[HN Gopher] Statistical challenges and misreadings of literature...
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Statistical challenges and misreadings of literature create
unreplicable science [pdf]
Author : luu
Score : 60 points
Date : 2024-11-05 06:19 UTC (6 days ago)
(HTM) web link (stat.columbia.edu)
(TXT) w3m dump (stat.columbia.edu)
| blackeyeblitzar wrote:
| On page 11 there is a mention of taking the result of self
| reporting (surveys) at their word. I've wondered about this issue
| not just in science but other situations. For example political
| polling, data point in time surveys, census, etc. Without
| verification, what good is the data? And yet you often see such
| self reported data quoted by articles or papers as if it were
| factual.
| fithisux wrote:
| It is a proxy.
| tsimionescu wrote:
| For many types of data you'd like, there is no alternative. For
| example for political polls, especially early ones where you're
| asking people who they would vote for if so and so were
| running, what other tool could you use? Or how would you verify
| the results?
|
| Of course, given the weakness of the data, you have to remember
| to place very little trust in the results. Even so, "X% of
| people _say_ they would vote for Y " can be an interesting
| information in itself, even if those people were lying.
| graemep wrote:
| Polls may or may not be carried out in a neutral way. A good
| many are deliberately biased, even more have known flaws that
| are glossed over. A humorous take on this I love (from the
| 80s BBC comedy Yes Minister, which is often scarily
| realistic):
|
| https://www.youtube.com/watch?v=G0ZZJXw4MTA
| tsimionescu wrote:
| Oh, absolutely, even in more subtle, unintentional ways
| (order of responses, whether you offer a neutral option or
| not, etc). But that doesn't mean polling is entirely
| useless, or that you need to verify the responses in the
| way that the original seemed to suggest.
| culebron21 wrote:
| Pollsters know about this, and the rule of thumb is that you
| can reliably ask people of what they currently do routinely,
| almost daily. E.g. if you ask them who they voted 10 years ago,
| they might remember it wrong. If you ask what they're gonna do,
| this is completely unreliable.
|
| In political polls, this is the only thing you can do. I guess,
| they have corrections for the real attendance of social/age
| strata.
|
| In deep interviews, the sociologists are more interested in how
| people explain their world picture and what they point at, and
| how react to questions, than to what they tell about
| themselves.
| michaelt wrote:
| _> For example political polling, data point in time surveys,
| census, etc. Without verification, what good is the data?_
|
| Political polls are regularly wrong
| https://www.bbc.co.uk/news/articles/cj4ve004llxo
|
| In my country, supporters of one political party seem a lot
| more likely to tell polling companies they're "undecided" - so
| the pollsters try to adjust their predictions, based on what
| they've seen in previous elections.
|
| But the problem is getting the adjustment right is very
| difficult. The political landscape can change a great deal
| between elections. If John McCain and Donald Trump appeal to
| very different sets of voters, why should voter behaviour
| observed with McCain hold true with Trump?
| HPsquared wrote:
| It's like the uncertainty principle: even the act of asking
| someone a question will likely change their views.
| jjk166 wrote:
| There are multiple levels to data. If 57% of respondents to a
| self reported survey say they are going to do X, it doesn't
| mean that 57% of people are actually going to do X. But if you
| do the same survey again and now only 43% of respondents say
| they will do X, then that is clear evidence _something_ has
| clearly changed, even if you don 't know exactly what effect
| that something will have on X. That is very useful.
|
| The problem is when people only look at self reporting. For
| example if in the previous scenario 57% responded X but only
| 54% actually did X, then someone might naively assume that
| after 43% respond X that 40% will actually do X. Or just as
| naively they could say 54% will still do X. Or they could apply
| any of an infinite number of other models. There exists a model
| that will spit out any given answer for any given input, so
| without that followup work to actually verify and understand
| the underlying mechanism, models are worthless.
| mjburgess wrote:
| We're increasingly aware today of how the media operates cycles
| of self-referential and self-justifying citations: a TV show will
| quote an article that reports "some people" taking an issue,
| which ends up being a quote from someone interviewed for another
| newspaper article.. and so on. This "legitimacy laundering" is
| rampant, and we're now getting towards media literacy levels
| which expose it for many people.
|
| However, most concerning: this is how academia has always worked.
