guardrails for stats are something that I've put a lot of thought towards. The fundamental challenge is that statistics is operating in a world of incomplete information. Statistical measures are almost never monotone with respect to new information, and so any new piece of context might completely invalidate an analysis. Going beyond the abstract requires an intimate knowledge of the domain being analyzed, and the limitations of statistical methods as applied to that domain. "Guardrails for stats" have to be domain- and even dataset-specific.
kqr 4y ago
The problem I see with guard rails is that it's very hard to know if you're doing statistics right, due to its nature. Inference is sometimes hard enough on its own (and I sometimes use computational methods in addition to theory just to double-check my results, but that's just the innermost layer. Outside of that you have to define appropriate and efficient samples, which is more difficult. You have to know what population you're actually interested in, which is less obvious than it sounds like, and on top of that you have to pick an experimental/observational method that minimises error and ideally lets you quantify it -- extremely hard in most practical cases. Add to that the fact that the outcome of statistical analysis might often be, "well, we still don't know anything meaningful!" But if you say that, someone else will sound more confident and guess who people will listen to? ---- The way out is not guard rails, it's much better training from earlier ages. This stuff is hard and we are not born with intuition for it. We need lots of practise. I still don't get why there's so much analysis and calculus in our curricula -- those are problems we can solve with numerical (sometimes statistical) methods. We ought to replace at least half off that with more probability and statistical inference and experimental design.
- tpoacher 4y ago
  Guardrails is actually an apt term. Just because you train people not to peer over the edge doesn't mean there's no need for safeguards. (Death Star laser operating platform, I'm looking at you!). And just because there's safeguards doesn't mean you stop warning people not to peer over the edge. The implicit assumption in your text is that if there's a guardrail, people will be leaning all day. Whereas my experience is that if there isn't, people will try that stuff regardless, and keep falling off the platform. Obligatory: https://www.youtube.com/watch?v=9bSZXucTH4A https://www.youtube.com/watch?v=9bSZXucTH4A

wirrbel 4y ago

Dunno whether I agree to this. I agree that both are acceptable ways to do statistics. However 1. Bayesian stats is an approach that tends to make model assumptions fairly explicity, whereas in frequentist approaches, many assumptions are fairly implicit (Normal distribution of data, etc.) 2. I would consider myself a Bayesianist but I am sceptical about too much mention of esoteric terminology like "Belief". Bayesian probabilities are probabilities following the Kolmogorov axioms, which is also the foundation of Frequentist stats. For decades, Bayesian inference was impractical because we need to resort to sampling methods and (a) computational power was insuffient and (b) we didn't have algorithms like No U-Turn Sampler (NUTS). Both aspects are 'solved', so why is Bayesianism not universally adopted? Of course it still has a reputational disadvantage, but I think more importantly its because * frequentist methods are good enough for purposes of publishing research [] for some problems we really have a hard time assembling bayesian graphs * some inference methods (e..g. Kalman filter) can both be seen as frequentist or Bayesian As a bayesianist I am amazed at how well frequentism can work, even when the 'traditional' way of applying it contradicts the derivations of founding fathers like Fisher/Pearson. It's almost as if we have an evolutionary process at play. [*] That is, if you use p-Values as publication thresholds

NohatCoder 4y ago
Scientific publishing has largely gone off the rails, thanks in no small part to the frequentist p-value obsession. It is not good enough, people just use it anyway. I think most people want to avoid the dance of picking a prior, that is why frequentism is still so widespread.

lalaithion 4y ago

This falls apart in higher dimensions, but in the example given in the article the two answers only differ because they have different priors. If you repeat the bayesian analysis using the prior \theta ~ N(0, x), and let x go to infinity, then you approach the frequentist answer. In my opinion, https://stats.stackexchange.com/questions/2272/whats-the-difference-between-a-confidence-interval-and-a-credible-interval https://stats.stackexchange.com/questions/2272/whats-the-dif... is a better explanation of the difference between confidence and credible intervals. Editing to add more commentary to my link: If you've read the link, one of the principal objections of the frequentist is "What if the jar is type B? Then your interval will be wrong 80% of the time, and only correct 20% of the time!" This is because, if you look at the original numbers, jar B has all types of cookies, and therefore any single draw from jar B can "look like" any other jar, and because other jars have more concentrated cookie types, they are "more likely" answers for each potential sample. This issue also comes up with the frequentist analysis! If you look at the confidence intervals, they all say "This jar could be jar B". Instead of being bad at detecting jar B, they are good at always considering jar B no matter the evidence – but it's the same uncertainty. The bayesian version of this objection is "when you pull a cookie with 3 chips your interval is only correct 41% of the time". This is because if we get jar A, we'll probably draw a 2-chip cookie, so it's outside of our confidence interval. But note that we probably don't really care about the confidence interval or credibility interval. It's basically a hack to take a probabilistic problem and turn our answer into black and white. To say "these hypotheses are valid and these are invalid". But this is statistics! If you just take the Bayesian approach, and throw out the need to create an arbitrary interval, you can just stop at the table titled P(Jar|Chips). That's all the information you need. If you draw a N-chip cookie, you can use that table to update your P(Chips_2) for a second draw, and you'll get a concrete probabilistic answer. Yes, you have to assume a prior. But frequentist statistics literally can't answer this question! Without a prior, there's no way to turn P(Chips | Jar) into a P(Jar | Chips) to update on, so you can't track your evidence to get better predictions. You just sit there saying "well, my interval meets the criterion even in the worst case".

