Measurement of the W boson mass reveals 7σ deviation from calculations
science.org
science.org
https://non-trivial-solution.blogspot.com/2022/04/do-we-have...
Another take many people in the field are expressing is that it's simply infeasible to reliably interpret statistical models at that level (especially one that is dominated by systematic uncertainty), since they are based on approximations and assumptions e.g. that certain nuisance parameters are "nicely" distributed and uncorrelated. See e.g. comments from Prof. Cranmer [1] who is one of the folks who developed the standard statistical formalism and methods used in modern particle physics experiments.
[1] https://twitter.com/kylecranmer/status/1512222463094140937?s...
In physics experiments they want to fix the structure of the model and know the assumptions. They want to know the distribution and parameters to hold. If assumptions don't hold, they must find out why, find better assumptions and fix the model.
To say it differently: physicists are not trying to discover statistical laws. They are trying to discover physical laws trough statistics.
Is there a concrete reason we can't be naive and just bootstrap confidence intervals for example? Of course I defer to the physicists here – but I'm curious whether there's some simple high-level reason the usual tricks don't work.
See also: https://twitter.com/pietrovischia/status/1512174848558219270
This does not seems like bleeding edge, it seems like "Gaussian approximations for everything."
In fact, this is the reference for the technique they used: https://cds.cern.ch/record/183996/files/OUNP-88-05.pdf. (Although the criticism seems to be leveled at the way they estimate the correlations, not that linear estimator specifically?)
I know these people are incredibly smart and conscientious. And the standard model is extremely successful and well confirmed. But that's a lot of degrees of freedom.
That's... cute. I doubt it will stop the theorists from flooding the arxiv with explanaitions in the coming days/weeks. Recall what happened when there was a barely 3 sigma (local) statistical fluctuation in LHC data:
https://resonaances.blogspot.com/2016/06/game-of-thrones-750...
Edit: Thank you for posting the excellent article!
> In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values.[1]
https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rul...
The "Simple English Wikipedia" is a really underrated resource for understanding jargon outside your field.
In particle physics, sigma denotes "significance", not standard deviation. Technically what we're quoting as "sigmas" are "z-values", where z=Phi^{-1}(1 - p), where Phi^{-1} is the inverse CDF of the Normal distribution and p is the p-value of the experimental result. So, 7 sigma is defined to be the level of significance (for an arbitrary distribution) corresponding to the same quantile as 7 standard deviations out in a Normal distribution.
In other words, "z sigma" means: That a result like this occurs as a statistical fluke, is just as likely as a standard-normal distributed variable giving a value above z.
(The author is a stats professor at CMU.)
Quoting: "The plot shows the first 50 simulations. In the first simulation I picked some distribution {F_1}. Let {\theta_1} be the median of {F_1}. I generated {n=100} observations from {F_1} and then constructed the interval. The confidence interval is the first vertical line. The true value is the dot. For the second simulation, I chose a different distribution {F_2}. Then I generated the data and constructed the interval. I did this many times, each time using a different distribution with a different true median. The blue interval shows the one time that the confidence interval did not trap the median. I did this 10,000 times (only 50 are shown). The interval covered the true value 94.33 % of the time. I wanted to show this plot because, when some texts show confidence interval simulations like this they use the same distribution for each trial. This is unnecessary and it gives the false impression that you need to repeat the same experiment in order to discuss coverage."
Nitpick: this is still a standard deviation in some (potentially very contrived and nonlinear) coordinate system. (As a simple example, a log-normal distribution might have a mean of 1 and a standard deviation effectively of multiplying or dividing by 2. Edit: also, multidimensional stuff might have to be shoehorned into a polar coordinate system.) But in practice you'd never bother to construct such a coordinate system, so that's more a mathematical artifact than anything useful.