| It's the great absurdity of _peer_ review, and of citation. This
| is how entire fields can sustain themselves with little or no
| scientific validity (esp. see, psychometrics).
|
| We are no where near the equivalent "academic literacy" for
| generally informed members of the public to understand this
| problem. _Entire_ fields can be sustained with zero "empirical
| pressure" to close down. So long as one can cite another who can
| cite another... and somewhere some government body will take
| these citations as prima facie evidence of a research programme,
| then funding will be given and more papers published.
| littlestymaar wrote:
| Economics works exactly like this, but at a fractal level:
| there are many different subfields living in the grand family
| of economists which barely communicate with one another, often
| have dramatically different theories that lead to opposing
| conclusion, none of which even trying to give an accurate
| description of the world but instead promote one particular
| ideological bias.
|
| Every once is a while someone from one chapel is going to
| actually do science by refuting one other chapel's bullshit,
| but it's almost never seen as a good thing by anyone, and the
| Nobel committee sometimes ends up giving the price together to
| people from different chapels even when their work contradict
| each other (Fama being empirically being proven wrong by
| Schiller and both being given the price in 2013 to appease the
| feud).
|
| Economic behaviors are so complex that studying economics
| seriously requires gigantic efforts for little results, and
| people who attempt to do that stand no chance of survival in
| the publish-or-perish competition against the grandiose
| bullshitters, and that's how we get where we are.
| light_hue_1 wrote:
| This happens everywhere. Let me tell you a story about who
| studies language and how.
|
| Linguistics studies language right? It has a few hundred year
| history of doing so seriously. But once AI people wanted to
| also have a look linguists became unhappy. They didn't like
| the idea of someone else saying something about their problem
| with a different set of tools. Those tools don't provide
| explanations that they approve of. Even the thought of
| measuring things over a dataset to check if your linguistic
| theories work made them bristle with Chomsky famously
| rejecting the idea of evidence at scale.
|
| So AI people created natural language processing. Their own
| field to study the same problem. One in a while someone tried
| to do something to bridge the gap which led to the famous
| "every time I fire a linguist my performance goes up".
|
| Many people in NLP see linguists as outdated fossils. Most
| would probably vote to just shut down linguistics departments
| I suspect.
|
| Then something funny happened. The LLM people did to NLP what
| they did to linguistics. And the NLP community rejected them.
| Because LLMs don't provide an explanation that they approve
| of. Recently just like in linguistics the President of ACL,
| the main NLP organization, said recently that NLP is not
| about machine learning! The absurdity of that statement just
| boggles the mind. So the LLM people now have their own
| conference and field.
|
| A lot of people cannot accept that things must change over
| time. They want to do research the way they did it 10 or 40
| years ago. So they create an immune response. It's human
| nature.
|
| As someone who doesn't feel this way and enjoys moving
| forward, I'm really sad about what's happened to NLP in the
| past few years. It would be nice if we could stop this rift.
| But it won't happen with the current hate filled President. I
| stopped sending any work there until they reform.
|
| Just the cycle of academic life. I wonder if science would
| move faster if we could select for people who are willing to
| change with the times.
| mjburgess wrote:
| The goal of physics isn't to produce video games -- that a
| video game can generate a frame using unphysical formulae
| and unphysical processes (eg., rasterisation) does not
| invalidate physics.
|
| There's linguistics which produces descriptive theories of
| the practice of language. There's "computational
| linguistics" which aims to build discrete algorithmic
| models of language. Both of these are interested in
| explanations of actual physical phenomena.
|
| Machine Learning is largely an engineering discipline
| which, like video games, can generate output using any
| method which will convince a human user. Indeed, often more
| radically, ML works by replaying variations of past
| convincing output rather than have any explicit simulation
| involved at all.
|
| Thus there's no sense in which LLMs supercede linguistics,
| nor somehow computational linguistics superseding this.
|
| This would be like claiming rasterisation means we should
| retire optics and the physics of light.
| HPsquared wrote:
| That's a nice dividing line, science and engineering are
| different for sure. But science makes use of engineering
| knowledge and engineering makes use of scientific
| knowledge.
| light_hue_1 wrote:
| That's like saying that it's not astronomy if I'm using a
| radio telescope, only optical telescopes are real
| astronomy. It makes zero sense. The tools are irrelevant.
| We don't have physics with and without ml journals.
