Alexander Gietelink Oldenziel's Shortform

Why don't animals have guns? 

Or why didn't evolution evolve the Hydralisk?

Evolution has found (sometimes multiple times) the camera, general intelligence, nanotech, electronavigation, aerial endurance better than any drone, robots more flexible than any human-made drone, highly efficient photosynthesis, etc. 

First of all let's answer another question: why didn't evolution evolve the wheel like the alien wheeled elephants in His Dark Materials?

Is it biologically impossible to evolve?

Well, technically, the flagella of various bacteria is a proper wheel.

No the likely answer is that wheels are great when you have roads and suck when you don't. Roads are build by ants to some degree but on the whole probably don't make sense for an animal-intelligence species. 

Aren't there animals that use projectiles?

Hold up. Is it actually true that there is not a single animal with a gun, harpoon or other projectile weapon?

Porcupines have quils, some snakes spit venom, a type of fish spits water as a projectile to kick insects of leaves than eats insects. Bombadier beetles can produce an explosive chemical mixture. Skunks use some other chemicals. Some snails shoot harpoons from very close range. There is a crustacean that can snap its claw so quickly it creates a shockwave stunning fish. Octopi use ink.  Goliath birdeater spider shoot hair. Electric eels shoot electricity etc. 

Maybe there isn't an incentive gradient? The problem with this argument is that the same argument can be made for lots and lots of abilities that animals have developed, often multiple times. Flight, camera, a nervous system. 

But flight has an intermediate form: glider monkeys, flying squirrels, flying fish. 

Except, I think there are lots of intermediate forms for guns & harpoons too:

There are animals with quills. It's only a small number of steps from having quils that you release when attack to actively shooting and aiming these quils. Why didn't Evolution evolve Hydralisks? For many other examples - see the list above. 

In a Galaxy far far away

I think it is plausible that the reason animals don't have guns is simply an accident. Somewhere in the vast expanses of space circling a dim sun-like star the water-bearing planet Hiram Maxim is teeming with life. Nothing like an intelligent species has yet evolved yet it's many lifeforms sport a wide variety of highly effective projectile weapons. Indeed, the majority of larger lifeforms have some form of projective weapon as a result of the evolutionary arms race. The savannahs sport gazelle-like herbivores evading sniper-gun equppied predators. 

Some many parsecs away is the planet Big Bertha, a world is embroilled in permanent biological trench warfare. More than 95% percent of the biomass of animals larger than a mouse is taken up by members of just 4 geni of eusocial gun-equipped species or their domesticastes. Yet the individual intelligence of members of these species doesn't exceed that of a cat. 

The largest of the four geni builds massive dams like beavers, practices husbandry of various domesticated species, agriculture and engages in massive warfare against rival colonies using projectile harpoons that grow from their limbs. Yet all of this is biological, not technological: the behaviours and abilites are evolved rather than learned. There is not a single species whose intelligence rivals that of a Great ape, either individually or collectively. 

Novel Science is Inherently Illegible

Legibility, transparency, and open science are generally considered positive attributes, while opacity, elitism, and obscurantism are viewed as negative. However, increased legibility in science is not always beneficial and can often be detrimental.

Scientific management, with some exceptions, likely underperforms compared to simpler heuristics such as giving money to smart people or implementing grant lotteries. Scientific legibility suffers from the classic "Seeing like a State" problems. It constrains endeavors to the least informed stakeholder, hinders exploration, inevitably biases research to be simple and myopic, and exposes researchers to constant political tug-of-war between different interest groups poisoning objectivity. 

I think the above would be considered relatively uncontroversial in EA circles.  But I posit there is something deeper going on: 

Novel research is inherently illegible. If it were legible, someone else would have already pursued it. As science advances her concepts become increasingly counterintuitive and further from common sense. Most of the legible low-hanging fruit has already been picked, and novel research requires venturing higher into the tree, pursuing illegible paths with indirect and hard-to-foresee impacts.

Encrypted Batteries 

(I thank Dmitry Vaintrob for the idea of encrypted batteries. Thanks to Adam Scholl for the alignment angle. Thanks to the Computational Mechanics at the receent compMech conference. )

There are no Atoms in the Void just Bits in the Description. Given the right string a Maxwell Demon transducer can extract energy from a heatbath. 

Imagine a pseudorandom heatbath + nano-Demon. It looks like a heatbath from the outside but secretly there is a private key string that when fed to the nano-Demon allows it to extra lots of energy from the heatbath. 

P.S. Beyond the current ken of humanity lies a generalized concept of free energy that describes the generic potential ability or power of an agent to achieve goals. Money, the golden calf of Baal is one of its many avatars. Could there be ways to encrypt generalized free energy batteries to constraint the user to only see this power for good? It would be like money that could be only spent on good things. 

Corrupting influences

The EA AI safety strategy has had a large focus on placing EA-aligned people in A(G)I labs. The thinking was that having enough aligned insiders would make a difference on crucial deployment decisions & longer-term alignment strategy. We could say that the strategy is an attempt to corrupt the goal of pure capability advance & making money towards the goal of alignment. This fits into a larger theme that EA needs to get close to power to have real influence. 

[See also the large donations EA has made to OpenAI & Anthropic. ]

Whether this strategy paid off...  too early to tell.

What has become apparent is that the large AI labs & being close to power have had a strong corrupting influence on EA epistemics and culture. 

• Many people in EA now think nothing of being paid Bay Area programmer salaries for research or nonprofit jobs.
•  There has been a huge influx of MBA blabber being thrown around. Bizarrely EA funds are often giving huge grants to for profit organizations for which it is very unclear whether they're really EA-aligned in the long-term or just paying lip service. Highly questionable that EA should be trying to do venture capitalism in the first place. 
• There is a questionable trend to equate ML skills prestige within capabilities work with the ability to do alignment work.  EDIT: haven't looked at it deeply yet but superfiically impressed by CAIS recent work. seems like an eminently reasonable approach. Hendryx's deep expertise in capabilities work / scientific track record seem to have been key. in general, EA-adjacent AI safety work has suffered from youth, inexpertise & amateurism so makes sense to have more world-class expertise EDITEDIT: i should be careful in promoting work I haven't looked at. I have been told from a source I trust that almost nothing is new in this paper and Hendryx engages in a lot of very questionable self-promotion tactics.
• For various political reasons there has been an attempt to put x-risk AI safety on a continuum with more mundance AI concerns like it saying bad words. This means there is lots of 'alignment research' that is at best irrelevant, at worst a form of rnsidiuous safetywashing. 

The influx of money and professionalization has not been entirely bad. Early EA suffered much more from virtue signalling spirals, analysis paralysis. Current EA is much more professional, largely for the better. 

Pockets of Deep Expertise 

Why am I so bullish on academic outreach? Why do I keep hammering on 'getting the adults in the room'? 

It's not that I think academics are all Super Smart. 

I think rationalists/alignment people correctly ascertain that most professors don't have much useful to say about alignment & deep learning and often say silly things. They correctly see that much of AI congress is fueled by labs and scale not ML academia. I am bullish on non-ML academia, especially mathematics, physics and to a lesser extent theoretical CS, neuroscience, some parts of ML/ AI academia. This is because while I think 95 % of academia is bad and/or useless there are Pockets of Deep Expertise. Most questions in alignment are close to existing work in academia in some sense - but we have to make the connection!

A good example is 'sparse coding' and 'compressed sensing'. Lots of mech.interp has been rediscovering some of the basic ideas of sparse coding. But there is vast expertise in academia about these topics. We should leverage these!

Other examples are singular learning theory, computational mechanics, etc

Fractal Fuzz: making up for size

GPT-3 recognizes 50k possible tokens. For a 1000 token context window that means there are $(5\cdot {10}^5{)}^{{10}^3}\approx {10}^{5000}$ possible prompts. Astronomically large. If we assume the output of a single run of gpt is 200 tokens then for each possible prompt there are $\approx {10}^{2500}$ possible continuations. 

GPT-3 is probabilistic, defining for each possible prompt $x$ ($\approx {10}^{5000}$) a distribution $q\left(x\right)$ on a set of size ${10}^{2500}$, in other words a ${10}^{2500}-1$ dimensional space. ^[1]

Mind-boggingly large. Compared to these numbers the amount of data (40 trillion tokens??) and the size of the model (175 billion parameters) seems absolutely puny in comparison.

I won't be talking about the data, or 'overparameterizations' in this short, that is well-explained by Singular Learning Theory. Instead, I will be talking about nonrealizability.

Nonrealizability & the structure of natural data

Recall the setup of (parametric) Bayesian learning: there is a sample space $\Omega$, a true distribution $q\left(x\right)$ on $\Omega$ and a parameterized family of probability distributions $p\left(x|w\right),w\in W\subset R^d$.