Anyway, as I mentioned elsewhere, the motivation for calling it sigma is that, by construction, it maps onto the quantiles of the standard Normal distribution. So an N-sigma result will have the same p-value as N standard deviations in a Normal distribution. So you can associate "sigmas" with "standard deviations of the Normal distribution". Perhaps this is what you are trying to say, but it does not make sigma a standard deviation in any statistical sense, i.e. it is not necessarily related to the variance of the relevant distribution.
Known unknowns and unknown unknowns, as Rumsfeld would put it.
About a decade ago I saw a very nice figure of estimates of the speed of light over time showing this effect. Unfortunately I haven't been able to find it since.
https://www.nhn.ou.edu/~johnson/Education/Juniorlab/C_Speed/...
Edit: here’s some error bars!
https://www.researchgate.net/figure/Uncertainties-in-Reporte...
I think error modeling doesn't get enough attention, nor does error due to model uncertainty per se, and the paper explains the consequences of those two things pretty well from an applied perspective. I also think it nicely integrates model uncertainty, error modeling, and complex systems modeling all at once.
I don't think there's anything really groundbreaking in it but I think it does a good job of explaining the importance of certain things that are often really overlooked.
My assumption is that turbulent changes in pressure cause diffraction, causing the light to not take a perfectly straight path. I don’t know enough about physics to know if that’s right.
https://www.physicsforums.com/threads/how-does-refraction-wo... is the thread I'm basing this on
So instead of the heavy theory, I'd like to see the stuff that made people scratch their heads in the first place.
This is more up to date and specifically on challenges to the SM. Where is physics going? | Sabine Hossenfelder, Bjørn Ekeberg and Sam Henry https://www.youtube.com/watch?v=b8npmtsfsTU&t=2306s
Rutherford [1] showed that atoms consist of a tiny, positively charged nucleus and rather large negatively charged shell. It was hypothesized that electrons are flying around the nucleus like planets around the sun. But we already knew at that point that moving charges emit radiation, which causes the electron to lose energy and move closer to the nucleus. So it should pretty much immediately collapse into a point. Bohr then showed that if you assume that only certain orbits were allowed, it works out pretty nicely. Nowadays we now that there is such a thing as a ground state, meaning the lowest amount of energy the electron can possibly have around a nucleus is enough to keep it moving.
The idea for quantizing things came from observing the black body spectrum. If you sum up all contributions classically, you get infinity. Planck tried to see what happens if you assume that energy comes in little packets instead of a continuous spectrum. He didn't have any justification for it, but it matched the observations pretty well.
Theoretical Concepts in Physics by Malcolm Longair is a mix of history and physics, by explaining how physicists came to discover their theories. I actually don't think it says much about modern particle physics though. It includes quantum mechanics.
Introduction to Elementary Particles by David Griffiths if you just want particle physics. Griffiths also has an intro book on quantum mechanics.
Anyway, the books you proposed look interesting, thanks.
As a sibling poster commented, the blackbody spectrum was also inexplicable from a classical point of view (see https://www.feynmanlectures.caltech.edu/I_41.html Section 41-2), but I think that the specific-heat problem was known before the blackbody problem.
You’re probably thinking of how a proton has a mass of 938 MeV/c^2. This is still a mass and not a voltage. 1 eV (electronvolt) is the amount of kinetic energy that an electron would have after being accelerated though an electric potential of one volt. By the mass-energy equivalence 1 eV is equivalent to a mass of ~1.783x10^-36 kg and a proton has a mass of ~1.673x10^−27 kg.
Yes, that’s what I was thinking. But it seems that there is a problem with the definition of the word “mass”. Clearly there are at least two definitions. First, the weight of an object. Here weight is measured and weight is called “mass”. There is no equivalence, same thing is called weight and mass. Weight and mass are synonyms. This mass has nothing to do with electricity and has nothing to do with motion.
The second definiton of mass is related to electricity and motion. It has no meaning outside electricity. In this case, they accelerate an electric current and measure its kinetic energy and call this kinetic energy “mass”. Again these words are synonyms. Why do physicists like these silly word plays so much, I have no idea.