|
| It's like saying AlphaFold isn't biology because it uses
| ml. It's absurd
|
| One goal of linguistics is to produce an account of how
| language works. If you understand how something works you
| can reproduce it. This was always a core part of
| linguistics. Just look at Chomsky's work. Or at what
| counts as evidence in a linguistics paper. It is a model
| that "explains" some feature of language.
|
| NLP does exactly the same thing as linguistics. It
| produces explanations and makes predictions about
| language. But it does so with ML.
|
| I think you're confused about optics and rasterization.
| There are countless ways to relate the two. Rasterization
| is not unphysical, no more so than say, finite element
| methods are unphysical. Heck physics is in the name of
| PBR. It's an approximation. And a fine one that people
| actually use in both science and games.
| mjburgess wrote:
| My master's project was applying non-gradient-based ML
| methods to parameter optimisation in quantum metrology --
| I'm aware that the reskin on curve-fitting algorithms
| called "Machine Learning" may be used as part of science.
| We've had taylor series approximations to functions for
| 300 years.
|
| But insofar as we're talking about LLMs, and most
| products you can use, we're talking about an engineering
| use. Engineers build systems that "perform" as you say
| explanatory accuracy is not a kind of engineering
| performance.
|
| It would equally well be said that, for a video game
| studio, "the more physicists I fired, the more the games
| perform better" -- for, of course, physicists have never
| studied 'making a video-game image convincing', and their
| techniques are best run on supercomputers, since they'd
| build explanatory simulations with explanatory accuracy.
| Video game developers do not do this. I have developed
| video games also, so I know.
|
| Convincingly placing pixels on a screen _as-if_ governed
| by the laws of physics does not require actually
| simulating them. It requires generating frames from a
| "point of view" that never expose the absense of real
| physics, or approximations thereof. The world isnt made
| of triangles, and you cannot clip thru solid objects.
|
| The goal of a person making an LLM is similar to the goal
| of a video game developer, these are engineering goals:
| to give a user an experience which they will pay for.
| Now, a 2d 80s video-game has more actual physics in it
| than an LLM has models of language use, but still, this
| is irrelevant to the goals of these creators.
|
| A linguist could analyse an LLM as the conditional
| probability structure of the english language, as
| captured largely recent electronic texts -- but this
| structure is partly what precisely linguists and others
| are trying to explain.
|
| This is why all curve-fitting to historical data isn't
| _in itself_ science: it is a restatement of the very
| target of explanation, the data. The job of scientific
| fields is to explain that curve.
|
| Its explotation by engineers is explanation-free. It isnt
| a science, and replaces no science.
| light_hue_1 wrote:
| And my PhD and dayjob is doing research on this.
|
| If a student told me they had this view of what ML is, I
| would tell them that we've failed to educate them.
|
| The thought that physics doesn't care about performance
| or approximation is silly. Just look at AlphaFold. Heck,
| I talk to climatologists and material scientists that
| want the equivalent all the time.
|
| Prediction is the heart of all science. Whether we're
| talking neroscience, linguistics, physics, etc.
|
| You think people who run things on supercomputers want to
| do so for some idealistic notion of what science is? No.
| They have to do so because they don't have good
| approximations. Just like with protein folding. Places
| like DESRES used to build supercomputers for that. This
| is over now.
|
| ML models learn representations of data which can then be
| reused for many tasks. There's a whole field where we try
| to understand those representations. Those are
| explanations of what's going on given that they're such
| good predictions. The embeddings you get from an LLM are
| better models of language than anything linguistics ever
| accomplished. Their goal should be to explain them and
| probe their limits instead of complaining. If linguists
| has come up with gpt they would have been celebrating,
| the method by which you do science is irrelevant as long
| as it works.
|
| I'll close by quoting Dawkins. "Science. It works,
| bitches". That's the value of science. Can we predict
| which molecule will cure cancer? Everything else is
| ideology and silly thoughts from before the paradigm
| shift.
| mjburgess wrote:
| Prediction is not the heart of science, this is early
| 20th C. mumbojumbo and humean nonesense that gets
| repeated by curve-fitters because it's all they do.
|
| Explanation is the heart of science, not prediction. All
| predictions newton would have made of the orbits of the
| planets would have been wrong (and so on). And this goes
| for the vast majority of textbooks physics when its
| applied to very many ordinary situations: no predictive
| power at all.
|
| ML models learn "representations" of the _data_ , yes.