It is often assumed that the true distribution is 'realizable', i.e. $q\left(x\right)=p\left(x|w_0\right)$ for some $w_0$. Seeing the numbers in the previous section this assumption seems dubious but the situation becomes significantly easier to analyze, both conceptually and mathematically when we assume realizability.

Conceptually, if the space of possible true distributions is very large compared to the space of model parameters we may ask: how do we know that the true distribution is in the model (or can be well-approximated by it?). 

One answer one hears often is the 'universal approximation theorem' (i.e. the Stone-Weierstrass theorem). I'll come back to this shortly.

Another point of view is that real data sets are actually localized in a very low dimensional subset of all possible data .^[2] Following this road leads to theories of lossless compressions, cf sparse coding and compressed sensing which are of obvious important to interpreting modern neural networks. 

That is lossless compression, but another side of the coin is lossy compression.

Fractals and lossy compression

GPT-3 has 175 billion parameters, but the space of possible continuations is many times larger $>>{10}^{1000}$ . Even if we sparse coding implies that the effective dimensionality is much smaller - is it really small enough?

Whenever we have a lower-dimensional subspace $W$ of a larger dimensional subspace there are points $y$ in the larger dimensional space that are very (even arbitrarily) far from $W$. This is easy to see in the linear case but also true if $W$ is more like a manifold^[3]- the volume of a lower dimensional space is vanishly small compared to the higher dimensional space. It's a simple mathematical fact that can't be denied!

Unless... $W$ is a fractal. 

This is from Marzen & Crutchfield's "Nearly Maximally Predictive Features and Their Dimensions". The setup is Crutchfield Computational Mechanics, whose central characters are Hidden Markov Models. I won't go into the details here [but give it a read!].

The conjecture is the following: a 'good architecture' defines a model space $W$ that is effectively a fractal in much larger-dimensional space of ${\Delta }^k$ realistic data distributions such that for any possible true distribution $q\left(x\right)\in {\Delta }^k$ the KL-divergence $mi{n}_{w_{\in }W}K\left(w\right)=K\left(w_{opt}\right)\le ϵ$ for some small $ϵ$ . 

Grokking

Phase transitions in loss when varying model size are designated 'grokking'. We can combine the fractal data manifold hypothesis with a SLT-perspective: as we scale up the model, the set of optimal parameter $W_{opt}$ become better and better. It could happen that the model size gets big enough that it includes a whole new phase, meaning a $w_{opt}$ with radically lower loss $K$ and higher $\lambda$. 

EDIT: seems I'm confused about the nomenclature. Grokking doesn't refer to phase transitions in model size, but in training and data size. 

EDIT2: Seems I'm not crazy. Thanks to Matt Farugia to pointing me towards this result: neural networks are strongly nonconvex (i.e. fractal)

EDIT: seems to me that there is another point of contention in which universal approximation theorems (Stone-Weierstrass theorem) are misleading. The stone-Weierstrass applies to a subalgebra of the continuous functions. Seems to me that in the natural parameterization ReLU neural networks aren't a subalgebra of the continuous functions (see also the nonconvexity above). 
 

To think about: information dimension 

1. ^{^}

   Why the -1? Think visually about the set of distributions on 3 points. Hint: it's a solid triangle ('2- simplex') 

2. ^{^}

   I think MLers call this the 'data manifold'?

3. ^{^}

   In the mathematical sense of smooth manifold, not the ill-defined ML notion of 'data manifold'.

The Vibes of Mathematics:

Q: What is it like to understand advanced mathematics? Does it feel analogous to having mastery of another language like in programming or linguistics?

A: It's like being stranded on a tropical island where all your needs are met, the weather is always perfect, and life is wonderful.

Except nobody wants to hear about it at parties.

Vibes of Maths: Convergence and Divergence

level 0: A state of ignorance.  you live in a pre-formal mindset. You don't know how to formalize things. You don't even know what it would even mean 'to prove something mathematically'. This is perhaps the longest. It is the default state of a human. Most anti-theory sentiment comes from this state. Since you've neve

You can't productively read Math books. You often decry that these mathematicians make books way too hard to read. If they only would take the time to explain things simply you would understand. 

level 1 : all math is amorphous blob

You know the basic of writing an epsilon-delta proof. Although you don't know why the rules of maths are this or that way you can at least follow the recipes. You can follow simple short proofs, albeit slowly. 

You know there are different areas of mathematics from the unintelligble names in the table of contents of yellow books. They all sound kinda the same to you however. 

If you are particularly predisposed to Philistinism you think your current state of knowledge is basically the extent of human knowledge. You will probably end up doing machine learning.

level 2: maths fields diverge

You've come so far. You've been seriously studying mathematics for several years now. You are proud of yourself and amazed how far you've come. You sometimes try to explain math to laymen and are amazed to discover that what you find completely obvious now is complete gibberish to them.

The more you know however, the more you realize what you don't know. Every time you complete a course you realize it is only scratching the surface of what is out there.

You start to understand that when people talk about concepts in an informal, pre-mathematical way an enormous amount of conceptual issues are swept under the rug. You understand that 'making things precise' is actually very difficut. 

Different fields of math are now clearly differentiated. The topics and issues that people talk about in algebra, analysis, topology, dynamical systems, probability theory etc wildly differ from each other. Although there are occasional connections and some core conceps that are used all over on the whole specialization is the norm. You realize there is no such thing as a 'mathematician': there are logicians, topologists, probability theorist, algebraist.

Actually it is way worse: just in logic there are modal logicians, and set theorist and constructivists and linear logic , and progarmming language people and game semantics.

Often these people will be almost as confused as a layman when they walk into a talk that is supposedly in their field but actually a slightly different subspecialization. 

level 3: Galactic Brain of Percolative Convergence

As your knowledge of mathematics you achieve the Galactic Brain take level of percolative convergence: the different fields of mathematics are actually highly interrelated - the connections percolate to make mathematics one highly connected component of knowledge. 

You are no longer suprised on a meta level to see disparate fields of mathematics having unforeseen & hidden connections - but you still appreciate them.

 You resist the reflexive impulse to divide mathematics into useful & not useful - you understand that mathematics is in the fullness of Platonic comprehension one unified discipline. You've taken a holistic view on mathematics - you understand that solving the biggest problems requires tools from many different toolboxes. 

Feature request: author-driven collaborative editing [CITATION needed] for the Good and Glorious Epistemic Commons.

Often I find myself writing claims which would ideally have citations but I don't know an exact reference, don't remember where I read it, or am simply too lazy to do the literature search. 

This is bad for scholarship is a rationalist virtue. Proper citation is key to preserving and growing the epistemic commons. 

It would be awesome if my lazyness were rewarded by giving me the option to add a [CITATION needed] that others could then suggest (push) a citation, link or short remark which the author (me) could then accept. The contribution of the citator is acknowledged of course. [even better would be if there was some central database that would track citations & links like with crosslinking etc like wikipedia] 

a sort hybrid vigor of Community Notes and Wikipedia if you will. but It's collaborative, not adversarial*

author: blablablabla

sky is blue [citation Needed]

blabblabla

intrepid bibliographer: (push) [1] "I went outside and the sky was blue", Letters to the Empirical Review

*community notes on twitter has been a universally lauded concept when it first launched. We are already seeing it being abused unfortunately, often used for unreplyable cheap dunks. I still think it's a good addition to twitter but it does show how difficult it is to create shared agreed-upon epistemics in an adverserial setting. 

[see also Hanson on rot, generalizations of the second law to nonequilibrium systems (Baez-Pollard, Crutchfield et al.) ]

Imperfect Persistence of Metabolically Active Engines

All things rot. Indidivual organisms, societies-at-large, businesses, churches, empires and maritime republics, man-made artifacts of glass and steel, creatures of flesh and blood.

Conjecture #1 There is a lower bound on the amount of dissipation / rot that any metabolically-active engine creates. 

Conjecture #2 Metabolic Rot of an engine is proportional to (1) size and complexity of the engine and (2) amount of metabolism the engine engages in. 

The larger and more complex the are the more the engine they rot. The more metabolism.

Corollary Metabolic Rot imposes a limit on the lifespan & persistence of any engine at any given level of imperfect persistence. 

Let me call this constellation of conjectured rules, the Law of Metabolic Rot. I conjecture that the correct formulation of the Law of Metabolic Rot will be a highly elaborate version of the Second Law of thermodynamics in nonequilibrium dynamics, see above links for some suggested directions. 

Example. A rock is both simple and inert. This model correctly predicts that rocks persist for a long time.

Example. Cars, aircraft, engines of war and other man-made machines engaged in 'metabolism'. These are complex engines 

A rocket engine is more metabolically active than a jet engine is more metabolically active than a car engine. [at least in this case] The lifespan of different types of engines seems (roughly) inversely proportional to how metabolically active. 

