Correcting for inflation (I used this tool by the US Bureau of Labor Statistics: https://www.bls.gov/data/inflation_calculator.htm), 30M USD in nov. 1995 would have a purchasing power equivalent to roughly 62M USD in feb. 2025. This is below half the budget of Moana 2 (150M USD, released in nov. 2024) for instance.
I would never use the official inflation numbers (they underestimate the actual inflation). It's easy to see that the most expensive movie ever made back in the day has a much lower budget that the most expensive movie made now, even adjusted for the official inflation rate.
For those interested in looking slightly more into the characteristic function, it may be worth pointing out that the characteristic function is equal to the Fourier-transform (with the sign of the argument being reversed) of the probability distribution in question.
In my own experience teaching teaching probability theory to physicists and engineers, establishing this connection is often a good way of helping people build intuition for why characteristic functions are so useful, why they crop up everywhere in probability theory, and why we can extract so much useful information about a distribution by looking at the characteristic function (since this group of students tends to already be rather familiar with Fourier-transforms).
Yes, this provides good intuition about why it is useful: the PDF of the sum of two random variables is the convolution of the original PDFs. A convolution is awkward to work with, but by the convolution theorem it is a multiplication in the Fourier ___domain. This immediately suggests that the Fourier transform of a PDF would be a useful thing to work with.
If you don't say that this is what you are doing then it all seems quite mysterious.
This is a good place to use cumulants. Instead of working with joint characteristic functions, which gets messy, it lets you isolate the effects of correlation into a separate term. The only limitation is that this doesn't work if the moment doesn't exist.
As a physicist, the moment when everything just clicked was when I realised that connected Feynman diagrams were basically the cumulants of that distribution.
Then almost everything in physics is about "what is the characteristic/moment/cumulant generating function?" and associated Legendre transforms
A little known bit of history is Feynman developed a diagrammatic method for expressing the moments of PGFs in his study of the stochastic theory of fission chains. This was before his work on QED. See:
Wow. I am not a physicist, but I use pdfs and moments and cumulants all the time. I came up with my own method to calculate cumulants for affine processes using some recursions, and they work. But if I hear you right, I might have stumbled upon something that Feynman did 70 years ago, and he probably did it better. Any good links you can recommend?
> As a physicist, the moment when everything just clicked was when I realised that connected Feynman diagrams were basically the cumulants of that distribution.
And the generating function of the cumulants is the logarithm of the generating function of the distribution (Fourier transform).
I feel like it's almost criminal of textbook writers not to mention this when introducing the characteristic function... At least as an aside or a footnote, for readers already familiar with Fourier transforms.
but isn't a characteristic function just "the" way to bridge the gap between sets, functions, and logic(? ...a 3way bridge!?)
I mean, it was useful for me to think about like a translation between sets and logic (this variable x is in the set xor not) into functions (a function f(x) that returns 1 or true whenever x is in set S)
You're thinking of a "characteristic function" in the sense of "indicator function" of a subset (https://en.wikipedia.org/wiki/Indicator_function), which is different thing to the characteristic function of a probability density function.
“Characterstic function” is (was) an overloaded term.
What you described is more often referred to as an “indicator function” these days, with “characteristic functions” denoting the transform (Fourier, laplace, z - depending on context). Closely related to “moment generating functions” to the point of being almost interchangeable.
so the same thing but, characterisic function as I knew them before these posts is a rudimentary 2-variable finite version. point and line (but the line is a curve, a circle because e).
but the new and improved 21st century characteristic functions are n-variable and have a full continious spectrum of variables between zero (false) and one (true) but only potentially lest infinite realizes itself (which would make the theories illogical).
- The characteristic function of a random variable X is defined as the function that maps t --> ExpectedValue[ exp( i * t * X ) ]
- Computing this expected value is the same as regarding t as a constant and integrating the function x --> exp( i * t * x) with respect to the distribution of X, i.e. if X has the density f, we compute the integral of f(x) * exp( i * t * x) with respect to x over the ___domain of f.
- on the other hand: computing the Fourier transform of f (here representing the density of X) and evaluating it at point t (i.e. computing (F(f))(t) if F represents the Fourier transform) is the same as fixing t and computing the integral of f(x) * exp( -i * t * x) with respect to x.
- Rearranging the integrand in the previous expression to f(x) * exp( i * -t * x), we see that it is the same as the integrand used in the characteristic function, only with a -t instead of a t.
The variable n comes out of nowhere in theorem 3.3, and they do not refer to it in the proof itself as far as I can tell. Is this just an editing error (I think the formula 3.4 needs the variable n if f is multidimensional and we are integrating over R^n, but since f is in L^1(R) I'm not sure what it signifies. I am however worried that there's something I'm missing).
What exactly is the argument that corporations are incapable of unbounded exponential growth contra a possible future AI? Is there just something magic about computers, or am I missing something obvious?
You can find discussions on this by googling "AI Foom" - the key idea is that an AGI that can recursively self-improve will be able to rapidly escape human control before any entity could check it, and very likely without humans even knowing that this had happened. Such an AI would likely consider humans an obstacle to its objectives, and would develop some sort of capacity to quickly destroy the world. Popular hypothesized mechanisms for doing so involve creating nanomachines or highly-lethal viruses.
