Problem solving using Markov chains (2007) [pdf]
math.uchicago.edu
math.uchicago.edu
NO SMOKING. NO SPITTING. MGMT
Is there any way to get a better outcome for the public here, or is “do good stuff then sell out” the way it’s always going to be?
In the end, incentives matter.
Outside software technology: there is a series of papers from Grossman (going back to the 80s!) that analyzes basic versus applied research in a macroeconomic framework. Basic research _can_ be a public good, applied research can be crowded out. Combined with microeconomic research that monopolies can be dynamically efficient (investing in applied and basic R&D, like Bell Labs) and you get several examples and theories that contradict your statement that "there is no private market entity with an incentive to provide research to the public."
Another real world example in hardware that contradicts this claim is the evolution of building control systems. Before the advent of IOT, so, circa 1980s - 2010s, you saw increasing sharing and harmonization of competing electronics standards because it turned out to be more efficient to be modular, not have to re-hire subcontractors at exorbitant rates to maintain or replace components that go haywire, etc.
Economic analysis? Another intelligence product that requires essentially no staff, no actual R&D, no equipment besides computers? Brother, you have to be kidding me.
The hardware thing is just companies evolving to a shared standard.
Do you have even a little bit of a clue how hard it is to do good pharmacological research? Toxicological? Biological? Chemical? Physical? You have mentioned intelligence products with 0 investment cost and 0 risk of failure.
This is perhaps one of the most fart-sniffing tech-centric perspectives I have ever been exposed to. Go read some actual research by actual scientists and come back when you can tell me why, for instance, Eli Lilley would ever make their data or internal R&D public.
Jonas Salk did it. He is an extremely rare exception, and his incentive was public health. Notice that his incentive was markedly not financial.
Market entities with a financial incentive, whose entire business model and success is predicated on their unique R&D results, have 0 incentive to release research to the public.
(1) FOSS is not only the next hyped front-end framework or modern data stack funnel. I encourage you to do more research for what European universities and organizations are doing. Not everyone follows the American or Chinese extractive approaches to software.
Further, while many corporations do indeed farm social capital and perform other appreciably maladative and cynic-inducing behaviors, the universe and the space of organizations is large. There are a great many examples of governments adopting public research and development released by private entities -- in FOSS and in other contexts.
Additionally, the fact that FOSS-product-focused companies tend to launch _after_ FOSS becomes successful to support the FOSS offering with associated services is quite a bit different from what is perhaps a FAANG-induced cynicism. To reiterate - the universe and the space of organizations is large.
(2) You interpreted that I did pointed to economic analysis as public vs. private R&D. This is a misinterpretation on your part and I encourage a re-read. I pointed to findings and studies to help you understand where the organizational and market frameworks for analysis stand.
(3) I am a researcher and regularly publish my findings, under the banner of the university I support, under non-profits I support, and under the company I run. I appreciate your experience has made you cynical. Lets break down this section.
> This is perhaps one of the most fart-sniffing tech-centric perspectives I have ever been exposed to.
This was not received as a good-faith statement, and further discussion on it will only engender argument. I suggest we move beyond trivial digs.
> Eli Lilley would ever make their data or internal R&D public.
Not to shill for them, but your point on Eli Lilly is incorrect. Eli Lilly has worked towards more transparent release of information -- they voluntarily launched an online clinical trial registry starting in 2002 (for Phase II–IV trials initiated on/after October 15, 2002) and extended full trial registration (including Phase I) from October 1, 2010.[0] Since 2014, Lilly has published clinical study results (Phase 2/3) regardless of outcome, adhering to PhRMA/EFPIA transparency principles. Patient-level data on marketed-approved indications is available to qualified researchers via a controlled-access third-party portal.[1] Beginning in 2021, Lilly has also produced plain‑language summaries of Phase 2–4 results in English, and more recently extended plain‑language summaries to Phase 1 trials in the EU in compliance with new regulations.[1]
Especially the third point is relevant -- good government regulation leads to better sharing and transparency. Smart companies take regulation as an innovation opportunity.
> Jonas Salk did it. He is an extremely rare exception, and his incentive was public health. Notice that his incentive was markedly not financial.