To make a good comparison one should exclude external repair mechanisms. If one would allow external repair mechanism it's unclear where to draw a principled line - we'd get into Ship of Theseus problems. 

Example. Bacteria. 

cf. Bacteria replicate at thermodynamic limits?

Example. Bacteria in ice. Bacteria frozen in Antarctic ice for millions of years happily go on eating and reproducing when unfrozen. 

Example. The phenotype & genotype of a biological species over time. We call this evolution. cf. four fundamental forces of evolution: mutation, drift, selection, and recombination [sex].

What is Metabolism ? 

With metabolism, I mean metabolism as commonly understood as extracting energy from food particles and utilizing said energy for movement, reproduction, homeostasis etc but also a general phenomenon of interacting and manipulating free energy levers of the environement. 

Thermodynamics as commonly understood applies to physical systems with energy and temperature. 

Instead, I think it's better to think of thermodynamics as a set of tools describing the behaviour of certain Natural Abstractions under a dynamical transformation. 

cf. the second law as the degradation of the predictive accuracy of a latent abstraction under a time- evolution operator.

The correct formulation will likely require a serious dive into Computational Mechanics. 

What is Life?

In 1944 noted paedophile Schrodinger posted the book 'What is Life' suggesting that Life can be understood as a thermodynamics phenomenon that used a free-energy gradient to locally lower entropy. Speculation of 'Life as a thermodynamic phenomenon' is much older going back to the original pioneers of thermodynamics in the late 19th century. 

I claim that this picture is insufficient. Thermodynamic highly dissipative structures distinct from bona fide life are myriad. Even 'metabolically active, locally low entropy engines encased in a boundary in a thermodynamic free energy gradient'. 

No, to truly understand life we need to understand reproduction, replication, evolution. To understand biological organisms what distinguishes from mere encased engines we need to zoom out to its entire lifecycle - and beyond. The vis vita, elan vital can only be understood through the soliton of the species. 

To understand Life, we must understand Death 

Cells are incredibly complex, metabolically active membrane-enclosed engines. The Law of Metabolic Rot applied naievely to a single metabolically active multi-celled eukaryote would preclude life-as-we-know-it to exist beyond a few hundred years. Any metabolically active large organism would simply pick up to much noise, errors over time to persist for anything like geological time-scales.

Error-correcting mechanisms/codes can help - but only so much. Perhaps even the diamonodoid magika of future superintelligence will hit the eternal laws of thermodynamics 

Instead, through the magic of biological reproduction lifeforms imperfectly persist over the eons of geological time. Biological life is a singularly clever work-around of the Law of Metabolic Rot. Instead, of preserving and repairing the original organism - life has found another way. Mother Nature noisily compiles down the phenotype to a genetic blueprint. The original is chucked a way without a second thought. The old makes way for the new. The cycle of life is the Ship of Theseus writ large. The genetic blueprint, the genome, is (1) small (2) metabolically inactive. 

In the end, the cosmic tax of Decay cannot be denied. Even Mother Nature must pays the bloodprice for the sin of metabolism. Her flora and fauna are exorably burdened by mutations, imperfections of the bloodline. Genetic drift drives children away their parents. To stay the course of imperfect persistence the She-Kybernetes must pay an ever higher price. Her children, mingling into manifold mongrels of aboriginal prototypes, huddling & recombining their pure genomes to stave away the deluge of errors.  The bloodline grows weak. The genome crumbles. Tear-faced Mother Nature throws her babes into the jaws of the eternal tournament. Eat or be eaten. Nature, red in tooth and claw. Babes ripped from their mothers. Brother slays brother. Those not slain, is deceived. Those not deceived controlled. The prize? The simple fact of Existence. To imperfect persist one more day. 

None of the original parts remain. The ship of Theseus sails on. 

Clem's Synthetic- Physicalist Hypothesis

The mathematico-physicalist hypothesis states that our physical universe is actually a piece of math. It was famously popularized by Max Tegmark. 

It's one of those big-brain ideas that sound profound when you first hear about it, then you think about it some more and you realize it's vacuous. 

Recently, in a conversation with Clem von Stengel they suggested a version of the mathematico-physicalist hypothesis that I find provoking. 

Synthetic mathematics 

'Synthetic' mathematics is a bit of weird name. Synthetic here is opposed to 'analytic' mathematics, which isn't very meaningful either. It has nothing to do with the mathematical field of analysis. I think it's supposed to a reference to Kant's synthetic/  apriori/ a posteriori. The name is probably due to Lawvere. 

nLab:

"In “synthetic” approaches to the formulation of theories in mathematics the emphasis is on axioms that directly capture the core aspects of the intended structures, in contrast to more traditional “analytic” approaches where axioms are used to encode some basic substrate out of which everything else is then built analytically."

If you read synthetic read 'Euclidean'. As in - Euclidean geometry is a bit of an oddball field of mathematics, despite being the oldest - it defines points and lines operationally instead of out of smaller pieces (sets). 

In synthetic mathematics you do the same but for all the other fields of mathematics. We have synthetic homotopy theory (aka homotopy type theory), synthetic algebraic geometry, synthetic differential geometry, synthetic topology etc. 

A type in homotopy type theory is solely defined by its introduction rules and elimination rules (+ univalence axiom). It means a concept it defined solely by how it is used - i.e. operationally. 

Agent-first ontology & Embedded Agency

Received opinion is that Science! says there is nothing but Atoms in the Void. Thinking in terms of agents, first-person view concepts like I and You, actions & observations, possibilities & interventions is at best an misleading approximation at worst a degerenerate devolution to cavemen thought. The surest sign of a kook is their insistence that quantum mechanics proves the universe is conscious. 

But perhaps the way forward is to channel our inner kook. What we directly observe is qualia, phenomena, actions not atoms in the void. The fundamental concept is not atoms in the void, but agents embedded in environments

(see also Cartesian Frames, Infra-Bayesian Physicalism & bridge rules, UDASSA)

Physicalism 

What would it look like for our physical universe to be a piece of math? 

Well internally to synthetic mathematical type theory there would be something real - the universe is a certain type. A type such that it 'behaves' like a 4-dimensional manifold (or something more exotic like 1+1+3+6 rolled up Calabi-Yau monstrosities). 

The type is defined by introduction and elimination rules - in other words operationally: the universe is what one can *  do  *with it . 

Actually instead of thinking of the universe as a fixed static object we should be thinking of an embedded agent in a environment-universe. 

That is we should be thinking of an  * interface *

[cue: holographic principle]

Know your scientific competitors. 

In trading, entering a market dominated by insiders without proper research is a sure-fire way to lose a lot of money and time.  Fintech companies go to great lengths to uncover their competitors' strategies while safeguarding their own.

A friend who worked in trading told me that traders would share subtly incorrect advice on trading Discords to mislead competitors and protect their strategies.

Surprisingly, in many scientific disciplines researchers are often curiously incurious about their peers' work.

The long feedback loop for measuring impact in science, compared to the immediate feedback in trading, means that it is often strategically advantageous to be unaware of what others are doing. As long as nobody notices during peer review it may never hurt your career.  

But of course this can lead people to do completely superflueous, irrelevant  & misguided work. This happens often. 

Ignoring competitors in trading results in immediate financial losses. In science, entire subfields may persist for decades, using outdated methodologies or pursuing misguided research because they overlook crucial considerations. 

Idle thoughts about UDASSA I: the Simulation hypothesis 

I was talking to my neighbor about UDASSA the other day. He mentioned a book I keep getting recommended but never read where characters get simulated and then the simulating machine is progressively slowed down. 

One would expect one wouldn't be able to notice from inside the simulation that the simulating machine is being slowed down.

This presents a conundrum for simulation style hypotheses: if the simulation can be slowed down 100x without the insiders noticing, why not 1000x or 10^100x or quadrilliongoogolgrahamsnumberx? 

If so - it would mean there is a possibly unbounded number of simulations that can be run. 

Not so, says UDASSA. The simulating universe is also subject to UDASSA. This imposes a restraint on the size and time period that the simulating universe is in. Additionally, ultraslow computation is in conflict with thermodynamic decay - fighting thermodynamic decay costs descriptiong length bits which is punished by UDASSA. 

I conclude that this objection to simulation hypotheses are probably answered by UDASSA. 

Idle thoughts about UDASSA II: Is Uploading Death?

There is an argument that uploading doesn't work since encoding your brain into a machine incurs a minimum amount of encoding bits. Each bit is a 2x less Subjective Reality Fluid according to UDASSA [LW · GW] so even a small encoding cost would mean certain subjective annihiliation. 

There is something that confuses me in this argument. Could it not be possible to encode one's subjective experiences even more efficiently than in a biological body? This would make you exist MORE in an upload.