| They are models of measures. The model is just an
| f=sample({(all possible measures,)}). Science provides
| representations of the data generating process, ie.,
| _reality_. It says _why_ those are the measures, _why_
| the temperature of gas has that value, not a report on
| what those values were. Nor even a compressed conditional
| probability model _of those values_ -- there is no atomic
| theory in a zip of temperatures.
|
| The purpose of a scientific model is to _explain_ these
| data-representations, not merely to predict them based on
| some naive regularity assumption about the measurement
| device: that it will always measure that way in the
| future.
|
| The reality of the ML is that it offers only intra-
| distribution generalisation, nothing of the kind of
| generalisation science offers where all possible
| distributions induced by intervention on the
| (explanatory) variables of scientific models are
| captured. And this is often a scam that only academics
| can get away with, ex hyp., just assuming that the test
| distribution "will turn out as expected" in premise.
|
| The reality is that this sort of repetition of historical
| data, requires extreme control over the data generating
| process which gives rise to the test distribution. How is
| that control delivered in practice? If its a medical lab,
| through untold amounts of toil delivering, say,
| histological slides "just right" so this dumb process
| almost works. If its a face tracking, well you'd be hope
| its not being used by the police -- because they aint
| orienting the camera at 3.001m at ISO 151.5 from the
| masses.
|
| This is the problem with models that obtain predictive
| power without explanatory content: they rely on
| prediction time being rigged with, in-practice, extreme
| control mechanisms that are wholly unstated and unknown
| by the "modellers". Because these are no models of
| reality at all, but mere repetitions of historical data
| with unknown, _unmodelled_ and hence _unexplained_
| similarity.
|
| Science concerns itself with explanation, that is its
| goal. Prediction is instrumental. Engineering's goal is
| the utility of the product, and so _any strategy
| whatsoever_ , even a dumb, "if its happened before, itll
| happen again" is permitted.
|
| There is no textbook of physics which models reality by
| saying, "Well, we suppose in the future, the positions
| and velocities will just follow the same distribution,
| but we've no idea why, and how dare you ask, and get out,
| and doesnt my Ideal-Gas-TransformerModel look pretty? It
| gets the pressure right for Argon at 20.0001 C in glass
| jars at about 2.002L"
| naasking wrote:
| Prediction and explanation are basically equivalent IMO.
| A predictive model entails an explanation, and an
| explanation entails a predictive model. Predictive models
| that are more accurate are more accurate explanations,
| and predictive models that are more precise are more
| precise explanations, and vice versa. They are not that
| distinct.
| mjburgess wrote:
| They are highly distinct.
|
| Compare reporting the temperature tomorrow as a mode of
| all temperatures in November at your location, with a
| climate & weather simulation involving: cloud layers, the
| ocean, etc.
|
| The former is likely to be vastly more predictively
| accurate than the latter, but explains nothing.
|
| Explanatory models are often less predictively accurate
| than these (weakly inductive) predictive models. Their
| purpose is to tell us how reality works, and that
| provides some insight as to when we can adopt merely
| predictive approaches. This is because merely predictive
| models capture accidental features of measurement which
| hold up for awhile in some environments, that we wouldn't
| wish to explain.
|
| Without explanatory insight we find merely predictive
| models catastrophically collapse, and are otherwise,
| highly fragile. Eg., consider the performance of "predict
| the mode" in a snow storm.
|
| If you want a midly formal analysis of the difference:
| explanatory models quantify over causal properties of
| reality, describe their relationship, and provide
| _necessary_ inferential methods for deducing conclusions
| from models. They permit arbitrary simulation across all
| relevant measuring systems.
|
| Merely predictive models quantify over historical
| measurements, _assume_ similarity conditions across them,
| and _assume_ future similarity to past cases. They
| provide only extremely weak inferential grounds for any
| inference. They can offer only repetition of one kind of
| measure, and cannot simulate the state of other relevant
| measuring devices or in different environments of
| measurement.
|
| Explanatory models describe reality. Predictive models
| describe the measurement device you happened to use, in
| the environment you specifically used it in.
| nialse wrote:
| I enjoyed your conversation and just want to chip in that
| there are as many definitions of science and knowledge as
| there are philosophers. One don't have to have only one
| definition, but usually one have to adhere to the ones
| within the realm of ones scientific paradigm to be
| accepted and to develop the science. Normal science as
| Kuhn called it.
