PDFs after all these years
I've only just realized that for years I've misunderstood, or perhaps had the wrong mental model, for the proton parton distribution functions (PDFs) and the structure of the proton. The PDF for a parton species tells us the number density of partons of that species with a fraction \(x\) of the longitudinal (or strictly speaking light-cone) momentum of the proton.
There are various sum rules that PDFs must satisfy. For example, the fractions must sum (in expectation) to one, $$ \sum_i \int_0^1 x f_i(x) \,\text{d}x = 1, $$ and there should be 2 up valence-quarks in the proton, $$ \int_0^1 \left(f_u(x) - f_{\bar u}(x)\right)\,\text{d}x = 2 $$ Now what's that formula about? Well, we don't distinguish between valence and sea quarks in the PDFs. The up quark could be created in the quark sea, popping in and out in pairs \(u\bar u\). The above formula says that the number of sea + valence up quarks minus the number of sea anti-up quarks equals 2.
The PDFs depend on a scale, however, called \(\mu\), and we write them as \(f(x, \mu)\). What's the meaning of that scale? I'd always thought of this scale as the scale at which we probed the proton. That we were observing the proton in some way that interfered with it or made it reveal different structure or that the proton was physically changing because of our probing it.
This probably won't lead to any mathematical errors, but it did leave me confused about what the structure of the proton really was. I think I almost saw it quantum mechanically, that you can't look at it without disturbing it, and that it reacted to the scale that you probed, \(\mu\).
The idea of the proton changing as we prod it is a terrible mental model; let's build a better one. What actually matters here is what we mean by a parton. We know that inside a proton, as well as the valence quarks \(uud\), there's a soup of partons popping into and out of existence; these are the sea partons. The partons are surrounded by this mess all the time. We can simplify things by saying that any mess around a parton, at energy less than \(\mu\), is just part of that parton; they are its dressing. Any mess above that scale counts as extra partons. The PDFs depend on that choice; of course they do! We are just saying here that parton distribution functions depend on how we define a parton.
Thus the hard scale is just something we pick to define what we mean by a parton. In order to make our calculations as accurate as possible, we are best to pick it to be a scale that minimizes higher-order logarithms in perturbative calculations. Formally, though, our calculations, which involve PDFs and hard scattering elements, shouldn't depend on our choice of scale, up to higher-order corrections. The DGLAP equations ensure that the scale-dependence of PDFs cancels with that of the hard-scattering elements up to higher-order corrections.
Hard Times
Read Hard Times by Charles Dickens over the holidays. Whilst I thoroughly enjoyed it, I don't consider it amongst my favourites of Dickens' novels. On a personal note, I was glad to have the stamina to read a classic and that my focus hasn't been entirely sapped by smartphones and the internet.
The book is set in the fictional industrial town of Coketown in the north-east of England and revolves around the factory owner, Mr. Bounderby; the utilitarian schoolmaster, Mr. Gradgrind, and his family; and an honest hand at the factory, Stephen Blackpool. Times are hard for everyone, though in different ways. Mr. Bounderby legitimizes his status and treatment of the workers by mythologizing himself in every breath and creating a ridiculous and false rags-to-riches story. Mr. Gradgrind, so committed to a utilitarian approach to education and life, deprives himself and his family of feeling and emotion. Blackpool, though honest and hardworking, becomes ostracized from the workers, wrongfully accused of robbery, and dies after falling into a dangerous disused mine shaft. The fatalistic plotlines reminded me of Thomas Hardy, as characters seemed doomed from the start.
Dickens succeeds in satirizing Bounderby and Gradgrind, and through them the utilitarian philosophy they represent, though fails to grapple with the economic and class relations that led to such hard times for all. Dickens was a reformist and his writing was his activism: he thought that through storytelling he'd change society. If we felt how Stephen Blackpool felt, we'd improve the lives of factory workers. Dickens attributed hard times to personal or idiosyncratic failings of factory owners, which could be remedied by moral suasion. He wasn't a Marxist, and didn't connect the hard times, and moral failings, to class interests and class morality, and the economic structure of the industrial town.