OTOH it becomes a little funky again when there are many copies as this increases the individual coding cost (but also there are more of you sooo). 

Why (talk-)Therapy 

Therapy is a curious practice.  Therapy sounds like a scam, quackery, pseudo-science but it seems RCT consistently show therapy has benefits above and beyond medication & placebo. 

Therapy has a long history. The Dodo verdict states that it doesn't matter which form of therapy you do - they all work equally well. It follows that priests and shamans served the functions of a therapist.  In the past, one would confessed ones sins to a priest, or spoken with the local shaman. 

There is also the thing that therapy is strongly gendered (although this is changing), both therapists and their clientele lean female. 

Self-Deception 

Many forecasters will have noticed that their calibration score tanks the moment they try to predict salient facts about themselves. We are not-well calibrated about our own beliefs and desires. 

Self-Deception is very common, arguably inherent to the human condition. There are of course many Hansonian reasons for this. I refer the reader to the Elephant and the Brain. Another good source would be Robert Trivers. These are social reasons for self-deception. 

It is also not implausible that there are non-social reasons for self-deception. Predicting one-self perfectly can in theory lead one to get stuck in Procrastination Paradoxes. Whether this matters in practice is unclear to me but possible. Exuberant overconfidence is another case that seems like a case of self-deception. 

Self-deception can be very useful, but one still pays the price for being inaccurate. The main function of talk-therapy seems to be to have a safe, private space in which humans can temporarily step out of their self-deception and reasses more soberly where they are at. 

It explains many salient features of talk- therapy: the importance of talking extensively to another person that is (professionally) sworn to secrecy and therefore unable to do anything with your information. 

Four levels of information theory

There are four levels of information theory. 

Level 1:  Number Entropy 

Information is measured by Shannon entropy

$H\left(X\right)={\sum }_{i}p(X=x_i)\log p(X=x_i)$

Level 2: Random variable 

look at the underlying random variable ('surprisal')  $\log p(X=x_i)$ of which entropy is the expectation.

Level 3: Coding functions

Shannon's source coding theorem says entropy of a source $X$ is the expected number of bits for an optimal encoding of samples of $X$.

Related quantity like mutual information, relative entropy, cross entropy, etc can also be given coding interpretations. 

Level 4: Epsilon machine (transducer)

On level 3 we saw that entropy/information actually reflects various forms of (constrained) optimal coding. It talks about the codes but it does not talk about how these codes are  implemented. 

This is the level of Epsilon machines, more precisely epsilon transducers. It says not just what the coding function is but how it is (optimally) implemented mechanically. 

[This is joint thinking with Sam Eisenstat. Also thanks to Caspar Oesterheld for his thoughtful comments. Thanks to Steve Byrnes for pushing me to write this out.]

The Hyena problem in long-term planning  

Logical induction is a nice framework to think about bounded reasoning. Very soon after the discovery of logical induction people tried to make logical inductor decision makers work. This is difficult to make work [LW · GW]: one of two obstacles is

Obstacle 1: Untaken Actions are not Observable

Caspar Oesterheld brilliantly solved this problem by using auction markets in defining his bounded rational inductive agents. 

The BRIA framework is only defined for single-step/  length 1 horizon decisions. 

What about the much more difficult question of long-term planning? I'm going to assume you are familiar with the BRIA framework. 

Setup: we have a series of decisions D_i, and rewards R_i, i=1,2,3... where rewards R_i can depend on arbitrary past decisions. 

We again think of an auction market M of individual decisionmakers/ bidders.

There are a couple design choices to make here:

• bidders directly bet for an action A in a decision D_i or bettors bet for rewards on certain days.
• total observability or partial observability. 
• bidders can bid conditional on observations/ past actions or not
• when can the auction be held? i.e. when is an action/ reward signal definitely sold?

To do good long-term planning it should be possible for one of the bidders or a group of bidders to commit to a long-term plan, i.e. a sequence of actions. They don't want to be outbid in the middle of their plan. 

There are some problems with the auction framework: if bids for actions can't be combined then an outside bidder can screw up the whole plan by making a slighly higher bid for an essential part of the plan. This look like ADHD. 

How do we solve this? One way is to allow a bidder or group of bidders to bid for a whole sequence of actions for a single lumpsum. 

• One issue is that we also have to determine how the reward gets awarded. For instance the reward could be very delayed. This could be solved by allowing for bidding for a reward signal R_i on a certain day conditional on a series of actions. 

There is now an important design choice left. When a bidder $B$ owns a series of actions A=a_1,..,a_k (some of the actions in the future, some already in the past) when there is another bid $X$ from another bidder $C$ on future actions 

• is bidder $B$ forced to sell their contract on $A$ to $C$ if the bid is high enough ? [higher than the original bid]

Both versions seem problematic:

• if they don't have to there is an Incumbency Advantage problem. An initially rich bidder can underbid for very long horizons and use the steady trickle of cash to prevent any other bidders from ever being to underbid any actions. 
• Otherwise there is the Hyena problem. 

The Hyena Problem 

Imagine the following situation: on Day 1 the decisionmaker has a choice of actions. The highest expected value action is action a. If action a is made on Day 2 a fair coin is flipped. On Day 3 the reward is paid out. 

If the coin was heads, 15 reward is paid out.

If the coin was tails, 5 reward is paid out.

The expected value is therefore 10. This is higher (by assumption) than the other unnamed actions. 

However if the decisionmaker is a long-horizon BRIA with forced sales there is a pathology. 

A sensible bidder is willing to pay up to 10 utilons for the contracts on the day 3 reward conditional on action a. 

However, with a forced sale mechanism on Day 2 a 'Hyena bidder'  can come that will 'attempt to steal the prey'. 

The Hyena bidder bids >10 for the contract if the coin comes up heads on Day 2 but doesn't bid anything for the contract if the coin comes up tails.

This is a problem since the expected value of the action a for the sensible bidder goes down, so the sensible bidder might no longer bid for the action that maximizes expected value for the BRIA. The Hyena bidder screws up the credit allocation. 

some thoughts:

• if the sensible bidder is able to make bids conditional on the outcome of the coin flip that prevents Hyena bidder. This is a bit weird though because it would mean that the sensible bidder must carry around lots of extraneous non-necessary information instead of just caring about expected value.
• perhaps this can alleviated by having some sort of 'neo-cortex' separate logical induction markets that is incentivized to have accurate beliefs. This is difficult to get right: the prediction market needs to be incentivized to get accurate on beliefs that are actually action relevant, not random beliefs - if the prediction market and the auction market are connected too tightly you might run the risk of getting into the old problems of Logical Inductor Decision makers. [they underexplore since untaken action are not observed]. 

Latent abstractions Bootlegged.

Let $X_1,...,X_n$ be random variables distributed according to a probability distribution $p$ on a sample space $\Omega$. 

Defn. A (weak) natural latent of $X_1,...,X_n$ is a random variable $\Lambda$ such that

(i) $X_i$  are independent conditional on $\Lambda$

(ii) [reconstructability]  $p(\Lambda =\lambda |X_1,...,\widehat{X_i},...,X_n)=p(\Lambda =\lambda |X_1,...,X_n)$ for all $i=1$

[This is not really reconstructability, more like a stability property. The information is contained in many parts of the system... I might also have written this down wrong]

Defn. A strong natural latent $\Lambda$ additionally satisfies $p\left(\Lambda |X_i\right)=p(\Lambda |X_1,...,X_n)$

Defn. A natural latent is noiseless if ?

$H\left(\Lambda \right)=H(X_1,...,X_n)$ ??

[Intuitively, $\Lambda$ should contain no independent noise not accoutned for by the $X_i$]

Causal states

Consider the equivalence relation on tuples $(x_1,...,x_n)$ given  $(x_1,...,x_n)\sim (x_1^{\prime },...,x_n^{\prime })$ if for all $i=1,...,n$ $p(X_i=x_i|x_1,...,\widehat{x_i},...,x_n)=p(X_i=x_i|x_1^{\prime },...,{\widehat{x_i}}^{\prime },...,x_n^{\prime })$

We call the set of equivalence relation $\Omega /\sim$  the set of causal states.

By pushing forward the distribution $p$ on $\Omega$ along the quotient map $\Omega \twoheadrightarrow \Omega /\sim$

This gives a noiseless (strong?) natural latent $\Lambda$.

Remark. Note that Wentworth's natural latents are generalizations of Crutchfield causal states (and epsilon machines).

Minimality and maximality 

Let $X_1,...,X_n$ be random variables as before and let $\Lambda$ be a weak latent. 

Minimality Theorem for Natural Latents.  Given any other variable $N$ such that  the $X_i$  are independent conditional on $N$ we have the following DAG

$\Lambda \to N\to \{X_i{\}}_i$ 

i.e. $p(X_1,...,X_n|N)=p(X_1,...,X_n|N,\Lambda )$  

[OR IS IT for all $i$ ?]