The definition of mass is subtler, but you seem to confusing units with the definition of the concept. Units are necessary because you need a scale to measure physical properties. You can't measure a length and say that it's "ten". You needs units attached like feet or meters. An eV (electronvolt) is a unit of energy. Just like a kilogram (unit of mass) originally was defined as the mass of a 10cm x 10cm x 10cm cube of water at room temperature, the eV (unit of energy) is defined as the increase in kinetic energy of an electron (which has a fixed and known charge) accelerated across 1 volt. But neither the definition of the kilogram or the eV define the concepts of mass or energy, they just merely define units, which humans chose, used to measure mass or energy.
Now how does mass and energy relate to each other? Simply put, the Special Theory of Relativity, developed by Einstein in 1905, states that mass and energy are equivalent to each other. Now the word "equivalent" has a precise but complicated meaning that I will not explain here (if you do want to understand it, take a course in special relativity). This relation is defined quantitatively by E=mc^2 (Energy E equals mass m times the speed of light c squared). Let's first look at another relationship, distance = velocity * time. This equation can be be rewritten as distance/time = velocity. If we use meters to measure distance and seconds to measure time, we can "divide" the units and define the units of velocity as a meters per second or m/s. Same thing with E=mc^2. We can rewrite the mass m as m = E/c^2, and let the units of mass be eV/c^2. Using eV/c^2 or kilograms or whatever to measure mass has no effect on the definition of the concept mass itself (which you can think of as an intrinsic property of objects independent of units which affects their behavior in known ways).
Why do physicists make all of this so complicated? They don't. It is reality that is subtle and complex and hard to understand. Because the purpose of physics is to describe reality, it has to be subtle and complicated.
(Like if you're trying to predict what happens when speeds approach those of light, you have to make weird relativistic corrections stemming from the observed speed of light being the same for all observers, regardless of their relative speeds.)
If you're lucky to not be in one of those weird cases, objects as spherical things with a sharp boundary in space and using the normal composition of speeds work just fine.
This is important because the weight of that particle was predicted by our generally-accepted theory of how the universe works. If the weight is different, it means the theory hasn't taken into account everything that it should.
https://news.fnal.gov/2022/04/cdf-collaboration-at-fermilab-...
do they stand by the result or is it more of a call for "hey, come have a look at this. we can't explain it."
it's got to be anxiety inducing! (and exciting, of course)
sqrt(6.4^2 + 6.9^2) ≈ 9.4
You can have a look here: http://ipl.physics.harvard.edu/wp-uploads/2013/03/PS3_Error_...
I know nothing about Quantum though, only maths.
If I have IID observations with finite 2nd moment (variance), then their average will pretty quickly converge to a Gaussian distribution. And I can relax a lot of this and still recover a variant of CLT.
Of course maybe the calculation is different, eg it’s not like there are N independent observations, but rather some other complex condition solved for the mean estimate.
I filled this widget https://www.wolframalpha.com/widgets/view.jsp?id=53fa34c5c66...
And got this result https://www.wolframalpha.com/input?i=mean%3D%5B%2F%2Fnumber%...
Note in the graphic that σ is 10000 smaller than μ so the probability to get a negative result is almost zero and you can just ignore it.
High Energy Physics sigma is calibrated to match normal distribution quantiles.
For years I've argued foreign symbols and single-letter variable names mainly seem to serve to keep a walled garden around the sciences, and this was cemented when I eventually went for a master's degree and I was expected to do this as well in compsci to get a better grade even if there is no advantage. If we could just write what we mean, I suspect people would find that more useful even if it makes it look less cultivated and more mainstream.
(To be clear, this is not criticism on the person I'm replying to, but split between the author of this specific title and most of the sciences as a whole because it's a universally supported barrier (if only ever implicitly), aside from a few science communicators.)