Aye, and I wish that all medical and life-enhancing research could be accomplished as relatively cheaply or as magnanimously as Jonas Salk.
> Market entities with a financial incentive, whose entire business model and success is predicated on their unique R&D results, have 0 incentive to release research to the public.
Please refer to (2) for studies and theory for why this is untrue.
The number of market entities who only are built on unique R&D tend to fail due to poor delivery of product, so their incentive to release their R&D to the world is more or less moot. I do acknowledge existence of market entities who are built solely on operationalizing R&D -- I challenge the implicit claim that all market entities fall into this category.
[0] https://www.lilly.com/au/policies-reports/transparency-discl...
[1] https://sustainability.lilly.com/governance/business-ethics
They were also forced in the 1950s to license all their innovations freely, as compensation for holding a monopoly. Which only strengthens the parent’s point that private institutions have little incentive to work for public benefit.
If we're talking about applied technology in the public goods space, then it can be a toss up. Sustainability research, for example, can be quite blurry as to whether the market is pricing it in or not as applied or basic research -- really depends on how a government handles externalities and regulatory capture!
I'll 100% agree to government entities as well as some well-chartered public entities being absolutely awesome at setting up incentive structures for desired outcomes. There is actually a whole field of research dedicated to the topic of incentive structuring called mechanism design -- think of it as the converse to Game Theory and strategic behavioral analysis -- that policy design and analysis learn from.
I'll also note that governments aren't structured to efficiently provide benefits or just-in-time delivery in most situations. Though the discussion has made me more curious about how operationally efficient the DOD is for civilian goods distribution, given it supports a massive population.
This kind of discussion is a bit off topic here, but I think it is important to remind people that the idea that private always is better than public is ideological dogma, not science. But your latest comment makes me believe you agree with that.
A moderate path, like what we see in the Scandinavian countries, looks to be a better model.
How that is in practice, I'm not sure, and I'm sure with some sleuthing it would be possible to find out at least some of it. But on the whole, I'm honestly not sure beyond that.
There was also the Waymo ad and the Rods from the Gods video where he couldn't bother to use a guide wire to aim.
There second one takes a mathematical model for the path integral for light and portrays it like that's actually what is happening, with plenty of phrases like light "chooses" the path of least action that imply something more going on. Also, the experiment at the end with the laser pointer is awful. The light we are seeing is scattering from the laser pointer's aperture, not some evidence that light is taking alternate paths.
Many people said this, but he set up an experiment to test it and the light does turn on instantly as claimed: https://www.youtube.com/watch?v=oI_X2cMHNe0
> There second one takes a mathematical model for the path integral for light and portrays it like that's actually what is happening
I know nothing about this. Is there a more accurate mathematical model available than the one he uses? Otherwise, I think it seems sensible to portray our best mathematical model as "what's really going on". And I didn't get the sense that light was "choosing" anything when watching the video, I got the sense that the amplitudes of all possible paths were cancelling out except for the shortest path (or something along those lines)
The words people like to use for the path integral is a sum over histories---that corresponds tightly with the ingredients in the path integral. So in this formulation it's what's "actually happening". But in other mathematical formulations other words are more appealing and what someone claims is "actually happening" sounds different.
See: Brian Keating licking Eric Weinstein's jock strap in public and then offering mild criticism on Piers Morgan.
If transparent enough (and not from an abhorrent source), I'd be ok with his product. He's even allowed to make the occasional mistake as long as he properly owns up to it.
Theres been a lot of valuable learning from him and it would be a pity to dismiss it all over a single fumble.
You can, actually, with a simple rule of thumb: if it's being advertised on YouTube, it's statistically low quality or a scam. The sheer number of brands that sponsor videos just to be exposed later for doing something shady is just too high.
I thought it goes without saying that I don't mean ads shown directly by YouTube, if you don't already block those in 2025 I don't know what to say.
Her latest video, showing her out of bed and going for short walks, is here: https://youtu.be/vqeIeIcDHD0?si=WoxpqZOuRTWD2XYd
> I wonder if Markov chains could predict how many times Veritasium changes the thumbnail and title of this video.