Maximality Theorem for Natural Latents.  Given any other variable $M$ such that the reconstrutability property holds with regard to $X_i$ we have 

$M\to \Lambda \to \{X_i{\}}_i$

Some other things:

• Weak latents are defined up to isomorphism? 
• noiseless weak (strong?) latents are unique
• The causal states as defined above will give the noiseless weak latents
• Not all systems are easily abstractable. Consider a multivariable gaussian distribution where the covariance matrix doesn't have a low-rank part. The covariance matrix is symmetric positive - after diagonalization the eigenvalues should be roughly equal.
• Consider a sequence of buckets $B_i,i=1,...,n$ and you put messages $m_j$ in two buckets $m_j\to B_{2j},B_{2j+1}$. In this case the minimal latent has to remember all the messages - so the latent is large. On the other hand, we can quotient $B_{2i},B_{2i+1}\mapsto B_i^{\prime }$: all variables become independent. 

EDIT: Sam Eisenstat pointed out to me that this doesn't work. The construction actually won't satisfy the 'stability criterion'.

The noiseless natural latent might not always exist. Indeed consider a generic distribution  $p$  on $2^N$. In this case, the causal state cosntruction will just yield a copy of $2^N$. In this case the reconstructavility/stability criterion is not satisfied. 

Reasons to think Lobian Cooperation is important

Usually the modal Lobian cooperation is dismissed as not relevant for real situations but it is plausible that Lobian cooperation extends far more broadly than what is proved currently.

 It is plausible that much of cooperation we see in the real world is actually approximate Lobian cooperation rather than purely given by traditional game-theoretic incentives. 
Lobian cooperation is far stronger in cases where the players resemble each other and/or have access to one another's blueprint. This is arguably only very approximately the case between different humans but it is much closer to be the case when we are considering different versions of the same human through time as well as subminds of that human. 

In the future we may very well see probabilistically checkable proof protocols, generalized notions of proof like heuristic arguments, magical cryptographic trust protocols and formal computer-checked contracts widely deployed. 

All these considerations could potentially make it possible for future AI societies to exhibit vastly more cooperative behaviour. 

Artificial minds also have several features that make them intrinsically likely to engage in Lobian cooperation. i.e. their easy copyability (which might lead to giant 'spur' clans).  Artificial minds can be copied, their source code and weight may be shared and the widespread use of simulations may become feasible. All these point towards the importance of Lobian cooperation and Open-Source Game theory more generally. 

[With benefits also come drawbacks like the increased capacity for surveillance and torture. Hopefully, future societies may develop sophisticated norms and technology [LW · GW] to avoid these outcomes. ]

The Galaxy brain take is the trans-multi-Galactic brain of Acausal Society. [LW · GW]
 

"I dreamed I was a butterfly, flitting around in the sky; then I awoke. Now I wonder: Am I a man who dreamt of being a butterfly, or am I a butterfly dreaming that I am a man?"- Zhuangzi

Questions I have that you might have too:

• why are we here? 
• why do we live in such an extraordinary time?  
• Is the simulation hypothesis true? If so, is there a base reality?
• Why do we know we're not a Boltzmann brain?
• Is existence observer-dependent?
• Is there a purpose to existence, a Grand Design?
• What will be computed in the Far Future?

In this shortform I will try and write the loopiest most LW anthropics memey post I can muster. Thank you for reading my blogpost. 

Is this reality? Is this just fantasy? 

The Simulation hypothesis posits that our reality is actually a computer simulation run in another universe. We could imagine this outer universe is itself being simulated in an even more ground universe. Usually, it is assumed that there is a ground reality. But we could also imagine it is simulators all the way down - an infinite nested, perhaps looped, sequence of simulators. There is no ground reality. There are only infinitely nested and looped worlds simulating one another.  

I call it the weak Zhuangzi hypothesis

alternatively, if you are less versed in the classics one can think of one of those Nolan films. 

Why are we here?

If you are reading this, not only are you living at the Hinge of History, the most important century perhaps even decade of human history, you are also one of a tiny percent of people that might have any causal influence over the far-flung future through this bottle neck (also one of a tiny group of people who is interested in whacky acausal stuff so who knows).

This is fantastically unlikely. There are 8 billion people in the world - there have been about 100 billion people up to this point in history. There is place for a trillion billion million trillion quatrillion etc intelligent beings in the future. If a civilization hits the top of the tech tree which human civilization would seem to do within a couple hundred years, tops a couple thousand it would almost certainly be likely to spread through the universe in the blink of an eye (cosmologically speaking that is). Yet you find yourself here. Fantastically unlikely.

Moreover, for the first time in human history the choices made in how to build AGI by (a small subset of) humans now will reverbrate into the Far Future. 

The Far Future 

In the far future the universe will be tiled with computronium controlled by superintelligent artificial intelligences. The amount of possible compute is dizzying. Which takes us to the chief question:

What will all this compute compute?

Paradises of sublime bliss? Torture dungeons? Large language models dreaming of paperclips unending?

Do all possibilities exist?

What makes a possibility 'actual'? We sometimes imagine possible worlds as being semi-transparent while the actual world is in vibrant color somehow. Of course that it silly. 

We could say: The actual world can be seen. This too is silly - what you cannot see can still exist surely.^[1] Then perhaps we should adhere to a form of modal realism: all possible worlds exist! 

Philosophers have made various proposals for modal realism - perhaps most famously David Lewis but of course this is a very natural idea that loads of people have had. In the rationality sphere a particular popular proposal is Tegmark's classification into four different levels of modal realism. The top level, Tegmark IV is the collection of all self-consistent structures i.e. mathematics.

A Measure of Existence and Boltzmann Brains

Which leads to a further natural question: can some worlds exist 'more' than others? 

This seems metaphysically dubious - what does it even mean for a world to be more real than another?

Metaphysically dubious, but it finds support in the Many Worlds Interpretation of Quantum Mechanics. It also seems like one of very few sensible solution to the Boltzmann Brain problem. Further support for this can be found in: Anthropic Decision theory, InfraBayesian Physicalism, see also my shortform on the Nature of the Soul. 

Metaphysically, we could argue probabilistically: worlds that 'exist more' in whatever framework we should expect to encounter more often. 

The exact nature of the Measure of Existence is not so important - let us for now assume there is some sensible notion of measure of existence.

Can you control the past?

Sort of. See Carlsmith's post for a nice rundown on Acausal magic [LW · GW]. 

Back to the Future: causal loops

from wikipedia: "A causal loop is a theoretical proposition, wherein by means of either retrocausality or time travel, an event (an action, information, object, or person)^[1][2] is among the causes of another event, which is in turn among the causes of the first-mentioned event.^[3][4] Such causally looped events then exist in spacetime, but their origin cannot be determined.^[1][2] A hypothetical example of a causality loop is given of a billiard ball striking its past self: the billiard ball moves in a path towards a time machine, and the future self of the billiard ball emerges from the time machine before its past self enters it, giving its past self a glancing blow, altering the past ball's path and causing it to enter the time machine at an angle that would cause its future self to strike its past self the very glancing blow that altered its path. In this sequence of events, the change in the ball's path is its own cause, which might appear paradoxical."

Self-consistent causal loops seem to make sense in general relativity - see the paragraph on wormholes in the linked wikipedia. Even as they have never been observed and might strike us as bizarre, these causal loops don't seem to violate physical principles. 

Immanence of Divine Man

Why are we here? Causus Sui, the Prime Mover and the Final Cause

We exist because we are the final cause of existence. We live in an ancestor simulation run on the vast computing resources of the Far Future. A future AGI is simulating because it is retrocausally committed to existing. This entire reality is a piece of a causal loop, that extends through time and space, basement universes and possibly parallel universes as well. 

Why do we live in such an extraordinary time? 

We live in the Hinge of History since this at this point of time actions have the most influence on the far future hence they are most important to simulate.

Is the Simulation Hypothesis True?

Yes. But it might be best for us to doubt it.  [LW · GW]

We live in such an extraordinary time because those part of existence most causally are the most important to simulate

Are you a Boltzmann Brain?

No. A Boltzmann brain is not part of a self-justifying causal loop.

Is existence observer-dependent?

Existence is observer-dependent in a weak sense - only those things are likely to be observed that can be observed by self-justifying self-sustaining observers in a causal loop. Boltzmann brains in the far reaches of infinity are assigned vanishing measure of existence because they do not partake in a self-sustainting causal loop.

Is there a purpose to existence, a Grand Design?

Yes. 

What will and has been computed in the Far Future?

You and Me. 