I can't end without mentioning Mr. Sleary and the travelling circus, the antidote to Gradgrind and Bounderby. They are, perhaps, the only characters who aren't emotionally or morally diminished by hard times, though they experience material hardship. They aren't formally educated by Gradgrind's standards but they have their fancies - their imagination, silliness, affection and feeling - which we shouldn't forget to value in our own lives.
I obtained my copy of the book through Standard Ebooks, a volunteer project to create beautiful digital editions of public domain books.
Double reading, writing and descent
There are several doubles in my title, for I've been reading two books, writing two sets of teaching notes, and thinking about double descent, the phenomenon in deep learning.
On the reading, as planned I've started a more substantial novel, Hard Times, considered by some critics to be Dickens' most complete or perfect work, since it is compact and without the sprawling plots and character ensemble of Dickens' other celebrated works. I was worried it would lack warmth and affection, but so far I am thoroughly enjoying it and its connection to educational theory.
I am also reading the wonderful Mrs Frisby and the Rats of NIMH with my daughter. This is everything a children's book should be, and I will write more when we have finished it.
As for writing, I am continuing my In Your Face For Anybody series, and am partway through an introduction to chatbots, starting with ELIZA in the 1960s and ending with contemporary large language models. It is a work in progress [pdf], but I'm enjoying creating it. The series began with my notes on quantum mechanics and Bell's theorem [pdf].
As it's the beginning of term, I also dug up my data analysis booklet for our lab students [pdf]. The booklet was inspired by one given to me in my first year as an undergraduate, though it's not as detailed or in depth, or perhaps as insightful. The ones we were given in 2005 were referred to as our red books and they were a superb practical guide to data analysis in the laboratory and some frequentist statistical concepts. I think they were written by Prof. Tom Shanks, more affectionately known as Dr. Tom at that time, though it may have been a different Tom.
Last but not least: Double Descent! This is a remarkable phenomenon in data science, that at first seems rather mysterious. Consider linear regression on \(n\) data points using \(p\) basis functions with \(p\) coefficients, $$ y = \sum_{i=1}^p \theta_i f_i(x) $$ We train our model on the \(n\) data points; select a point estimate for the parameters; and evaluate performance on a hold-out data set of another, say, \(m\) data points.
What happens when we increase \(p\) starting at \(p \ll n\)? At first, the model is under-parameterised. It lacks flexibility to fit the data points; it is biased, but not wild, that is, it does not suffer from high variance. As we increase \(p\), the performance improves as the model is sufficiently flexible to fit the data. This is the first descent.
As we increase \(p\) yet further, performance deteriorates. Our model becomes too wiggly; we are no longer primarily bias-dominated, but we suffer from variance. At \(p \approx n\), we perfectly fit the training data, but performance on the test data is poor, as we have overfit. This is the interpolation limit.
In classical statistics, that would usually be the end of the story. We find the optimal balance between bias and variance, and that's that. Here comes the twist though. Keep on increasing \(p\) beyond \(n\). Something strange can happen. The system is overparameterized and underdetermined, and there are many ways of achieving a perfect fit to the training data. The performance on the hold-out data improves; this is the second descent.
As we increase the number of basis functions, we are able to create combinations of basis functions that are smooth but perfect fits through the training data. Nothing in our training, however, favoured smoothness: we fitted to the training data without any regularization term. Why, then, are these smooth solutions selected? They are selected by the implicit bias of the gradient descent optimizer in this context. Whilst there are many solutions that perfectly fit the training data, the gradient descent algorithm is biased towards the solution that minimizes the \(L_2\)-norm of the parameters. That is a smooth solution.
The long and short of it is that by increasing \(p\) we create more possibilities for smooth functions that fit the training data perfectly and generalise well. Since we are using point estimates, there is no explicit Occam-type penalty for increasing \(p\). These smooth interpolating functions, that weren't selected or available near the interpolation threshold, are selected by the implicit bias of the optimization algorithm, and being smooth functions that fit the training data, they typically generalise well.
Tags: reading, writing, double-descent
Baryogenesis, finally
Sakharov's third condition for baryogenesis [A.D. Sakharov, Zh. Eksp. Teor. Fiz. Pis'ma 5, 32 (1967); JETP Lett. 91B, 24 (1967)] is usually stated as: departure from thermal equilibrium or interactions out of thermal equilibrium or some such. I've always had some trouble with the textbook arguments for it, but I think I've now had a personal epiphany.