Consider the nuclear reaction metaphor. It's clearly not memoryless. Eventually you'll run out of fissile material.
The diagram for that example is bad as well. Do arrows correspond to state transitions? Or do they correspond to forks in the process where one neutron results in two?
I think no real process is memoryless: time passes/machines degrade/human behaviors evolve. It is always an approximation that is found/assumed to hold within the modeled timeframe.
It feels like bishops pattern recognition but with a clearer tone (and a different field of course)
The PGM book is also structured very clearly for researchers in PGMs. The book is laid out in 3 major section: the models, inference techniques (the bulk of the book), and learning. Which means, if you follow the logic of the book, you basically have to work through 1000+ pages of content before you can actually start running even toy versions of these models. If you do need to get into the nitty-gritty of particular inference algorithms, I don't believe there is another textbook with nearly the level of scope and detail.
Bishop's section on PGMs from Pattern Recognition and Machine Learning is probably a better place to start learning about these more advanced models, and if you become very interested then Koller+Friedman will be an invaluable text.
It's worth noting that the PGM course taught by Koller was one of the initial, and still very excellent, Coursera courses. I'm not sure if it's still free, but it was a nice way to get a deep dive into the topic in a reasonably short time frame (I do remember those homeworks as brutal though!)[0].
0. https://www.coursera.org/specializations/probabilistic-graph...
The data never fits the graph. Real-world tables are messy and full of hidden junk, so you either spend weeks arguing over structure or give up the nice causal story.
DL stole the mind-share. A transformer is a one-liner with a mature tooling stack; hard to argue with that when deadlines loom.
That said, they’re not completely dead - reportedly Microsoft’s TrueSkill (Xbox ranking), a bunch of Google ops/diagnosis pipelines, some healthcare diagnosis tools by IBM Watson built on Infer.NET.
Anyone here actually shipped a PGM that beat a neural baseline? Would really love to appreciate your war stories.
Kind of like flow-based programming. I don't think there are any fundamental reason why it can't work, it just hasn't yet.
Could you link me to where I could learn more about this?
"Causality: Models, Reasoning and Inference", https://a.co/d/6b3TKhQ, is the technical and researcher audience book.
[0]: https://mattmahoney.net/dc/dce.html#Section_421 [1]: https://mattmahoney.net/dc/text.html
As in go into the first open source entry which is #2 in this list, cmix, unzip the files, go into paq8.cpp and search for DMC. See "Model using DMC (Dynamic Markov Compression)" and associated code. In these cases DMC is one model mixed in with others and the best model for the current context is used.
Hook exclusively uses DMC for outstanding results but the others use DMC as one of the prediction models.
By the way, does anyone know which model or type of model was used in winning gold in IMO?
Might be a reference to this[1] blog post which was posted here[2] a year ago.
There has also been some academic work linking the two, like this[3] paper.
[1]: https://elijahpotter.dev/articles/markov_chains_are_the_orig...
It's not an unreasonable view, at least for decoder-only LLMs (which is what most popular LLMs are). While it may seem they violate the Markov property since they clearly do make use of their history, in practice that entire history is summarized in an embedding passed into the decoder. I.e.just like a Markov chain their entire history is compressed into a single point which leaves them conditionally independent of their past given their present state.
It's worth noting that this claim is NOT generally applicable to LLMs since both encoder/decoder and encoder-only LLMs do violate the Markov property and therefore cannot be properly considered Markov chains in a meaningful way.
But running inference on decoder only model is, at a high enough level of abstraction, is conceptually the same as running a Markov chain (on steroids).
Physics models of closed systems moving under classical mechanics are deterministic, continuous Markov processes. Random walks on a graph are non deterministic, discrete Markov processes.
You may further generalize that if a process has state X, and the prior N states contribute to predicting the next state, you can make a new process whose state is an N-vector of Xs, and the graph connecting those states reduces the evolution of the system to a random walk on a graph, and thus a Markov process.
Thus any system where the best possible model of its evolution requires you to examine at most finitely many consecutive states immediately preceding the current state is a Markov process.