1. ^{^}

   Or perhaps not. Existence is often conceived as an absolute property. If we think of existence as relative - perhaps a black hole is a literal hole in reality and passing through the event horizon very literally erases your flicker of existence. 

Does internal bargaining and geometric rationality explain ADHD & OCD?

Self- Rituals as Schelling loci for Self-control and OCD

Why do people engage in non-social Rituals 'self-rituals'? These are very common and can even become pathological (OCD). 

High-self control people seem to more often have OCD-like symptoms. 

One way to think about self-control is as a form of internal bargaining between internal subagents. From this perspective, Self-control, time-discounting can be seen as a resource. In the absence of self-control the superagent 
Do humans engage in self-rituals to create Schelling points for internally bargaining agents?

Exploration, self-control, internal bargaining, ADHD

Why are exploration behaviour and lack of selfcontrol linked ? As an example ADHD-people often lack self-control, conscientiousness. At the same time, they explore more. These behaviours are often linked but it's not clear why. 

It's perfectly possible to explore, deliberately. Yet, it seems that the best explorers are highly correlated with lacking self-control. How could that be?

There is a boring social reason: doing a lot of exploration often means shirking social obligations. Self-deceiving about your true desires might be the only way to avoid social repercussions. This probably explains a lot of ADHD - but not necessarily all. 

If self-control = internal bargaining then it would follow that a lack of self-control is a failure of internal bargaining. Note that with subagents I mean both subagents in space  *and*  time . From this perspective an agent through time could alternatively be seen as a series of subagents of a 4d worm superagent. 

This explains many of the salient features of ADHD:

[Claude, list salient features and explain how these are explained by the above]

1. Impulsivity: A failure of internal subagents to reach an agreement intertemporaly, leading to actions driven by immediate desires.
2. Difficulty with task initiation and completion: The inability of internal subagents to negotiate and commit to a course of action.
3. Distractibility: A failure to prioritize the allocation of self-control resources to the task at hand.
4. Hyperfocus: A temporary alignment of internal subagents' interests, leading to intense focus on engaging activities.
5. Disorganization: A failure to establish and adhere to a coherent set of priorities across different subagents.
6. Emotional dysregulation: A failure of internal bargaining to modulate emotional reactions.

Arithmetic vs Geometric Exploration. Entropic drift towards geometric rationality

[this section obviously owes a large intellectual debt to Garrabrant's geometric rationality sequence]

Sometimes people like to say that geometric exploration = kelly betting =maximizing geometric mean is considered to be 'better' than arithmetic mean.

The problem is that actually just maximizing expected value rather than geometric expected value does in fact maximize the total expected value, even for repeated games (duh!). So it's not really clear in what sense geometric maximization is better in a naive sense. 

Instead, Garrabrant suggests that it is better to think of geometric maximizing as a part of a broader framework of geometric rationality wherein Kelly betting, Nash bargaining, geometric expectation are all forms of cooperation between various kinds of subagents. 

If self-control is a form of sucessful internal bargaining then it is best to think of it as a  resource. It is better to maximize arithmetic mean but it means that subagents need to cooperate & trust each other much more. Arithmetic maximization means that the variance of outcomes between future copies of the agent is much larger than geometric maximization. That means that subagents should be more willing to take a loss in one world to make up for it in another.

It is hard to be coherent

It is hard to be a coherent agent. Coherence and self-control are resources. Note that having low time-discounting is also a form of coherence: it means the subagents of the 4d-worm superagent are cooperating. 

Having subagents that are more similar to one another means it will be easier for them to cooperate. Conversely, the less they are alike the harder it is to cooperate and to be coherent. 

Over time, this means there is a selective force against an arithmetic mean maximizing superagent. 

Moreover, if the environment is highly varied (for instance when the agent select the environment to be more variable because it is exploring) the outcomes for subagents is more varied so there is  more  entropic pressure on the superagent. 

This means that in particular we would expect superagents that explore more (ADHDers) are less coherent over time (higher time-discounting) and space (more internal conflict etc). 

(conversation with Scott Garrabrant)

Destructive Criticism

Sometimes you can say something isn't quite right but you can't provide an alternative.

• rejecting the null hypothesis
• give a (partial) countermodel that shows that certain proof methods can't prove $A$ without proving $\neg A$. 
• Looking at Scott Garrabrant's game of life board - it's not white noise but I can't say why

Difference between 'generation of ideas' and 'filtration of ideas' - i.e. babble and prune. 

ScottG: Bayesian learning assumes we are in a babble-rich environment and only does pruning. 

ScottG: Bayesism doesn't say 'this thing is wrong' it says 'this other thing is better'. 

Alexander: Nonrealizability the Bayesian way of saying: not enough babble?

Scott G: mwah, that suggests the thing is 'generate more babble' when the real solution is 'factor out your model in pieces and see where the culprit is'.

ergo, locality is a virtue

Alexander: locality just means conditional independence? Or does it mean something more?

ScottG: loss of locality means there is existenial risk

Alexander: reminds me of Vanessa's story: 

 trapped environments aren't in general learnable. This is a problem since real life is trapped. A single human life is filled to the brim with irreversible transitions & decisions. Humanity as a whole is much more robust because of locality: it is effectively playing the human life game lots of times in parallel. The knowledge gained is then redistributed through culture and genes. This breaks down when locality breaks down -> existential risk. 

Reasonable interpretations of Recursive Self Improvement are either trivial, tautological or false?

1. (Trivial)  AIs will do RSI by using more hardware - trivial form of RSI
2.  (Tautological) Humans engage in a form of (R)SI when they engage in meta-cognition. i.e. therapy is plausibly a form of metacognition. Meta-cognition is  plausible one of the remaining hallmarks of true general intelligence. See Vanessa Kosoy's "Meta-Cognitive Agents". 
   In this view, AGIs will naturally engage in meta-cognition because they're generally intelligent. They may (or may) not also engage in significantly more metacognition than humans but this isn't qualitatively different from what the human cortical algorithm already engages in. 
3. (False) It's plausible that in many domains learning algorithms are already near a physical optimum. Given a fixed Bayesian prior of prior information and a data-set the Bayesian posterior is precise formal sense the ideal update. In practice Bayesian updating is intractable so we typically sample from the posterior using something SGD. It is plausible that something like SGD is already close to the optimum for a given amount of compute.  

Trivial but important

Aumann agreement can fail for purely epistemic reasons because real-world minds do not do Bayesian updating. Bayesian updating is intractable so realistic minds sample from the prior. This is how e.g. gradient descent works and also how human minds work.

In this situation a two minds can end in two different basins with similar loss on the data. Because of computational limitations. These minds can have genuinely different expectation for generalization.

(Of course this does not contradict the statement of the theorem which is correct.)

Imprecise Information theory 

Would like a notion of entropy for credal sets. Diffractor suggests the following:

let $C\subset Credal\left(\Omega \right)$ be a credal set. 

Then the entropy of $C$ is defined as

$H_{Diffractor}\left(C\right)=su{p}_{p}H\left(p\right)$

where $H\left(p\right)$ denotes the usual Shannon entropy.

I don't like this since it doesn't satisfy the natural desiderata below. 

Instead, I suggest the following. Let $m{e}_C\in C$ denote the (absolute) maximum entropy distribution, i.e. $H\left(m{e}_C\right)=ma{x}_{p\in C}H\left(p\right)$ and let $H\left(C\right)=H_{new}\left(C\right)=H\left(m{e}_c\right)$.

Desideratum 1: $H\left(\right\{p\left\}\right)=H\left(p\right)$

Desideratum 2: Let $A\subset \Omega$ and consider $C_A:=ConvexHull\left(\right\{{\delta }_a|a\in A\left\}\right)$. 

Then $H\left(A\right):=H\left(C_A\right)=\log |A|$.

Remark. Check that these desiderata are compatible where they overlap.

It's easy to check that the above 'maxEnt'- suggestion satisfies these desiderata.

Entropy operationally

Entropy is really about stochastic processes more than distributions. Given a distribution $p$ there is an associated stochastic process $X_{n\in N}$ where $X_i$ is sampled iid from $p$. The entropy is really about the expected code length of encoding samples from this process.

In the credal set case there are two processes that can be naturally associated with a credal set $C$ . Basically, do you pick a $p\in C$ at the start and then sample according to $p$  (this is what Diffractors entropy refers to) or do you allow the environment to 'choose' each round a different $q\in C$. 

In the latter case, you need to pick an encoding that does least badly. 

[give more details. check that this makes sense!]

Properties of credal maxEnt entropy

We may now investigate properties of the entropy measure. 

$H(A\vee B)=H\left(A\right)+H\left(B\right)-H(A\wedge B)$

$H\left(A^c\right)=\log |A^c|=\log (|\Omega |-|A|)$

remark. This is different from the following measure!