That's key to the story of the paperclip maximizer - the paperclip maximizer will go about its task by trying to improve itself to be able to best solve the problem, and once it's improved enough it will decide that paperclips would be maximized by destroying humanity and would come up with a plan to achieve this outcome. However, humans may not realize that the AI is planning this until it's too late.
> What exactly is the argument that corporations are incapable of unbounded exponential growth
Individual corporations aren't capable of unbounded exponential growth because they can't keep the interests of the humans that make them up aligned with the "interests" of the corporation indefinitely. They develop cancer of the middle management and either die or settle into a comfortable steady-state monopoly.
Market systems as a whole can and do grow exponentially - and this makes them extremely dangerous. But they're not intelligent and so can't effectively resist when a world power decides to shorten the leash, as occasionally happens.
Corporations innovate through the work of human minds. Corporations improving doesn't cause the human minds to improve, so there's no recursive self-improvement. Corporations today still have the same kind of human brains trying to innovate them as they did in the past.
A human+ level AI would be able to understand and improve its own hardware and software in a way we can't with our own brains. As it improves itself, it will get compounding benefits from its own improvements.
How is "The circumferance of an idealized circle divided by its diameter" not a finite expression of π? Saying something cannot be expressed finitely in an integer-based numeral system, and saying that it admits no finite representation are two radically different statements.
Despite it being a non-starter from a pragmatic standpoint, we could for instance easily imagine a novel numeral type that encodes the set S = {a + b·π where a and b are integers} (we can encode integers quite easily and all we need to reposesent such a number in silico is to encode a and b). Using such a numeral type, we are able to do exact arithmetic if our operations are restricted to addition and subtraction (and if we are content with fractional representation of numbers as being considered "exact", we can also do division and multiplication although we would have to work within the larger set S' = { (a + b·π) / (c + d·π) where, a, b, c, and d are integers and c·d ≠ 0} rather than within S).
Unrelated to the article in question, but using ℝ over 𝕽 for the reals is more of a modern development. If you read older articles and textbooks, many will use 𝕽 rather than the sleeker ℝ (most in my experience, but results will be heavily affected by your cut-off for 'old').
I don't have direct evidence for my speculations, but I presume the reason [fraktur](https://en.wikipedia.org/wiki/Fraktur) was more common in mathematics back in the days is largely down to articles having to be type-set using movable type. If you insisted on using ℝ over 𝕽, you were likely to make life considerably harder for your printer (which in turn meant higher printing costs), since they would be considerably more likely to have to cast new types. As printing was modernized and movable type was replaced by more flexible printing-technologies, this pragmatic reason for preferring one glyph over the other went away. Another explanation/contributing factor is that the switch seems to have occurred in tandem with the barycenter of mathematics switching away from continental Europe and towards the US in the post-WWII period (at least if we disregard Soviet mathematics which also flourished in this period, but which was largely published in Russian). The average American would probably be less familiar with 𝕽 and other fraktur glyphs than the average German.
Part of the issue is that after a while we tend to forget that the cases that turned out to be true were dismissed as conspiracy theories at the time. In recent memory for example, a lot of claims about the capabilities and application of the US signal intelligence apparatus abroad (especially in allied coununtries) were dismissed as conspiracy theories prior to Snowden. If you talk to a lot of people today, they will tend to remember it more as a "we kinda' always knew, but just didn't have confirmation" situation than a "I'm sure the NSA is doing _something_, but there is no way it would be this extensive" situation.
Great example, this was the first time I noticed the phenomena my original comment was referring to. Everyone "normal" described this stuff as tinfoil hat and even the people slightly outside overton window never suspected it was as bad as the reality turned out. But, everyone just instantly switched to "yeah we knew that" and didn't update any beliefs or priors
>everyone just instantly switched to "yeah we knew that"
This is an example of a very common mistake that leads to a lot of animosity, which is that "those people who said X" and "these people saying they never said X" are the same people (this is fueled by the misunderstanding that there are essentially only two groups of people, which is an illusion that most political systems reinforce).
I suspect OP may have been going for a variation on the old "Programmer returns with zero eggs and 12 gallons of milk after having been asked to get one gallon of milk and if they have eggs to buy a dozen"-joke, but it falls flat in this instance since it relies on an interpretation bordering on deliberate misconstrual (i.e. applying the modifier "for each year of service" to the whole phrase "16 weeks plus two additional weeks" rather than just to the latter fragment "two additional weeks").
Without knowing the exact approval history in the US, I doubt that it is a lax approval process as much as it is an absence of better alternatives.
There are not really any known effective clinical interventions (be they in the form of medicine, therapy, or things like more exercise), and a 10% improvement is better than nothing (clinical therapy (notably Cognitive Behavioral Therapy) is generally believed to be more effective but not by that much and it is financially out of reach for many people who are able to afford medication).
> The problem with treating depressed people is to get them to actually do the things that will help them. That can be incredibly difficult without medication.
The problem with talking about doing "things that will help them" is that we don't really have a lot of effective clinical interventions. Even the most common interventions (SSRIs, Congnitive Behavioral Therapy, physical exercise, etc.) are not really that effective at treating depression and have little to no effect in large parts of the affected population. That being said, these treatments _do_ work for some people so they should definitely not be dismissed out of hand (although it may in some cases be regression to the mean more than anything else, i.e. if you get better after a while on your own but have undergone treatments of one form or another you may erroneously believe your most recent treatment was effective).