Edit: scrolled further in the thread. Looks like I'm not the only one, though this person at least knew to name the sigma: https://news.ycombinator.com/item?id=30955621
Also I'm not sure what you mean by "back", is it referring to what we iirc called story exercises in Dutch primary school ("Jan goes to the store and buys seven ladders, then sells three..." etc.) or was this a thing a few hundred years ago or so?
Mathematical notation really isn't that hard as long as you treat it as its own thing and learn it properly rather than trying to use a likely imperative model of computing programming as a reference point.
And I don't really understand the "I didn't do math and Greek in School". I barely had a foreign language, but if you're actually learning the concept you memorize the letter as well. You can't understand what a wave function is and then not remember that its symbol is Psi. And if you don't know what a wave function is, it won't help to write derivate_2nd_order(waveFunction, time).
EDIT: obviously we're not talking about stories to teach newcomers, you're talking about writing equations in scientific articles and books with words.
Sure this is the "state of the art", but despite the fact that pure language notations might be even worse, i cant help to think that people thinking like the parent might find something even better.
Maybe something inspired by braille notation or something that is invented while trying to understand how our brain works (just speculating here) will be even more expressive.
I actually like seeing an adult be bothered by the fact that the same symbols that turn science more expressive are also the reason that there's a big ladder for newcomers to understand whats being expressed given its all very arbitrary (someone in the XVI century choose a random greek letter to represent X).
Imagine how much science would improve with more "brain power" being also able to try to solve some problems given there are less arbitrarity..
... that's why I said "I'm not proposing to turn everything into English prose. Rather, using (abbreviated) names for variables"
Actually, that might be a good exercise: try doing some moderately abstract equations with variable names such as you'd write in a programming language and you'll find yourself shortening them pretty quickly. We literally do it sometimes when modeling an equation for a new domain: we start by writing words and at the end of the blackboard they already became a symbol.
Using symbols reduces the amount of text your brain has to parse. It makes it much easier to reach consensus on a shared understanding of things. The price to pay is to learn this new notation or language.
It may not make sense for layperson but that’s not really the audience.
Chinese people learn dozens of thousands of ideograms, I am pretty sure the problem with understanding the science has nothing to do with a few Greek symbols.
Life would be hell for any practitioner without single-letter abbreviations. In fact, we like them so much, that's why we adopted the greek letters (we ran out of alphabet). And, for better or for worse, convention runs deep in scientific literature. In practice it reduces a lot of redundancy, makes it more efficient for researchers to skim and understand results. But the cost is a years-long learning curve to break into any scientific field's literature.
FWIW, the linked article is from the journal Science, which is a technical publication. Often "sigma" is omitted in sci-comm articles, or at least is translated for the reader. They will say something like "there is a one in X million chance this is a fluke".
It's much easier to draw a fancy symbol by hand than write several simple letters quickly and legibly, and it also takes much less space.
We've been having the privilege to write using computers for last 20-25 years, when PCs became widespread, relatively cheap, and running good enough software. And this is outside the lecture hall settings anyway.
That is honestly the best argument I've ever heard (you're the first I see mention it). With as much as I hate writing rather than typing, I can see the point there actually. Maybe this practice is not as wholly stemming from elitism as it first seemed.
That's a very small price if you're actually involved with physics regularly, but HN is a relatively mainstream place.
I had physics for 4 years in school but this wasn't part of the curriculum. At some point I asked why we were told (seemingly-to-me falsely) that there were only 3 phases of matter when on google videos I had seen something about superfluidity. The teacher made a joke about my stumbling over that word and then the buzzer went so... that's the kind of physics we had.
And that's for someone who went to school in one of the richest (GDP per capita) and most-developed (HDI) countries in the world. I don't know what it's like for anyone tuning in from a less well-off place, or for someone who had physics decades ago without refreshers (for me it's only a bit more than one decade now).
Something tells me I should have looked for a statistics paper that replaced GDP and HDI with some random symbol and used that instead. That's the kind of thing you're promoting and I just don't see why. TLAs aren't everything but they're better than single letters.
