I really enjoyed both blog posts, thank you for sharing! And I have to say, your explanation of the subexponential distribution property was remarkably clear for someone without a background in statistics :)
Would you mind sharing the flashcards you generated to build this intuition? I've been using Anki for a while and really trying to focus now on improving my prompt writing; would love to see how you managed it for this problem.
As much as I would like to, I think getting to that understanding required at least 500 flashcards on general statistical and probability concepts, ranging from fundamentals to extreme value theory. Most of those are only barely relevant at face value, but still contribute to understanding.
It's not that I set out to understand this specific thing but that I had studied statistics with flashcard support for a year and that happened to work after a few attempts.
I've long wanted to write about this but never been able to think of anything original to say, but your question forced me to face this with effort. Thanks!
When making flashcards I draw a lot from the softer type of theory-building they do in social sciences. I ask questions like
- What are the properties of this?
- What variants of this exist? I.e. how would I recognise this in the wild, or in other shapes?
- What subcomponents can this be deconstructed into?
- Into which bigger picture does this fit?
- What are the consequences of this? What are its antecedents?
- What is this a special case of? What would a generalisation of this look like?
- Which are other related things? What are their similarities and differences?
- In what context might I need to know this?
Whenever I encounter what seems like a significant thing I loosely ask some of these questions, and try to construct atomic, focused flashcards from the answers.
I say loosely because it would take forever to to through all questions for all flashcards I make, so there's some bit of intuition that attracts me to which I think are the most significant questions for any given thing.
-----
One trick to make flashcards more specific that I use (maybe even abuse) is putting part of the answer into the prompt. Instead of prompting "What is the property of subexponential distributions I found meaningful in this book?" I might prompt "What behaviour do subexponential distributions have around high barriers that others don't?" -- I'm giving away part of the answer by including "high barrier" in the prompt, but I'm okay with that.
If I'm concerned about that, I might create a second flashcard prompting something like "What can a subexponential distribution do in one step that a more well-behaved distribution needs many steps to do?" with the answer "clear a high barrier". That captures both sides of the property without making too general a prompt.
I also do this a lot with "why" questions. Instead of prompting "what is the definition of y?" I might prompt "why is the definition of y=f(x)?" That gives away essentially the entire answer but focuses on the why instead.
Yeah, there are a lot of interesting theories regarding this (which were far more common in the halcyon days of 2021, when it seemed like ZIRP was the new normal).
I actually think an underrated piece of this is that investments have become far less capital intensive. Business in previous economic cycles required a huge amount of capital to begin and maintain: railroads, oil, manufacturing all require enormous sums of capital to continue. Conversely, the dominant businesses in this economic cycle are all Internet based. Google, Meta, etc could run for 1,000 years without substantial cash need, it's a far less capital intensive model. You can see this reflected statistically: the FCF yield of the S&P 500 is 2x what it was in 1990.
If businesses need far less cash than before yet remain highly productive, it stands to reason real interest rates would drop: the demand for capital by economic drivers has gone down, while the supply of capital has increased through FCF gains.
I wonder if something like space colonization/resource extraction would reverse or change the trend toward lower interest rates? I imagine (perhaps excluding some sort of sci-fi esque replicator robots) such endeavors would be very capital intensive.
Read up on the onion futures markets; due to a (rather fascinating) historical decision, they are one of the few commodities without speculative futures. As a result, price volatility in the market is massive, with far worse swings compared to the efficient markets where speculators operate.
Caveat: I'm the founder of a VC-backed startup, so my perspective is likely skewed.
The VC mode of thinking becomes a lot clearer when you realize that their returns are completely driven by moonshots. Something like 1% of the portfolio will drive 56% of the returns. As a result:
(1) It costs them nearly nothing if a startup goes to 0. Most of their portfolio will go to 0.
(2) Conversely, it will cost them a ton if they miss a potential moonshot. Even if the idea sounds dumb on face, if there's a chance it becomes big, the VC needs to be in it in order to survive.
(3) There's reputational risk at play for the individual VC investor. Every VC will have a low hit rate, so they're not worried about looking dumb losing money on some startup going to 0. What they are worried about is being known as the person who passed on the next Airbnb despite getting a look.
(4) Later stage VC funding becomes an access game — there's more capital than good ideas, and so these companies can have their pick of the litter. This means, as a VC, you need to prioritize being founder friendly and having a pre-existing relationship with a company. As such, you'll see VCs throw in $1m to $10m checks without much though, with the hope that if the company every does scale, they'll be able to write a sizable follow-on.