| GPerson wrote:
| There's no single forward direction in science. We're in a
| machine learning hype cycle. That doesn't mean we must fire
| everyone in every other department.
| pacman001 wrote:
| Every single time (some 4 times in my past life) when I'm
| familiar with/close to the background story of something that
| played out in the news I come to the same conclusion: news
| doesn't report the facts. I'm in Western Europe, btw. I advise
| to run this experiment yourself.
| littlestymaar wrote:
| That's the problem when you have people responsible for news
| having the business model of ad-sposored entertainment and
| not fact reporting.
| pacman001 wrote:
| The conclusion I'm coming to is that this, too, is an
| exponent of an inherent weakness: we like to be told
| stories. As children we like to be told stories and this
| remains. We now have a complete spectrum of story-telling:
| science fiction, fiction, non-fiction, ..., and news. News
| reporting itself is a continuum, with tabloid gossip on one
| side and political coverage on the other. We have so much
| stories to choose from.
| zorked wrote:
| In Western Europe a lot of the media isn't ad-sponsored.
| aeim wrote:
| Just "bad-sponsored"
| littlestymaar wrote:
| I live in France, and saying that 90% of the media are
| ad-sponsored is a conservative estimate.
|
| So I don't know which part of western Europe you're
| talking about, but it clearly doesn't apply to all of it.
| pbrowne011 wrote:
| Another name for a similar but slightly more pernicious
| problem is the Gell-Mann Amnesia effect: "the phenomenon of
| experts reading articles within their fields of expertise and
| finding them to be error-ridden and full of misunderstanding,
| but seemingly forgetting those experiences when reading
| articles in the same publications written on topics outside
| of their fields of expertise, which they believe to be
| credible."
|
| https://en.wikipedia.org/wiki/Michael_Crichton#Gell-
| Mann_amn...
| mrguyorama wrote:
| No journalist will ever have the expertise required to
| accurately report scientific results.
|
| Instead we need to help average people understand that when
| the news says "Scientists say drinking red wine is healthy!"
| that _No actual scientist ever said that!_
|
| Instead, the journalist writing the segment cribbed notes
| from the University's PR page about the study, which was also
| written by someone with zero science background, and in fact
| almost always has a _marketing_ background.
|
| Oh, and those PR releases are outright stating false things
| that the actual scientific paper doesn't even discuss like
| half the time.
|
| And now you have decades of people insisting that nutrition
| science is awful, even though nobody in academia or science
| is saying any of the things the average person thinks they
| have.
| dash2 wrote:
| I think science journalists often have _some_ science
| background, and university press releases are in my
| experience collaborations between the scientist and the PR
| person.
| voidhorse wrote:
| Totally, and I think there is a fundamental, deeper, inherent
| problem in using statistics to determine how you want to
| _manipulate_ an given object of study.
|
| Researches find that, on average, consumers want X. Companies
| decide they want to maximize reach and so begin to produce X.
| Consumers soon have little choice except for X, reaffirming, to
| future researchers, that consumers want X.
|
| This is why I think a plurality of data is so necessary
| especially when it comes to anything in the social domain, and
| why it's imperative that we begin to invest more into cross-
| disciplinary research. Specialization has gotten us far, but
| it's starting to lead to breakdowns. The only thing that might
| offset the statistical observation of consumer behavior is a
| statistical study of consumer _opinion_ that proves to refute
| or contradict that behavior... but then in response some people
| will say "people don't know what they want" further
| reaffirming the conclusions that the action based on the data
| itself caused (behavior or observation bias). The application
| of statistics to social problems essentially becomes a self-
| fulfilling prophecy.
| mjburgess wrote:
| I fully agree. How many IQ effects in studied populations are
| actually created by the use of IQ tests?
|
| Consider how infiltrated IQ-like assessment in throughout
| society, selecting for doctors, lawyers, postgrads --
| military, police, etc. Then consider what data is offered as
| evidence that IQ 1) exists, and 2) causes observable measures
| in real-world outcomes. Filtering on the test becomes
| evidence the test is a measure of anything.
|
| The application of "statistics" I dislike the most is where
| these feedback cycles exist, and large swathes of academia
| have some extreme responsibility here.
