Allan Birnbaum died 40 years ago today. He lived to be only 53 [i]. From the perspective of philosophy of statistics and philosophy of science, Birnbaum is best known for his work on likelihood, the Likelihood Principle [ii], and for his attempts to blend concepts of likelihood with error probability ideas to arrive at what he termed “concepts of statistical evidence”. Failing to find adequate concepts of statistical evidence, Birnbaum called for joining the work of “interested statisticians, scientific workers and philosophers and historians of science”–an idea I have heartily endorsed. While known for a result that the (strong) Likelihood Principle followed from sufficiency and conditionality principles (a result that Jimmy Savage deemed one of the greatest breakthroughs in statistics), a few years after publishing it, he turned away from it, perhaps discovering gaps in his argument. A post linking to a 2014 Statistical Science issue discussing Birnbaum’s result is here. Reference  links to the Synthese 1977 volume dedicated to his memory. The editors describe it as their way of “paying homage to Professor Birnbaum’s penetrating and stimulating work on the foundations of statistics”. Ample weekend reading! Continue reading
Philosophy of Statistics
Today is Allan Birnbaum’s birthday. In honor of his birthday this year, I’m posting the articles in the Synthese volume that was dedicated to his memory in 1977. The editors describe it as their way of “paying homage to Professor Birnbaum’s penetrating and stimulating work on the foundations of statistics”. I paste a few snippets from the articles by Giere and Birnbaum. If you’re interested in statistical foundations, and are unfamiliar with Birnbaum, here’s a chance to catch up.(Even if you are,you may be unaware of some of these key papers.)
HAPPY BIRTHDAY ALLAN!
Synthese Volume 36, No. 1 Sept 1977: Foundations of Probability and Statistics, Part I
This special issue of Synthese on the foundations of probability and statistics is dedicated to the memory of Professor Allan Birnbaum. Professor Birnbaum’s essay ‘The Neyman-Pearson Theory as Decision Theory; and as Inference Theory; with a Criticism of the Lindley-Savage Argument for Bayesian Theory’ was received by the editors of Synthese in October, 1975, and a decision was made to publish a special symposium consisting of this paper together with several invited comments and related papers. The sad news about Professor Birnbaum’s death reached us in the summer of 1976, but the editorial project could nevertheless be completed according to the original plan. By publishing this special issue we wish to pay homage to Professor Birnbaum’s penetrating and stimulating work on the foundations of statistics. We are grateful to Professor Ronald Giere who wrote an introductory essay on Professor Birnbaum’s concept of statistical evidence and who compiled a list of Professor Birnbaum’s publications.
Today is Allan Birnbaum’s Birthday. Birnbaum’s (1962) classic “On the Foundations of Statistical Inference,” in Breakthroughs in Statistics (volume I 1993), concerns a principle that remains at the heart of today’s controversies in statistics–even if it isn’t obvious at first: the Likelihood Principle (LP) (also called the strong likelihood Principle SLP, to distinguish it from the weak LP ). According to the LP/SLP, given the statistical model, the information from the data are fully contained in the likelihood ratio. Thus, properties of the sampling distribution of the test statistic vanish (as I put it in my slides from my last post)! But error probabilities are all properties of the sampling distribution. Thus, embracing the LP (SLP) blocks our error statistician’s direct ways of taking into account “biasing selection effects” (slide #10).
Intentions is a New Code Word: Where, then, is all the information regarding your trying and trying again, stopping when the data look good, cherry picking, barn hunting and data dredging? For likelihoodists and other probabilists who hold the LP/SLP, it is ephemeral information locked in your head reflecting your “intentions”! “Intentions” is a code word for “error probabilities” in foundational discussions, as in “who would want to take intentions into account?” (Replace “intentions” (or the “researcher’s intentions”) with “error probabilities” (or the method’s error probabilities”) and you get a more accurate picture.) Keep this deciphering tool firmly in mind as you read criticisms of methods that take error probabilities into account. For error statisticians, this information reflects real and crucial properties of your inference procedure.
Memory Lane: 3 years ago. Oxford Jail (also called Oxford Castle) is an entirely fitting place to be on (and around) Halloween! Moreover, rooting around this rather lavish set of jail cells (what used to be a single cell is now a dressing room) is every bit as conducive to philosophical reflection as is exile on Elba! (It is now a boutique hotel, though many of the rooms are still too jail-like for me.) My goal (while in this gaol—as the English sometimes spell it) is to try and free us from the bogeymen and bogeywomen often associated with “classical” statistics. As a start, the very term “classical statistics” should, I think, be shelved, not that names should matter.