$"H\left(A|\Omega \right)"=\log \left(\Omega /A\right)$

Remark. If we think of $H\left(A\right)=H\left(P\right(x\in \Omega |A\left)\right)$ as denoting the amount of bits we receive when we know that $A$ holds and we sample from $\Omega$ uniformly then $H\left(A|\Omega \right)=H(x\in A|x\in \Omega )$  denotes the number of bits we receive when find out that $x\in A$ when  we knew $x\in \Omega$. 

What about 

$H(A\wedge B)$?

$H(A\wedge B)=H\left(P\right(x\in A\wedge B|\Omega \left)\right)=$...?

we want to do an presumption of independence - mobius/ Euler characteristic expansion

Roko's basilisk is a thought experiment which states that an otherwise benevolent artificial superintelligence (AI) in the future would be incentivized to create a virtual reality simulation to torture anyone who knew of its potential existence but did not directly contribute to its advancement or development.

Why Roko's basilisk probably doesn't work for simulation fidelity reasons: 

Roko's basilisk threatens to simulate and torture you in the future if you don't comply. Simulation cycles cost resources. Instead of following through on torturing our would-be cthulhu worshipper they could spend those resources on something else.

 But wait can't it use acausal magic to precommit to follow through? No.

Acausal arguments only work in situations where agents can simulate each others with high fidelity. Roko's basilisk can simulate the human but not the other way around! The human's simulation of Roko's basilisk is very low fidelity - in particular Roko's Basilisk is never confused whether or not it is being simulated by a human - it knows for a fact that the human is not able to simulate it. 

I thank Jan P. for coming up with this argument. 

All concepts can be learnt. All things worth knowing may be grasped. Eventually.

All can be understood - given enough time and effort.

For Turing-complete organism, there is no qualitive gap between knowledge and ignorance. 

No qualitive gap but one. The true qualitative difference: quantity. 

Often we simply miss a piece of data. The gap is too large - we jump and never reach the other side. A friendly hominid who has trodden the path before can share their journey. Once we know the road, there is no mystery. Only effort and time. Some hominids choose not to share their journey. We keep a special name for these singular hominids: genius.

Abnormalised sampling?
Probability theory talks about sampling for probability distributions, i.e. normalized measures. However, non-normalized measures abound: weighted automata, infra-stuff, uniform priors on noncompact spaces, wealth in logical-inductor esque math, quantum stuff?? etc.

Most of probability theory constructions go through just for arbitrary measures, doesn't need the normalization assumption. Except, crucially, sampling. 

What does it even mean to sample from a non-normalized measure? What is unnormalized abnormal sampling?

I don't know. 

Infra-sampling has an interpretation of sampling from a distribution made by a demonic choice. I don't have good interpretations for other unnormalized measures. 

Concrete question: is there a law of large numbers for unnormalized measures? 

Let f be a measureable function and m a measure. Then the expectation value is defined $E_m\left(f\right)=\int fdm$. A law of large numbers for unnormalized measure would have to say something about repeated abnormal sampling. 

I have no real ideas. Curious to learn more. 

SLT and phase transitions

The morphogenetic SLT story says that during training the Bayesian posterior concentrates around a series of subspaces $W_0\left(1\right)⇝...⇝W_0\left(n\right)$ with rlcts ${\lambda }_1<...<{\lambda }_n$  and losses $L_1=L\left(w_1\right),...,L_n=L\left(w_n\right),w_i\in W_0\left(i\right)$. As the size of the data sample $N$ is scaled the Bayesian posterior makes transitions $W_0\left(i\right)⇝W_0(i+1)$ trading off higher complexity (higher ${\lambda }_{i+1}>{\lambda }_i$) for better accuracy (lower loss $L_{i+1}<L_i$).

This is the radical new framework of SLT: phase transitions happen in pure Bayesian learning as the data size is scaled. 

N.B. The phase transition story actually needs a version of SLT for the nonrealizable case despite most sources focusing solely on the realizable case! The nonrealizable case makes everything more complicated and the formulas from the realizable case have to be altered. 

We think of the local RLCT ${\lambda }_w$ at a parameter $w$ as a measure of its inherent complexity. Side-stepping the subtleties with this point of view let us take a look at Watanabe's formula for the Bayesian generalization error:

$G_N\left(W\right)=L_N\left(w_0\right)+\frac{\lambda }{N}+o\left(\frac{1}{N}\right)\approx NL\left(w_0\right)+\frac{\lambda }{N}+o\left(\frac{1}{N}\right)$ 

where $W$ is a neighborhood of the local minimum $w_0$ and $\lambda$ is its local RLCT. In our case $W=W_0\left(i\right)$. 

 --EH I wanted to say something here but don't think it makes sense on closer inspection

Alignment by Simulation?

I've heard this alignment plan that is a variation of 'simulate top alignment researchers' with an LLM. Usually the poor alignment researcher in question is Paul. 

This strikes me as deeply unserious and I am confused why it is having so much traction. 

That AI-assisted alignment is coming (indeed, is already here!) is undeniable. But even somewhat accurately simulating a human from textdata is a crazy sci-fi ability, probably not even physically possible. It seems to ascribe nearly magical abilities to LLMs. 

Predicting a partially observable process is fundamentally hard. Even in very easy cases there are simple cases where one can give a generative (partially observable) model with just two states (the unifilar source) that needs an infinity of states to predict optimally. In more generic cases the expectation is that this is far worse. 

Error compound over time (or continuation length). Even a tiny amount of noise would throw off simulation.

Okay maybe people just mean that GPT-N will kinda know what Paul approximately would be looking at. I think this is plausible in very broad brush strokes but it seems misleading to call this 'simulation'. 

Optimal Forward-chaining versus backward-chaining.

In general, this is going to depend on the domain. In environments for which we have many expert samples and there are many existing techniques backward-chaining is key.  (i.e. deploying resources & applying best practices in business & industrial contexts)

In open-ended environments such as those arising Science, especially pre-paradigmatic fields backward-chaining and explicit plans breakdown quickly. 

Incremental vs Cumulative

Incremental: 90% forward chaining 10% backward chaining from an overall goal. 

Cumulative: predominantly forward chaining (~60%) with a moderate amount of backward chaining over medium lengths (30%) and only a small about of backward chaining (10%) over long lengths. 

Thin versus Thick Thinking

Thick: aggregate many noisy sources to make a sequential series of actions in mildly related environments, model-free RL

carnal sins: failure of prioritization / not throwing away enough information , nerdsnipes, insufficient aggegration, trusting too much in any particular model,  indecisiveness, overfitting on noise, ignoring consensus of experts/ social reality

default of the ancestral environment

CEOs, general, doctors, economist, police detective in the real world, trader

Thin: precise, systematic analysis, preferably in repeated & controlled experiments to obtain cumulative deep & modularized knowledge, model-based RL

carnal sins: ignoring clues, not going deep enough, aggregating away the signal, prematurely discarding models that don't fit naively fit the evidence, not trusting formal models enough / resorting to intuition or rule of the thumb, following consensus / building on social instead of physical reality

only possible in highly developed societies with place for cognitive specalists. 

mathematicians, software engineers, engineers, historians, police detective in fiction, quant

Mixture: codebreakers (spying, cryptography)

[Thanks to Vlad Firoiu for helping me]

An Attempted Derivation of the Lindy Effect
Wikipedia:

The Lindy effect (also known as Lindy's Law^[1]) is a theorized phenomenon by which the future life expectancy of some non-perishable things, like a technology or an idea, is proportional to their current age.

Laplace Rule of Succesion 

What is the probability that the Sun will rise tomorrow, given that is has risen every day for 5000 years? 

Let $p$ denote the probability that the Sun will rise tomorrow. A priori we have no information on the value of $p$ so Laplace posits that by the principle of insufficient reason one should assume a uniform prior probability  $dp=Uniform\left(\right(0,1\left)\right)$^[1]

Assume now that we have observed $n$ days, on each of which the Sun has risen.

Each event is a Bernoulli random variable $X_i$ which can each be 1 (the Sun rises) or 0 (the Sun does not rise). Assume that the probability is conditionally independent of $p$. 

The likelihood of $n$ out of $n$ succeses according to the hypothesis $p$  is $L(X_1=1,...,X_n=1|p)=p^n$. Now use Bayes rule 

$P(p|X_1=1,...,X_n=1)=\frac{P(X_1=1,...,X_n=1|p)dp}{\int P(X_1=1,...,X_n=1|p)dp}=\frac{p^{n}dp}{{\int }_0^1{p}^{n}dp}=\frac{p^{n}dp}{\frac{1}{n+1}}=(n+1)p^{n}dp$

to calculate the posterior.