It's weird to think about if you're used to public markets, where hit rate matters a lot more and diligence is centered on "why should I invest". In VC-land, the burden of proof is flipped.
Gotcha, I think that makes sense! Just to make sure I understand -- on server-side, they're pulling in all relevant information from their own database and whatever 3rd parties are needed (maybe for injury reports, etc.). The server then generates an HTML page that it sends as-is to the client, which just faithfully renders the code? Thanks for the answer!
Somewhat ironic given I suspect Postman would disagree with the majority of this article. This article asserts that the amount of engagement with the written word has gone up dramatically; "Amusing Ourselves To Death" has an entire chapter dedicated to demonstrating how much more early Americans engaged with written work. Consumption of books per capita was far higher in the 18th century than it is today, the literacy rate for men in some of the colonies was ~92% (and for women 62%), the number of people who read Common Sense on a per head basis is roughly equivalent to the number of people who watch the Super Bowl today, etc.
The argument in "Amusing Ourselves to Death" is not that the written word enhances reality, it is that it engages critical faculties by forcing the reader to contend with the argument advanced by an author. This article suggests that the written word is a crutch for thought, and thereby diminishes our ability to get at reality. These ideas are not exactly opposed, but there is a certain discordance.
When Postman refers to written word he refers mainly to books. The article mentions the increase of hours spent reading text as a result of texting, social media etc. This form of written word is actually close to Postman's concerns.
It's written word, but its metaphor is the same as the metaphor of mediums like TV.
I agree in part. The texting, social media, etc. has the same disjointed tone as the "now, this" attitude of television that Postman wrote about. However, he had a specific concern with images versus text (whatever kind of text) -- namely, that they pass too quickly for people to dwell upon and can never advance an argument in the way that writing can.
To be honest, I think this is a general weakness in Postman's work. He talks a bit about the difference between pre and post-telegram writing, but he lionizes all 18th century writing despite there being tons of different forms (pamphlets, books, novels) that don't have the same intellectual value imo.
Elon didn't agree to "deal contingent on everything checking out", he agreed to deal. Normally, this would have been vetted in due diligence, but Musk waived that period -- this is why the turnaround from "I might buy Twitter" to "I'm buying Twitter" was much, much faster than your typical acquisition talk. Matt Levine covered this better than I did, so I'll just quote his relevant section below:
“Temporarily on hold” is not a thing. Elon Musk has signed a binding contract requiring him to buy Twitter. Legions of bankers and lawyers and Twitter employees and special-purpose-vehicle promoters are working to fulfill his legal obligation to get the deal closed. “The parties hereto will use their respective reasonable best efforts to consummate and make effective the transactions contemplated by this Agreement,” says the merger agreement. (Section 6.3(a).) He can’t just put that “on hold.”
That contract does not allow Musk to walk away if it turns out that “spam/fake accounts” represent more than 5% of Twitter users. We discussed this last month, when Twitter admitted in a securities filing that it had (slightly) overestimated its daily active users for years. The merger agreement contains a provision that allows Musk to walk away if Twitter’s securities filings are wrong — and this 5% number is in its securities filings — but only if the inaccuracy would have a “Material Adverse Effect” on the company. (See Sections 4.6(a) and 7.2(b).) That is an incredibly high standard: Delaware courts have almost never found an MAE. An MAE has to be something that would “substantially threaten the overall earnings potential of the target in a durationally-significant manner,” the courts have said; there is a rule of thumb that an MAE requires a 40% decrease in long-term profitability. If it turned out that 6% or 20% or 50% of Twitter accounts are bots, that will be embarrassing and might even reduce Twitter’s future advertising revenue, but will it be an MAE? No.
“Pending details supporting calculation” is not how this works. This disclosure — that “the average of false or spam accounts ... represented fewer than 5% of” Twitter’s monetizable daily active users — has been in Twitter’s securities filings for many years, always with a caveat that “in making this determination, we applied significant judgment, so our estimation of false or spam accounts may not accurately represent the actual number of such accounts, and the actual number of false or spam accounts could be higher than we have estimated.” Musk had the opportunity to read these filings before offering to buy Twitter, and he had the opportunity to do due diligence on these numbers before signing the deal. (He declined.) He can’t now go to Twitter and say “actually now you need to prove that your user numbers are right.” If he wants to walk, he has to prove that they’re wrong, and also that they’re wrong in a way that has a material adverse effect on the business. Which he obviously can’t do.
Would you mind sharing the flashcards you generated to build this intuition? I've been using Anki for a while and really trying to focus now on improving my prompt writing; would love to see how you managed it for this problem.