In appraising statistical accounts at the foundational level, we need to realize the extent to which accounts are viewed through the eyeholes of a mask or philosophical theory. Moreover, the mask some wear while pursuing this task might well be at odds with their ordinary way of looking at evidence, inference, and learning. In any event, to avoid non-question-begging criticisms, the standpoint from which the appraisal is launched must itself be independently defended. But for (most) Bayesian critics of error statistics the assumption that uncertain inference demands a posterior probability for claims inferred is thought to be so obvious as not to require support. Critics are implicitly making assumptions that are at odds with the frequentist statistical philosophy. In particular, they assume a certain philosophy about statistical inference (probabilism), often coupled with the allegation that error statistical methods can only achieve radical behavioristic goals, wherein all that matters are long-run error rates (of some sort)
Criticisms then follow readily: the form of one or both:
- Error probabilities do not supply posterior probabilities in hypotheses, interpreted as if they do (and some say we just can’t help it), they lead to inconsistencies
- Methods with good long-run error rates can give rise to counterintuitive inferences in particular cases.
- I have proposed an alternative philosophy that replaces these tenets with different ones:
- the role of probability in inference is to quantify how reliably or severely claims (or discrepancies from claims) have been tested
- the severity goal directs us to the relevant error probabilities, avoiding the oft-repeated statistical fallacies due to tests that are overly sensitive, as well as those insufficiently sensitive to particular errors.
- Control of long run error probabilities, while necessary is not sufficient for good tests or warranted inferences.
Link to complete discussion:
Mayo, Deborah G. On the Birnbaum Argument for the Strong Likelihood Principle (with discussion & rejoinder). Statistical Science 29 (2014), no. 2, 227-266.
Links to individual papers:
Mayo, Deborah G. On the Birnbaum Argument for the Strong Likelihood Principle. Statistical Science 29 (2014), no. 2, 227-239.
Dawid, A. P. Discussion of “On the Birnbaum Argument for the Strong Likelihood Principle”. Statistical Science 29 (2014), no. 2, 240-241.
Evans, Michael. Discussion of “On the Birnbaum Argument for the Strong Likelihood Principle”. Statistical Science 29 (2014), no. 2, 242-246.
Martin, Ryan; Liu, Chuanhai. Discussion: Foundations of Statistical Inference, Revisited. Statistical Science 29 (2014), no. 2, 247-251.
Fraser, D. A. S. Discussion: On Arguments Concerning Statistical Principles. Statistical Science 29 (2014), no. 2, 252-253.
Hannig, Jan. Discussion of “On the Birnbaum Argument for the Strong Likelihood Principle”. Statistical Science 29 (2014), no. 2, 254-258.
Bjørnstad, Jan F. Discussion of “On the Birnbaum Argument for the Strong Likelihood Principle”. Statistical Science 29 (2014), no. 2, 259-260.
Mayo, Deborah G. Rejoinder: “On the Birnbaum Argument for the Strong Likelihood Principle”. Statistical Science 29 (2014), no. 2, 261-266.
Abstract: An essential component of inference based on familiar frequentist notions, such as p-values, significance and confidence levels, is the relevant sampling distribution. This feature results in violations of a principle known as the strong likelihood principle (SLP), the focus of this paper. In particular, if outcomes x∗ and y∗ from experiments E1 and E2 (both with unknown parameter θ), have different probability models f1( . ), f2( . ), then even though f1(x∗; θ) = cf2(y∗; θ) for all θ, outcomes x∗ and y∗may have different implications for an inference about θ. Although such violations stem from considering outcomes other than the one observed, we argue, this does not require us to consider experiments other than the one performed to produce the data. David Cox [Ann. Math. Statist. 29 (1958) 357–372] proposes the Weak Conditionality Principle (WCP) to justify restricting the space of relevant repetitions. The WCP says that once it is known which Ei produced the measurement, the assessment should be in terms of the properties of Ei. The surprising upshot of Allan Birnbaum’s [J.Amer.Statist.Assoc.57(1962) 269–306] argument is that the SLP appears to follow from applying the WCP in the case of mixtures, and so uncontroversial a principle as sufficiency (SP). But this would preclude the use of sampling distributions. The goal of this article is to provide a new clarification and critique of Birnbaum’s argument. Although his argument purports that [(WCP and SP), entails SLP], we show how data may violate the SLP while holding both the WCP and SP. Such cases also refute [WCP entails SLP].