> without them you'd down in re-used letters
eh, literally the opposite? Using (abbreviated) names you'd not drown in re-used letters.
I should clarify, though, that I was thinking of college physics classes, which are definitely more mature, both about exploring new knowledge instead of memorizing facts, and about learning to actually speak in the experts' language.
Using symbols for common concepts without defining them is, however, absurd. (Not counting a few -- c, e, hbar, m, maybe q?)
I personally don't see why greek letters are such a big sticking point, there's only 24 of them, and unlike Greek children you don't have to learn them all in one go.
I’ve seen an increasingly worrying trend of using downvotes to voice disagreement, rather than as the intended purpose as a kind of crowd-based moderation. And before anyone lambasts me for complaining about downvotes, I’m complaining about the trend, where the above comment is just a exemplar.
Actually Paul Graham did intend downvotes to express disagreement. The theory was that if people could express disagreement by downvoting there would be less people posting insubstantial comments to disagree.
It seems to me that brevity is the real excuse here. Moreover, if it were just about symbols but papers were otherwise accessibly written (to reasonable extents, obviously), that would be different still. This is not the case.
Appearances are probably also important for funding. I'd bet that if you submitted same proposal twice, once phrased in a convoluted way and once phrased in a "we're gonna blow up some material multiple times and see how far the shards fly" style, a number of times to independent funding committees, there would be a statistically significant correlation with which proposal gets funded.
And let’s be real. If you couldn’t understand sigma notation in school, the chances that you would comprehend complex science are very low no matter what kind of verbiage or, as it often would be more apt to say, verbal garbage it is wrapped in.
I absolutely agree with you that oftentimes bad research is disguised with ten dollar words. And oftentimes it is disguised with convenient agenda (no matter how true or good this agenda is by itself). But I don’t believe it has anything to do with Greek letters.
This is from the editor's comment at the top of the article, I'm guessing it was a mistake, but that might be why people are getting thrown off by it
Neither is great, actually. With all due respect for Asimov, who I love.
When you add in the "10% chance that some scientist messed up the maths or something in the experiment", then it's impossible to ever reach 7 sigma...
EDIT: Yes, because the Gaussian distribution extends to +/- infinity; davrosthedalek explains it best, below.
However, the 7 here is basically (x - mu)/sigma, so it is normalised (in that sense), anyway.
A good example is efficiency measurements. I can't count how often I have seen students say something like: Our detector is 99%+-3% efficient. Obviously a detector can't be 102% efficient.
I have a master's degree in statistics and this is the first I'm hearing about it.
> Our detector is 99%+-3% efficient. Obviously a detector can't be 102% efficient.
In the absence of any other context I'd guess that they're using an approximation to a confidence interval that might be perfectly fine if the estimated value was nearer the center of the allowable range.
Hearing 99+-3% is a very strong indication that the person used an incorrect way to determine the uncertainty, most likely by taking the square-root of counts. But you are right, if the efficiency would be around 50%, that approximation is not so bad.
(One should not confuse a CI with a range of plausible values, in other words.)
It's more interesting if you calculate the distribution of the sum of rolling 100 dices. It's easy to calculate, becuase μ=100*3.5=35, σ=sqrt(100*105/36)~=17.07... But now the distribution is very similar to a Gaussian with μ=100*3.5=35 and σ=sqrt(100*105/36)~=17.07... https://en.wikipedia.org/wiki/Central_limit_theorem They are not equal because the sum of the roll of 100 dices is bounded between 100 and 600 and the Gaussian is not bounded. For most applications, you can just use the Gaussian instead of the exact distribution.
Alternate reply: Gaussian approximation to the binomial is perfectly valid in all sorts of cases.
I would assume that the implication is that its 7 sigma assuming the measurements were done correctly.
Probability is subjective, in this case because it's dependant on the design of the experiment / quality of the analysis of that experiment to determine a p-value of a given result.
The book "Bayesian analysis in high energy physics" is a short and sweet introduction. If I got the title wrong I'll update it later.