Pretty sure Duff was a heavily filtered macro beer.
Not saying engineered beer is necessarily bad- Sapporo and Asahi never disappoint- but I imagine you would want to stick to unfiltered and unpasteurized to retain some of the more… alive compounds.
I started dressing nice at work, reasoning that looking sharp would buy me a few seconds or minutes of grace to allow my social deficiencies to catch up - just in case an executive decided to ask me a question.
Of course, that never happened for months, years until the one day I went in wearing cargo pants and a gothy synth band shirt and was greeted by a delegation of executives from out of town engaging everyone in small talk…
I worked for a downtown firm for a while which loosened up dress code a little bit so I didn’t always wear my jacket in—though cargo pants and rock T would definitely have led to an HR meeting. One day I had to borrow a jacket from someone when I had to go to a nearby studio for a TV interview:-)
Great article. Personally I have been learning more about the mathematics of beyond-CLT scenarios (fat tails, infinite variance etc)
The great philosophical question is why CLT applies so universally. The article explains it well as a consequence of the averaging process.
Alternatively, I’ve read that natural processes tend to exhibit Gaussian behaviour because there is a tendency towards equilibrium: forces, homeostasis, central potentials and so on and this equilibrium drives the measurable into the central region.
For processes such as prices in financial markets, with complicated feedback loops and reflexivity (in the Soros sense) the probability mass tends to ends up in the non central region, where the CLT does not apply.
The key principle is that you get CLT when a bunch of random factors add. Which happens in lots of places.
In finance, the effects of random factors tend to multiply. So you get a log-normal curve.
As Taleb points out, though, the underlying assumptions behind log-normal break in large market movements. Because in large movements, things that were uncorrelated, become correlated. Resulting in fat tails, where extreme combinations of events (aka "black swans") become far more likely than naively expected.
I know you know that and were just simplifying. Just wanted this fact to be better known for practitioners. Your comment on multiplicative processes is spot on.
Absolutely. The effect of straightforward correlations is a change in the variance, which can be measured in finance.
The effect of the nonlinear changing correlations is that future global behavior can't be predicted from local observations without a very sophisticated model.
The standard framing defines the Gaussian as this special object with a nice PDF, then presents the CLT as a surprising property it happens to have. But convolution of densities is the fundamental operation. If you keep convolving any finite-variance distribution with itself, the shape converges, and we called the limit "normal." The Gaussian is a fixed point of iterated convolution under √n rescaling. It earned its name by being the thing you inevitably get, not by having elegant closed-form properties.
The most interesting assumptions to relax are the independence assumptions. They're way more permissive than the textbook version suggests. You need dependence to decay fast enough, and mixing conditions (α-mixing, strong mixing) give you exactly that: correlations that die off let the CLT go through essentially unchanged. Where it genuinely breaks is long-range dependence -fractionally integrated processes, Hurst parameter above 0.5, where autocorrelations decay hyperbolically instead of exponentially. There the √n normalization is wrong, you get different scaling exponents, and sometimes non-Gaussian limits.
There are also interesting higher order terms. The √n is specifically the rate that zeroes out the higher-order cumulants. Skewness (third cumulant) decays at 1/√n, excess kurtosis at 1/n, and so on up. Edgeworth expansions formalize this as an asymptotic series in powers of 1/√n with cumulant-dependent coefficients. So the Gaussian is the leading term of that expansion, and Edgeworth tells you the rate and structure of convergence to it.
It is the not knowing, the unknown unknowns and known unknowns which result in the max entropy distribution's appearance. When we know more, it is not Gaussian. That is known.
Exactly this. From this perspective, the CLT then can be restated as: "it's interesting that when you add up a sufficiently large number of independent random variables, then even if you have a lot of specific detailed knowledge about each of those variables, in the end all you know about their sum is its mean and variation. But at least you do reliably know that much."
Came here basically looking to see this explanation. Normal dist is [approximately] common when summing lots of things we don't understand, otherwise, it isn't really.
>natural processes tend to exhibit Gaussian behaviour
to me it results of 2 factors - 1. Gaussian is the max entropy for a distribution with a given variance and 2. variance is the model of energy-limited behavior whereis physical processes are always under some energy limits. Basically it is the 2nd law.
AFAIK they still dominate on clock rate, which I was surprised to see when doing some back of the envelope calculations regarding core counts.
I felt my 8 core i9 9900K was inadequate, so shopped around for something AMD, and IIRC the core multiplier of the chip I found was dominated by the clock rate multiplier so it’s possible that at full utilization my i9 is still towards the best I can get at the price.
Not sure if I’m the typical consumer in this case however.
Your 9900k at 5ghz does work slower than a Ryzen 9800X3D at 5ghz. A lot slower (1700 single core geekbench vs 3300, and just about any benchmark will tell the same story). Clock speed alone doesn't mean anything.
>8 Cores and 16 processing threads, based on AMD "Zen 5" architecture
which is the same thread geometry as my 9900K.
My main concerns at the time were:
1. More cores for running large workloads on k8s since I had just upgraded to 128G RAM
2. More thread level parallelism for my C++ code
Naively I thought that, ceteris paribus and assuming good L1 cache utilization, having more physical cores with a higher clock rate would be the ticket for 2.
Does the 9800X3D have a wider pipeline or is it some other microarchitectural feature that makes it faster?
You don't even need to go into the pipeline details. The 9800X3D has 8x more L2 cache, 6x more L3 cache, 2x the memory bandwidth than the now 8 years old i9 9900K. 3D V-cache is pretty cool.
I purposely picked a CPU with the same thread geometry as your 9900K to avoid calls of "apples & oranges" or whatever. If you want more threads, the 9950X is right there in the same socket. Or Core Ultra 9 285k. Either of which will run circles around a 9900K in code compilation.
I think my i9 was released right after the Spectre and Meltdown mitigations in 2019, but I seem to remember even more recent vulns in that family… so that could also be a factor.
I replied to the sibling comment: I was making simplifying assumptions for two specific use cases and naively treated physical cores and clock rate as my variables.
Making funny memes of my friends mainly. ChatGPT won’t touch that, I haven’t tried with Claude yet, but grok keeps the group chat flush with laughing emojis.
That’s all I use it for really- things out of alignment with the other platforms- which IMO are better on every other metric (except having a sense of humour of course)
Perhaps the lesson here is upgrade your use case for AI's! All that power and that's your stumbling block? LOL, no disrespect.
Sure, I have no problem with what you're doing, and as things evolve I'm sure there'll be no problem, but there's countless other apps designed to do exactly what you've said.
Not saying engineered beer is necessarily bad- Sapporo and Asahi never disappoint- but I imagine you would want to stick to unfiltered and unpasteurized to retain some of the more… alive compounds.
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