spekcular 4y ago
Honestly, I think Wasserman does a better job. The cookie interval example gets the fact that frequentists require uniform coverage properties with respect to the unknown parameter right. But the "when you pull a cookie with 3 chips your interval is only correct 41% of the time" thing isn't really an essential Bayesian vs. Frequentist issue. As Wasserman notes, coverage is a minimal requirement for something being a confidence interval; we usually also construct them to avoid obvious deficiencies. All this objection shows is that the particular procedure in the example might not be the best one; it's not an argument that can be applied to all CIs in general. (And it's clear, if you play with the numbers, that the example can be improved.) Also, regarding "But note that we probably don't really care about the confidence interval or credibility interval." Many times we do - giving a point estimate and an associated quantification of its uncertainty is one of the most basic statistical tasks. Further, it's somewhat misleading to critique frequentists by saying they don't give probabilities for P(Jar | Chips), because in the frequentist setup the jar is a fixed and unknown parameter, not stochastic. For the two-cookie setting, it's trivial to generalize the construction in the M.SE post, so saying "frquentists can't track evidence to get better predictions" is simply wrong.

dekhn 4y ago

Whilst successful in my career and user of probability, statistics, and inference on a regular basis, I simply cannot understand what's being discussed here. I don't even want to understand it. Just like quantum, half the argument seems to be the a mismatch between mental models and actual reality.

ordu 4y ago
> half the argument seems to be the a mismatch between mental models and actual reality. Which half seems to be a mismatch to you? A bayesian half or a frequentist one?
- dekhn 4y ago
  Every time I've tried to understand the entire argument it just raises more questions to me. For example as I was first introduced to it, frequentists simple count frequencies observed in nature and then compute stats on them, and then build inferential models using those stats without assuming any complex underlying distribution. While Bayesians count frequencies, apply a prior correction (say, adding a pseudocount of one for every unobserved possible event, or any other way of assuming the generative process has a distribution that we've previously estimated), some stats,then build models from that. however, after I was told that, I've seen several other arguments that quickly dive into: the distribution of the underlying events (I've heard that frequentists assume one type while bayesian assume another). Other folks just sort of give the example of the base rate fallacy. Throughout all of this I've realized: I don't understand stats at all. I came to the scientific world with a view much more like physics: there is a microscopic event system (a particle simulation, or whatever) that we are observing, but due to limitations, we can only make macroscopic observations, which represent biased aggregations of the underlying microscopic event system. We can figure out those biases and use the aggregate data to build predictive models of the underlying systems- without ever really knowing the true details of the microscopic model. From what I can tell, everything about what physicists do to model the world mentally is more Bayesian than Frequentist, if I understand what the hell people mean when they argue about it. However, as I said, every time I look at the arguments, I realize I don't understand stats, while I understand the physics approach which seems to be fairly obvious.

dang 4y ago

Discussed at the time: What Is Bayesian/Frequentist Inference? - https://news.ycombinator.com/item?id=4800449 https://news.ycombinator.com/item?id=4800449 - Nov 2012 (27 comments)

sega_sai 4y ago

In the end I believe the Bayesian inference is more straightforward to implement and understand if you can afford computationally sampling of the posterior. So I think at least in physics there is a shift towards Bayesian approaches.

enaaem 4y ago

The difference between Bayesian and Frequentist is in the interpretation of randomness. In Bayesian statistics 'randomness' is not a property of nature but a description of our knowledge. What's randomness in a coin toss? If we had all the information we could perfectly predict the result of a toss. But if we know nothing then at most we can say is that both outcomes are equally probable. Another example, if you had no idea who the next presidential winner will be between two candidates, than saying it's 50-50 is an accurate description of your knowledge. If anyone is more interested I would refer to you to [1]. Here, probability theory is interpreted as an extension of logic. Very interesting stuff. [1] http://www.med.mcgill.ca/epidemiology/hanley/bios601/GaussianModel/JaynesProbabilityTheory.pdf http://www.med.mcgill.ca/epidemiology/hanley/bios601/Gaussia...

angrais 4y ago
>> What's randomness in a coin toss? If we had all the information we could perfectly predict the result of a toss. But if we know nothing then at most we can say is that both outcomes are equally probable. Perhaps I'm being dense or overall don't understand, but how is this possible? What is "all the information"? Isn't it at most likely they the outcome is 50/50?
- idontpost 4y ago
  Each coin-toss is a deterministic physical process governed by laws of motion. If we had perfect information about the motion of all components in the system (hand, coin, air, floor, etc.), then we could, in principle, perfectly predict the outcome of every toss. Each individual toss would have a 100% probability of its predicted outcome. Since we typically lack any of that information, we are stuck with the 50 / 50 prior distribution.
  - davrosthedalek 4y ago
    This is very debatable if you throw QM into the mix. From all what we know, we cannot predict everything with 100% success rate -- QM cannot be explained by a hidden variables model.

tpoacher 4y ago

Fully agree with every single word in this article. Particularly the bit about "identity statistics". Also, regarding failure of notation: I've been arguing for a while that the notation we use for probability is highly problematic and effectively eggregious abuse of notation. And not just the whole "belief vs frequency of hypotheticals", but even the simple fact of what it represents. Does p(x) denote an event? The whole distribution? The distribution over a particular space? An apologist might rightly point out that p(x) is actually shorthand for p(x=X), where x is the event and X is the distribution. But even this screams confusion. For me the ideal notation would have been something that makes it explicit that the probability describes a set of 'sampling events' from a 'pool', e.g. Prob_{x}(X), where x is the set of events, and X is the nature of the distribution (i.e. a function which returns frequencies/beliefs over a domain). And probability density functions would be denoted as actual derivatives, e.g. d/dx P_x (Cum.Normal)