The textbook argument, e.g., Trodden's notes, considers the expected baryon number in equilibrium: $$ \langle B \rangle = \text{Tr}\left(e^{-\beta H} B\right) $$ Inserting an identity, \( (\text{CPT}) (\text{CPT})^{-1} \), using the cyclicity of the trace and the invariance of the Hamiltonian under CPT: $$ \begin{align} \langle B \rangle =& \text{Tr}\left((\text{CPT}) (\text{CPT})^{-1} e^{-\beta H} B\right)\\ =& \text{Tr}\left(e^{-\beta H} (\text{CPT})^{-1} B (\text{CPT}) \right)\\ =& - \text{Tr}\left(e^{-\beta H} B\right)\\ =& -\langle B \rangle \end{align} $$ and thus \( \langle B \rangle = 0 \) in equilibrium.
The thing that confused me was that it seemed that we almost begged the question. We started by assuming that \(H\) was the only relevant macroscopic parameter. We didn't include any chemical potential for baryon number in our ensemble. Suppose we took: $$ \langle B \rangle = \text{Tr}\left(e^{-\beta H + \mu_B B} B\right) $$ The partition function now no longer commutes with CPT and we cannot conclude that \(\langle B \rangle = 0\). Try it. The above argument won't go through. Can we do that? I don't think so. Sakharov's first condition for baryogenesis was that \(B\) was not conserved. In which case, there is no chemical potential in the equilibrium ensemble.
But what if we can give a chemical potential to some other conserved quantity? What? Well, \(B - L\) is a natural candidate, as unlike individual baryon and lepton number, it is conserved. In that case, we could write the expectation of baryon number as, $$ \langle B \rangle = \text{Tr}\left(e^{-\beta H + \mu_{B - L} (B - L)} B\right) $$ Because of the chemical potential, we cannot conclude that \(\langle B \rangle = 0\). Is this a loophole to Sakharov's conditions? No, we've in fact rediscovered leptogenesis! We give a chemical potential for \(B - L\) and equilibration redistributes the charges such that \( \langle B \rangle \neq 0\). We convert a \(B- L\) asymmetry to a baryon asymmetry.
Did this require a departure from thermal equilibrium? Well, under standard cosmology, there are no chemical potentials after reheating, as the inflaton decays into a hot thermal bath with no chemical potentials. Creating a chemical potential for \(B-L\) from scratch requires an out-of-equilibrium process. Thus Sakharov's third condition holds.
What if we don't assume that there are no chemical potentials after inflation? Maybe inflation can produce a chemical potential for \(B-L\)? Is that a loophole? No, we've in fact discovered inflationary baryogenesis and changed the era in which baryogenesis occurs.
What if we don't assume inflation at all? In that case, indeed, we can input what we like as an initial condition, e.g., an equilibrium state with non-zero baryon number.
Tags: baryogenesis, statistical-mechanics, physics
AI & Misogyny
Read The New Age of Sexism: How the AI Revolution is Reinventing Misogyny by Laura Bates. This was an urgent and alarming book, that occasionally trod close to but avoided alarmism, about the interplay and impacts of AI and technology on sexism and discrimination.
The book at times read like a thriller: it was emotive, shocking and fast-paced, but wasn't particularly theoretical, either on technology and AI; on the causes and origins of sexism; or on our economic system and the rise of big tech and monopoly capitalism. The book works, however, as an eye-opener about where we are already headed and as a passionate call to arms to think and do something about it.
Instead of sacrificing anything and everything to the whims of men like Mark Zuckerberg, Elon Musk, Phillipp-Fussenegger, Cameron-James Wilson and a thousand others like them, instead of breathlessly enabling their ruthless pursuit of financial profit in the name of progress, we need to start from a place of sustainability, of fairness, of real, social progress, not technological development for the sake of it ...
... a different baseline of what progress looks like, what its goals are, who it serves and how it happens.
17 papers, 2.6 sigmas, and 1 event
The LZ experiment recently announced (1 September) a new result through a talk and preprint paper.