|
| Whole fields of gene-traits studies were created over decade+
| and then disappeared overnight as actual sequencing took
| place. All the rigour and splendour of "statistics", and then
| poof when _science_ was done, it disappeared.
|
| Since there are basically no scientific theories of human
| psychology, society, and the like -- gluing together
| correlations here should be seen as prima facie absurd. The
| alternative? Rely on expertise, and build resilience-to-
| failure into the system and tolerance for higher variability.
|
| Human expertise obtained in domain-specific environments is
| vastly superior to the species correlations of surveys
| written by idiots who've never done any actual science.
| InDubioProRubio wrote:
| Could one create a proof of pseudo-science, by injecting a
| faked fundamental corner stone paper, that becomes proof by
| inheritance that a full field is rotten?
|
| Also why does this remind me of european politicans, claiming
| everyone wants to life european lifes, meanwhile whole
| countries goto war and atrocities without big counter-
| demonstrations by those western valued citizens .. narrative
| glider guns going ad absurdum..
| kenjackson wrote:
| > Could one create a proof of pseudo-science, by injecting a
| faked fundamental corner stone paper, that becomes proof by
| inheritance that a full field is rotten?
|
| I don't think there's any doubt that pseudoscience exists,
| even amongst the most optimistic of scientists.
|
| The problem is identifying what is bad science vs what is
| good. The fact that I can send this from a small phone from a
| parking lot is proof that someone did good science at some
| point in time. Or that I've seen therapies like CBT turn
| someone who was struggling mentally on a daily basis to
| thrive, or that I've seen valve replacement surgery give
| someone years of great life after a "six months to live
| diagnosis" -- all show that there is good science.
|
| I think we almost need a "discipline" of people who validate
| scientific results, and people should be held accountable for
| results that validate or don't.
|
| Peer review is great. It's not a farce (I've gotten some
| incredible feedback on papers from it), but it is also
| extremely limited by design. We need more.
| jjk166 wrote:
| Even a broken clock is right twice a day. I can publish a
| paper with results I think are wrong, but it's entirely
| possible that follow up studies confirm it was accidentally
| correct. While this is improbable in general, if we restrict
| ourselves to publishing claims that are sufficiently
| plausible that people in the field would accept them as a
| corner stone paper, then there is a very decent chance it
| sounds plausible because it's true. Even if I test the claim
| myself before publishing to confirm it's false, I may have
| made an error.
|
| Possibly repeatedly publishing bogus papers in a certain
| manner might be able to confidently weed out poor academic
| hygeine, but it's not a trivial thing.
| brabel wrote:
| Great video about that by Kurzgesagt:
| https://www.youtube.com/watch?v=bgo7rm5Maqg&ab_channel=Kurzg...
|
| They show how deep they had to go to find the original source
| of the claim that a single human's blood vessels, if lined up,
| would stretch 100,000km... and how that was quoted by so many
| that no one really know where the claim came from. And of
| course, it was wrong.
| naasking wrote:
| Damn it, I had just posted this video too. Off to delete it.
| renewiltord wrote:
| This is the danger with non-empirical government funded
| "science". Empiricism allows verification. Private funding
| means that things that matter get studied.
|
| If you trace the oft-cited claim from "studies" that cats kill
| so and so animals a year you will find it's just someone's
| Fermi estimate. And you'll find that the person has a personal
| distaste for housecats.
| rscho wrote:
| The idea that private funding means that what is studied
| matters is highly dubious at best. Most private research
| pertains to application and product development, and often
| relies on fundamental discoveries coming from academic labs.
| Only academia can afford to let scientists run loose. It
| certainly results in a good amount of inapplicable theories,
| but is nevertheless very much essential.
| renewiltord wrote:
| I doubt gender studies is essential.
| ykonstant wrote:
| Besides the sociological problems listed, we must always be
| conscious of how counterintuitive and difficult statistical
| inference itself can be. Good things to search for are
| statistical fallacies, probabilistic paradoxes and books like
| Counterexamples in Probability.
|
| And it is not sufficient to read about them once or twice;
| researchers who use statistical inference regularly must revisit
| these caveats at least as regularly.
|
| Myself, I have taught Probability and Statistics many times,
| discussed and dispelled many misconceptions by students. Would I
| be 100% sure I will not be caught up in a fallacy while
| informally thinking about probability? I wouldn't even be 10%
| sure; any intuition I conjure up, I would triple check as
| rigorously as possible.
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