Key words: Birnbaumization, likelihood principle (weak and strong), sampling theory, sufficiency, weak conditionality
Regular readers of this blog know that the topic of the “Strong Likelihood Principle (SLP)” has come up quite frequently. Numerous informal discussions of earlier attempts to clarify where Birnbaum’s argument for the SLP goes wrong may be found on this blog. [SEE PARTIAL LIST BELOW.[i]] These mostly stem from my initial paper Mayo (2010) [ii]. I’m grateful for the feedback.
In the months since this paper has been accepted for publication, I’ve been asked, from time to time, to reflect informally on the overall journey: (1) Why was/is the Birnbaum argument so convincing for so long? (Are there points being overlooked, even now?) (2) What would Birnbaum have thought? (3) What is the likely upshot for the future of statistical foundations (if any)?
I’ll try to share some responses over the next week. (Naturally, additional questions are welcome.)
[i] A quick take on the argument may be found in the appendix to: “A Statistical Scientist Meets a Philosopher of Science: A conversation between David Cox and Deborah Mayo (as recorded, June 2011)”
- Midnight with Birnbaum (Happy New Year).
- New Version: On the Birnbaum argument for the SLP: Slides for my JSM talk.
- Don’t Birnbaumize that experiment my friend*–updated reblog.
- Allan Birnbaum, Philosophical Error Statistician: 27 May 1923 – 1 July 1976 .
- LSE seminar
- A. Birnbaum: Statistical Methods in Scientific Inference
- ReBlogging the Likelihood Principle #2: Solitary Fishing: SLP Violations
- Putting the brakes on the breakthrough: An informal look at the argument for the Likelihood Principle.
- Forthcoming paper on the strong likelihood principle.
UPhils and responses
- U-PHIL: Gandenberger & Hennig : Blogging Birnbaum’s Proof
- U-Phil: Mayo’s response to Hennig and Gandenberger
- Mark Chang (now) gets it right about circularity
- U-Phil: Ton o’ Bricks
- Blogging (flogging?) the SLP: Response to Reply- Xi’an Robert
- U-Phil: J. A. Miller: Blogging the SLP
- Mayo, D. G. (2010). “An Error in the Argument from Conditionality and Sufficiency to the Likelihood Principle” in Error and Inference: Recent Exchanges on Experimental Reasoning, Reliability and the Objectivity and Rationality of Science (D Mayo and A. Spanos eds.), Cambridge: Cambridge University Press: 305-14.
1.An Assumed Law of Statistical Evidence (law of likelihood)
Nearly all critical discussions of frequentist error statistical inference (significance tests, confidence intervals, p- values, power, etc.) start with the following general assumption about the nature of inductive evidence or support:
Data x are better evidence for hypothesis H1 than for H0 if x are more probable under H1 than under H0.
Ian Hacking (1965) called this the logic of support: x supports hypotheses H1 more than H0 if H1 is more likely, given x than is H0:
Pr(x; H1) > Pr(x; H0).
[With likelihoods, the data x are fixed, the hypotheses vary.]*
x is evidence for H1 over H0 if the likelihood ratio LR (H1 over H0 ) is greater than 1.
It is given in other ways besides, but it’s the same general idea. (Some will take the LR as actually quantifying the support, others leave it qualitative.)
In terms of rejection:
“An hypothesis should be rejected if and only if there is some rival hypothesis much better supported [i.e., much more likely] than it is.” (Hacking 1965, 89)
2. Barnard (British Journal of Philosophy of Science )
But this “law” will immediately be seen to fail on our minimal severity requirement. Hunting for an impressive fit, or trying and trying again, it’s easy to find a rival hypothesis H1 much better “supported” than H0 even when H0 is true. Or, as Barnard (1972) puts it, “there always is such a rival hypothesis, viz. that things just had to turn out the way they actually did” (1972 p. 129). H0: the coin is fair, gets a small likelihood (.5)k given k tosses of a coin, while H1: the probability of heads is 1 just on those tosses that yield a head, renders the sequence of k outcomes maximally likely. This is an example of Barnard’s “things just had to turn out as they did”. Or, to use an example with P-values: a statistically significant difference, being improbable under the null H0 , will afford high likelihood to any number of explanations that fit the data well.