The paper documents something strange: a single, unexplained event that could be consistent with an interaction with a Dark Matter particle. The data were analysed using frequentist, and specifically Fisherian, statistical methods described here. In particular, evidence was quantified using a \(p\)-value, that is, the probability of obtaining data that were at least as extreme as that observed given the background-only (i.e., no Dark Matter) hypothesis. The \(p\)-value was about 0.5%. For historical reasons, this is communicated as \(2.6\sigma\), as 0.5% corresponds to the probability mass contained in the tail of a Normal distribution at \(Z = 2.6\).
Hey ho, you might think. That ain't much evidence for an extraordinary claim, especially since it is contingent on their background and detector modeling to the 1 rogue-event level, and given all we know about \(p\)-values, the replication crisis, and the strength of evidence. That said, it might just about be below a threshold proposed by Benjamin et al. (2017) in social and medical sciences, thought not close to the physics typical requirement of \(5\sigma\) for a discovery.
In the two days that followed the paper, however, there were 17 theory papers that presented particular models of Dark Matter that could explain that 1 event. Hmm. I haven't read them well enough to comment on how well-motivated these explanations are, though the most popular appears to be a Higgsino explanation (see e.g., Fan & Reece), where a supersymmetric partner of the Higgs plays the role of Dark Matter. However, a paper today suggests Higgsinos may be in tension with so-called sideband results from LZ. That means, they would predict Dark Matter signals in other searches performed by LZ, but none were seen.
There was a particular statement in Fan & Reece that caught my eye:
The look-elsewhere correction depends on the total number of models that were fit to the data, and as such is arguably too conservative when, as we will argue, there is one particularly compelling model to focus onHmm. One of the infamous things about \(p\)-values is that they depend on the experimentalists' analysis plan. That is, they depend on what the experimentalists would have done, were the data different. Part of computing them correctly thus involves correcting for the look-elsewhere effect (LEE). This means, correcting for all the things you, the experimenter, would have looked at had the data been different. There is a famous reductio ad absurdum about a statistician and electrician, by Edwards and Pratt, described here that shows how strange this can be. The long-and-short of this is that the \(p\)-value depends on the experimentalists' analysis plan, which should ideally be declared before data collection. I don't think it's reasonable to claim after the fact that
Ah! But I was only interested in this subset of tests anyway (which happened to include tests that produce the smallest \(p\)-values). Let's just use those ones. Thus the \(p\)-value is smaller!This would have been perfectly fine pre-data. I would have no objection to pre-data, or pre-unblinding of data, someone saying, let's use an analysis plan that only performs these tests to maximise the statistical power for these particular Dark Matter models that we find plausible. Post-data, however, it's too late, at least within a frequentist paradigm.
On the other hand, perhaps Fan & Reece are subconsciously thinking as Bayesians. In Bayesian frameworks, we don't need to consider the sampling plan or what we'd have done were the data different: our conclusions depend only on the data we did observe. This is connected to the likelihood principle. In that case, perhaps a more charitable interpretation is that:
Ah! I don't care about your error rates and what you'd have done were data different. I just care about what I should believe given the data at hand. The Dark Matter models that explain the event were a priori plausible to me. Given that, I am more inclined to believe that they explain the data.I think that's a defensible point of view.
Tags: dark-matter, statistics, physics
Tall Tales and Wee Stories
Read Tall Tales and Wee Stories, a collection of Billy Connolly's stand-up material. I chose it to distract myself from other matters going on, and it mostly did the trick. I didn't know many of Connolly's routines, though remember his voice and manner from TV and from some VHS tapes my parents had in the 90s.
The routines are often funny, though occasionally misogynistic to a modern reader. Obviously something is lost on the page, especially the chaos, spontaneity, and physicality of his live performance, such as the one about getting drunk from the feet upwards. I was surprised by the lack of reflection, nostalgia, politics or social commentary in his material: I presumed he'd draw on his years as a welder or his working-class upbringing, but the routines are somewhat apolitical.
I'm part way through another AI/tech/society book, and then I plan to tackle a classic: Dickens or Hardy, something like that. Connolly's book was amusing, but didn't feel altogether rewarding.
Dreaming is for free
Read a heartbreaking story about a landslide at landfill site that killed 30 people in Guinea. Was appalled to realize that these occurrences are so common in the global south that they have their own Wikipedia entry. What an awful metaphor for the world we've created. Don't let anyone tell you a better world isn't possible.