For example, an LLM that will process a finite context window of tokens and then emit a weighted random token is most definitely a Markov process.
>In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event
Edit: I could be missing important nuance that other people are pointing out in this thread!
It's often a matter of asking the right person what technique works. It's often a matter of making a measurement before getting lost in the math. It's often a matter of finding the right paper in the literature.
For example, https://news.ycombinator.com/item?id=44574033
Is this still possible with the latest models being trained on synthetic data? And if it possible, what would that one phrase be?
> We find that indiscriminate use of model-generated content in training causes irreversible defects in the resulting models, in which tails of the original content distribution disappear. [0]
In practice nobody is "indiscriminately" using model output to fine-tune models since that doesn't even make sense. Even if you're harvesting web data generated by LLMs, that data has in fact been curated by it's acceptance on whatever platform you found it on is a form of curation.
There was a very recent paper Supervised Fine Tuning on Curated Data is Reinforcement Learning (and can be improved) [1] whose content is pretty well summarized by the title. So long as the data is curated in some way, you are providing more information to the model and the results should improve somewhat.
0. https://www.nature.com/articles/s41586-024-07566-y
1. https://www.arxiv.org/pdf/2507.12856
edit: updated based on cooksnoot's comment
If you just mean its risk has been over exaggerated and/or over simplified then yea, you'd have a point.
Having spent quite a bit of time diving into many questionable "research" papers (the original model collapse paper is not actually one of these, it's a solid paper), there's a very common pattern of showing that something does or does not work under special conditions but casually making generalized claims about those results. It's so easy with LLMs to find a way to get the result you want that there are far too many papers out there that people quickly take as fact when the claims are much, much weaker than the papers let on. So I tend to get a bit reactionary when addressing many of these "facts" about LLMs.
But you are correct that with the model collapse paper this is much more the public misunderstanding the claims of the original paper than any fault with that paper itself.
Define a square of some known size (1x1 should be fine, I think)
Inscribe a circle inside the square
Generate random points inside the square
Look at how many fall inside the square but not the circle, versus the ones that do fall in the circle.
From that, using what you know about the area of the square and circle respectively, the ratio of "inside square but not in circle" and "inside circle" points can be used to set up an equation for the value of pi.
Somebody who's more familiar with this than me can probably fix the details I got wrong, but I think that's the general spirit of it.
For Markov Chains in general, the only thing that jumps to mind for me is generating text for old school IRC bots. :-)
[1]: which is probably not the point of this essay. For for muddying the waters, I have both concepts kinda 'top of mind' in my head right now after watching the Veritasium video.
[1] https://claude.ai/public/artifacts/1b921a50-897e-4d9e-8cfa-0...
Back in like 9th grade, when Wikipedia did not yet exist (but MathWorld and IRC did) I did this with TI-Basic instead of paying attention in geometry class. It's cool, but converges hilariously slowly. The in versus out formula is basically distance from origin > 1, but you end up double sampling a lot using randomness.
I told a college roommate about it and he basically invented a calculus approach summing pixels in columns or something as an optimization. You could probably further optimize by finding upper and lower bounds of the "frontier" of the circle, or iteratively splitting rectangle slices in infinitum, but thats probably just reinventing state of the art. And all this skips the cool random sampling the monte carlo algorithm uses.
In the sample programs there's a big red one... https://www.dangermouse.net/esoteric/piet/samples.html
There's also the IOCCC classic https://www.ioccc.org/1988/westley/index.html
Monte Carlo Value for Pi
Each successive sequence of six bytes is used as 24 bit X and Y co-ordinates within a square. If the distance of the randomly-generated point is less than the radius of a circle inscribed within the square, the six-byte sequence is considered a “hit”. The percentage of hits can be used to calculate the value of Pi. For very large streams (this approximation converges very slowly), the value will approach the correct value of Pi if the sequence is close to random. A 500000 byte file created by radioactive decay yielded:
Monte Carlo value for Pi is 3.143580574 (error 0.06 percent).http://math.uchicago.edu/~shmuel/Network-course-readings/Mar...
(there's a copyright 2007 at the bottom of the linked page, which isn't explicitly "published in 2007" in my mind)