3.Breaking the law (of likelihood) by going to the “second,” error statistical level:
How does it fail our severity requirement? First look at what the frequentist error statistician must always do to critique an inference: she must consider the capability of the inference method that purports to provide evidence for a claim. She goes to a higher level or metalevel, as it were. In this case, the likelihood ratio plays the role of the needed statistic d(X). To put it informally, she asks:
What’s the probability the method would yield an LR disfavoring H0 compared to some alternative H1 even if H0 is true?
Today is Egon Pearson’s birthday: 11 August 1895-12 June, 1980.
E. Pearson rejected some of the familiar tenets that have come to be associated with Neyman and Pearson (N-P) statistical tests, notably the idea that the essential justification for tests resides in a long-run control of rates of erroneous interpretations–what he termed the “behavioral” rationale of tests. In an unpublished letter E. Pearson wrote to Birnbaum (1974), he talks about N-P theory admitting of two interpretations: behavioral and evidential:
“I think you will pick up here and there in my own papers signs of evidentiality, and you can say now that we or I should have stated clearly the difference between the behavioral and evidential interpretations. Certainly we have suffered since in the way the people have concentrated (to an absurd extent often) on behavioral interpretations”.
(Nowadays, some people concentrate to an absurd extent on “science-wise error rates in dichotomous screening”.)
When Erich Lehmann, in his review of my “Error and the Growth of Experimental Knowledge” (EGEK 1996), called Pearson “the hero of Mayo’s story,” it was because I found in E.S.P.’s work, if only in brief discussions, hints, and examples, the key elements for an “inferential” or “evidential” interpretation of N-P statistics. Granted, these “evidential” attitudes and practices have never been explicitly codified to guide the interpretation of N-P tests. If they had been, I would not be on about providing an inferential philosophy all these years.[i] Nevertheless, “Pearson and Pearson” statistics (both Egon, not Karl) would have looked very different from Neyman and Pearson statistics, I suspect. One of the few sources of E.S. Pearson’s statistical philosophy is his (1955) “Statistical Concepts in Their Relation to Reality”. It begins like this: Continue reading
score years ago (!) we held the conference “Statistical Science and Philosophy of Science: Where Do (Should) They meet?” at the London School of Economics, Center for the Philosophy of Natural and Social Science, CPNSS, where I’m visiting professor  Many of the discussions on this blog grew out of contributions from the conference, and conversations initiated soon after. The conference site is here; my paper on the general question is here.
My main contribution was “Statistical Science Meets Philosophy of Science Part 2: Shallow versus Deep Explorations” SS & POS 2. It begins like this:
1. Comedy Hour at the Bayesian Retreat
Overheard at the comedy hour at the Bayesian retreat: Did you hear the one about the frequentist… Continue reading
Today is Allan Birnbaum’s Birthday. Birnbaum’s (1962) classic “On the Foundations of Statistical Inference” is in Breakthroughs in Statistics (volume I 1993). I’ve a hunch that Birnbaum would have liked my rejoinder to discussants of my forthcoming paper (Statistical Science): Bjornstad, Dawid, Evans, Fraser, Hannig, and Martin and Liu. I hadn’t realized until recently that all of this is up under “future papers” here . You can find the rejoinder: STS1404-004RA0-2. That takes away some of the surprise of having it all come out at once (and in final form). For those unfamiliar with the argument, at the end of this entry are slides from a recent, entirely informal, talk that I never posted, as well as some links from this blog. Happy Birthday Birnbaum! Continue reading
At the start of our seminar, I said that “on weekends this spring (in connection with Phil 6334, but not limited to seminar participants) I will post some of my ‘deconstructions‘ of articles”. I began with Andrew Gelman‘s note “Ethics and the statistical use of prior information”[i], but never posted my deconstruction of it. So since it’s Saturday night, and the seminar is just ending, here it is, along with related links to Stat and ESP research (including me, Jack Good, Persi Diaconis and Pat Suppes). Please share comments especially in relation to current day ESP research. Continue reading
Central Identification Laboratory
Guest, March 27, PHil 6334
“Statistical Considerations of the Histomorphometric Test Protocol for Determination of Human Origin of Skeletal Remains”
“Philosophy majors rule” according to this recent article. We philosophers should be getting the word out. Admittedly, the type of people inclined to do well in philosophy are already likely to succeed in analytic areas. Coupled with the chuzpah of taking up an “outmoded and impractical” major like philosophy in the first place, innovative tendencies are not surprising. But can the study of philosophy also promote these capacities? I think it can and does; yet it could be far more effective than it is, if it was less hermetic and more engaged with problem-solving across the landscape of science,statistics,law,medicine,and evidence-based policy. Here’s the article: Continue reading
“Frequentist Statistics as a Theory of Inductive Inference”
“Probability/Statistics Lecture Notes 4: Hypothesis Testing”
*This is Day #5 on the Syllabus, as Day #4 had to be made up (Feb 24, 2014) due to snow. Slides for Day #4 will go up Feb. 26, 2014. (See the revised Syllabus Second Installment.)
Day #1 slides are here.
BOSTON COLLOQUIUM FOR PHILOSOPHY OF SCIENCE
54th Annual Program
REVISITING THE FOUNDATIONS OF STATISTICS IN THE ERA OF BIG DATA: SCALING UP TO MEET THE CHALLENGE
Cosponsored by the Department of Mathematics & Statistics at Boston University.
Friday, February 21, 2014
10 a.m. – 5:30 p.m.
Photonics Center, 9th Floor Colloquium Room (Rm 906)
8 St. Mary’s Street
- Computational Challenges in Genomic Medicine
Jill Mesirov Computational Biology and Bioinformatics, Broad Institute
- Selection, Significance, and Signification: Issues in High Energy Physics
Kent Staley Philosophy, Saint Louis University
- Multi-Resolution Inference: An Engineering (Engineered?) Foundation of Statistical Inference
Xiao-Li Meng Statistics, Harvard University
- Is the Philosophy of Probabilism an Obstacle to Statistical Fraud Busting?
Deborah Mayo Philosophy, Virginia Tech
- Targeted Learning from Big Data
Mark van der Laan Biostatistics and Statistics, UC Berkeley
On weekends this spring (in connection with Phil 6334, but not limited to seminar participants) I will post relevant “comedy hours”, invites to analyze short papers or blogs (“U-Phils”, as in “U-philosophize”), and some of my “deconstructions” of articles. To begin with a “U-Phil”, consider a note by Andrew Gelman: “Ethics and the statistical use of prior information,”[i].
U-Phil (2/10/14): In section 3 Gelman comments on some of David Cox’s remarks in a (highly informal and non-scripted) conversation we recorded:
“A Statistical Scientist Meets a Philosopher of Science: A Conversation between Sir David Cox and Deborah Mayo,” published in Rationality, Markets and Morals [iii] (Section 2 has some remarks on Larry Wasserman, by the way.)
Here’s the relevant portion of the conversation:
COX: Deborah, in some fields foundations do not seem very important, but we both think foundations of statistical inference are important; why do you think that is?
MAYO: I think because they ask about fundamental questions of evidence, inference, and probability. I don’t think that foundations of different fields are all alike; because in statistics we’re so intimately connected to the scientific interest in learning about the world, we invariably cross into philosophical questions about empirical knowledge and inductive inference.
COX: One aspect of it is that it forces us to say what it is that we really want to know when we analyze a situation statistically. Do we want to put in a lot of information external to the data, or as little as possible. It forces us to think about questions of that sort.
MAYO: But key questions, I think, are not so much a matter of putting in a lot or a little information. …What matters is the kind of information, and how to use it to learn. This gets to the question of how we manage to be so successful in learning about the world, despite knowledge gaps, uncertainties and errors. To me that’s one of the deepest questions and it’s the main one I care about. I don’t think a (deductive) Bayesian computation can adequately answer it.…..
COX: There’s a lot of talk about what used to be called inverse probability and is now called Bayesian theory. That represents at least two extremely different approaches. How do you see the two? Do you see them as part of a single whole? Or as very different? Continue reading
I will post seminar slides here (they will generally be ragtag affairs), links to the papers are in the syllabus.
This follows up on yesterday’s deconstruction:
I’m happy to play devil’s advocate in commenting on Larry’s very interesting and provocative (in a good way) paper on ‘how recent developments in statistical modeling and inference have [a] changed the intended scope of data analysis, and [b] raised new foundational issues that rendered the ‘older’ foundational problems more or less irrelevant’.
The new intended scope, ‘low assumptions, high dimensions’, is delimited by three characteristics:
“1. The number of parameters is larger than the number of data points.
2. Data can be numbers, images, text, video, manifolds, geometric objects, etc.
3. The model is always wrong. We use models, and they lead to useful insights but the parameters in the model are not meaningful.” (p. 1)
In the discussion that follows I focus almost exclusively on the ‘low assumptions’ component of the new paradigm. The discussion by David F. Hendry (2011), “Empirical Economic Model Discovery and Theory Evaluation,” RMM, 2: 115-145, is particularly relevant to some of the issues raised by the ‘high dimensions’ component in a way that complements the discussion that follows.
My immediate reaction to the demarcation based on 1-3 is that the new intended scope, although interesting in itself, excludes the overwhelming majority of scientific fields where restriction 3 seems unduly limiting. In my own field of economics the substantive information comes primarily in the form of substantively specified mechanisms (structural models), accompanied with theory-restricted and substantively meaningful parameters.
In addition, I consider the assertion “the model is always wrong” an unhelpful truism when ‘wrong’ is used in the sense that “the model is not an exact picture of the ‘reality’ it aims to capture”. Worse, if ‘wrong’ refers to ‘the data in question could not have been generated by the assumed model’, then any inference based on such a model will be dubious at best! Continue reading
Larry Wasserman (“Normal Deviate”) has announced he will stop blogging (for now at least). That means we’re losing one of the wisest blog-voices on issues relevant to statistical foundations (among many other areas in statistics). Whether this lures him back or reaffirms his decision to stay away, I thought I’d reblog my (2012) “deconstruction” of him (in relation to a paper linked below)[i]
Deconstructing Larry Wasserman [i] by D. Mayo
The temptation is strong, but I shall refrain from using the whole post to deconstruct Al Franken’s 2003 quip about media bias (from Lies and Lying Liars Who Tell Them: A Fair and Balanced Look at the Right), with which Larry Wasserman begins his paper “Low Assumptions, High Dimensions” (2011) in his contribution to Rationality, Markets and Morals (RMM) Special Topic: Statistical Science and Philosophy of Science:
Wasserman: There is a joke about media bias from the comedian Al Franken:
‘To make the argument that the media has a left- or right-wing, or a liberal or a conservative bias, is like asking if the problem with Al-Qaeda is: do they use too much oil in their hummus?’
According to Wasserman, “a similar comment could be applied to the usual debates in the foundations of statistical inference.”
Although it’s not altogether clear what Wasserman means by his analogy with comedian (now senator) Franken, it’s clear enough what Franken meant if we follow up the quip with the next sentence in his text (which Wasserman omits): “The problem with al Qaeda is that they’re trying to kill us!” (p. 1). The rest of Franken’s opening chapter is not about al Qaeda but about bias in media. Conservatives, he says, decry what they claim is a liberal bias in mainstream media. Franken rejects their claim.
The mainstream media does not have a liberal bias. And for all their other biases . . . , the mainstream media . . . at least try to be fair. …There is, however, a right-wing media. . . . They are biased. And they have an agenda…The members of the right-wing media are not interested in conveying the truth… . They are an indispensable component of the right-wing machine that has taken over our country… . We have to be vigilant. And we have to be more than vigilant. We have to fight back… . Let’s call them what they are: liars. Lying, lying, liars. (Franken, pp. 3-4)
When I read this in 2004 (when Bush was in office), I couldn’t have agreed more. How things change*. Now, of course, any argument that swerves from the politically correct is by definition unsound, irrelevant, and/ or biased. [ii]
But what does this have to do with Bayesian-frequentist foundations? What is Wasserman, deep down, really trying to tell us by way of this analogy (if only subliminally)? Such are my ponderings—and thus this deconstruction. (I will invite your “U-Phils” at the end[a].) I will allude to passages from my contribution to RMM (2011) http://www.rmm-journal.de/htdocs/st01.html (in red).
A.What Is the Foundational Issue?
Wasserman: To me, the most pressing foundational question is: how do we reconcile the two most powerful needs in modern statistics: the need to make methods assumption free and the need to make methods work in high dimensions… . The Bayes-Frequentist debate is not irrelevant but it is not as central as it once was. (p. 201)
One may wonder why he calls this a foundational issue, as opposed to, say, a technical one. I will assume he means what he says and attempt to extract his meaning by looking through a foundational lens. Continue reading