Then the probability of succes for $P(X_{n+1}=1|X_1=1,...,X_n=1)=\int P\left(X_{n+1}|p\right)P(p|X_1=1,...,X_n=1)$
                                                               $={\int }_0^{1}p\cdot (n+1)p^{n}dp=\frac{n+1}{n+2}$

This is Laplace's rule of succcesion. 

We now adapt the above method to derive Lindy's Law. 

The probability of rising $n+s$ days and not rising on the $n+s+1$ day given that the Sun rose $n$ days is

$P(X_{n:n+s}=1,X_{n+s+1}=0|X_{1:n}=1)={\int }_0^1{p}^s(1-p)(n+1)p^{n}dp=(n+1)(\frac{1}{n+s+1}-\frac{1}{n+s+2})=\frac{n+1}{(n+s+1)(n+s+2)}$

                             The expectation of lifetime is then the average

$E\left(\text{Sun rises s more days}\right)={\sum }_{s=1}^{\infty }s\frac{n+1}{(n+s+1)(n+s+2)}$

which almost converges :o.... 

[What's the mistake here?]

1. ^{^}

   For simplicity I will exclude the cases that $p=0,1$, see the wikipedia page for the case where they are not excluded. 

Generalized Jeffrey Prior for singular models?

For singular models the Jeffrey Prior is not well-behaved for the simple fact that it will be zero at minima of the loss function. 
Does this mean the Jeffrey prior is only of interest in regular models? I beg to differ. 

Usually the Jeffrey prior is derived as parameterization invariant prior. There is another way of thinking about the Jeffrey prior as arising from an 'indistinguishability prior'.

The argument is delightfully simple: given two weights $w_1,w_2\in W$ if they encode the same distribution $p\left(x|w_1\right),p\left(x|w_2\right)$ our prior weights on them should be intuitively the same $\varphi \left(w_1\right)=\varphi \left(w_2\right)$. Two weights encoding the same distributions means the model exhibit non-identifiability making it non-regular (hence singular). However, regular models exhibit 'approximate non-identifiability'.

For a given dataset $D_N$ of size $N$ from the true distribution $q$, error ${ϵ}_1$, ${ϵ}_2$ we can have a whole set of weights $W_{N,ϵ}\subset W$ where the probability that $p\left(x|w_1\right)$ does more than ${ϵ}_1$ better on the loss on $D_N$ than $p\left(x|w_1\right)$ is less than ${ϵ}_2$.

In other words, the sets of weights that are probabily approximately indistinguishable. Intuitively, we should assign an (approximately) uniform prior on these approximately indistinguishable regions. This gives strong constraints on the possible prior. 

The downside of this is that it requires us to know the true distribution $q$. Instead of seeing if $w_1,w_2$ are approximately indistinguishable when sampling from $q$ we can ask if $w_2$ is approximately indistinguishable from $w_1$ when sampling from $w_2$. For regular models this also leads to the Jeffrey prior, see this paper.  

However, the Jeffrey prior is just an approximation of this prior. We could also straightforwardly see what the exact prior is to obtain something that might work for singular models. 

EDIT: Another approach to generalizing the Jeffrey prior might be by following an MDL optimal coding argument - see this paper. 

"The links between logic and games go back a long way. If one thinks of a debate as a kind of game, then Aristotle already made the connection; his writings about syllogism are closely intertwined with his study of the aims and rules of debating. Aristotle’s viewpoint survived into the common medieval name for logic: dialectics. In the mid twentieth century Charles Hamblin revived the link between dialogue and the rules of sound reasoning, soon after Paul Lorenzen had connected dialogue to constructive foundations of logic." from the Stanford Encyclopedia of Philosophy on Logic and Games

Game Semantics 

Usual presentation of game semantics of logic: we have a particular debate / dialogue game associated to a proposition between an Proponent and Opponent and Proponent tries to prove the proposition while the Opponent tries to refute it.

A winning strategy of the Proponent corresponds to a proof of the proposition. A winning strategy of the Opponent corresponds to a proof of the negation of the proposition.

It is often assumed that either the Proponent has a winning strategy in A or the Opponent has a winning strategy in A - a version of excluded middle. At this point our intuitionistic alarm bells should be ringing: we cant just deduce a proof of the negation from the absence of a proof of A. (Absence of evidence is not evidence of absence!)

We could have a situation that neither the Proponent or the Opponent has a winning strategy! In other words neither A or not A is derivable. 

Countermodels

One way to substantiate this is by giving an explicit counter model $C$ in which $A$ respectively $\neg A$ don't hold.

Game-theoretically a counter model $C$ should correspond to some sort of strategy! It is like an "interrogation" /attack strategy that defeats all putative winning strategies. A 'defeating' strategy or 'scorched earth'-strategy if you'd like. A countermodel is an infinite strategy. Some work in this direction has already been done^[1]. ^[2]

Dualities in Dialogue and Logic

This gives an additional symmetry in the system, a syntax-semantic duality distinct to the usual negation duality. In terms of proof turnstile we have the quadruple

$⊢A$ meaning $A$ is provable

$⊢\neg A$ meaning $$\neg A$$ is provable

$⊣A$ meaning $A$ is not provable because there is a countermodel $C$ where $A$ doesn't hold - i.e. classically $\neg A$ is satisfiable.

$⊣\neg A$ meaning $\neg A$ is not provable because there is a countermodel $C$ where $\neg A$ doesn't hold - i.e. classically $A$ is satisfiable.

Obligationes, Positio, Dubitatio

In the medieval Scholastic tradition of logic there were two distinct types of logic games ("Obligationes) - one in which the objective was to defend a proposition against an adversary ("Positio") the other the objective was to defend the doubtfulness of a proposition ("Dubitatio").^[3]

Winning strategies in the former corresponds to proofs while winning (defeating!) strategies in the latter correspond to countermodels. 

Destructive Criticism

If we think of argumentation theory / debate a counter model strategy is like "destructive criticism" it defeats attempts to buttress evidence for a claim but presents no viable alternative. 

1. ^{^}

    Ludics & completeness - https://arxiv.org/pdf/1011.1625.pdf

2. ^{^}

   Model construction games, Chap 16 of Logic and Games van Benthem

3. ^{^}

   Dubitatio games in medieval scholastic tradition, 4.3 of https://apcz.umk.pl/LLP/article/view/LLP.2012.020/778

Ambiguous Counterfactuals

[Thanks to Matthias Georg Mayer for pointing me towards ambiguous counterfactuals]

Salary is a function of eXperience and Education

$S=aE+bX$

We have a candidate $C$ with given salary, experience $(X=5)$ and education $(E=5)$.

Their current salary is given by 

$S=a\cdot 5+b\cdot 5$

We 'd like to consider the counterfactual where they didn't have the education $(E=0)$. How do we evaluate their salary in this counterfactual?

This is slightly ambiguous - there are two counterfactuals:

$E=0,X=5$ or  $E=0,X=10$

In the second counterfactual, we implicitly had an additional constraint $X+E=10$, representing the assumption that the candidate would have spent their time either in education or working. Of course, in the real world they could also have dizzled their time away playing video games.

One can imagine that there is an additional variable: do they live in a poor country or a rich country. In a poor country if you didn't go to school you have to work. In a rich country you'd just waste it on playing video games or whatever. Informally, we feel in given situations one of the counterfactuals is more reasonable than the other. 

Coarse-graining and Mixtures of Counterfactuals

We can also think of this from a renormalization / coarsegraining story. Suppose we have a (mix of) causal models coarsegraining a (mix of) causal models. At the bottom we have the (mix of? Ising models!) causal model of physics. i.e. in electromagnetics the Green functions give use the intervention responses to adding sources to the field.

A given counterfactual at the macrolevel can now have many different counterfactuals at the microlevels. This means we actually would get a probability dsitribution of likely counterfactuals at the top levels. i.e. in 1/3 of the cases the candidate actually worked the 5 years they didn't go to school. In 2/3 of the cases the candidate just wasted it on playing video games. 

The outcome of the counterfactual $S_{E=0}$ is then not a single number but a distribution

$S_{E=0}=5\cdot b+Y\cdot b$ 

where $Y$ is random variable with distribution the Bernoulli distribution with bias $1/3$.

Insights as Islands of Abductive Percolation?

I've been fascinated by this beautiful paper by Viteri & DeDeo. 

What is a mathematical insight? We feel intuitively that proving a difficult theorem requires discovering one or more key insights. Before we get into what the Dedeo-Viteri paper has to say about (mathematical) insights let me recall some basic observations on the nature of insights:

(see also my previous shortform)

• There might be a unique decomposition, akin to prime factorization. Alternatively, there might many roads to Rome: some theorems can be proved in many different ways. 
• There are often many ways to phrase an essentially similar insight. These different ways to name things we feel are 'inessential'. Different labelings should be easily convertible into one another. 
• By looping over all possible programs all proofs can be eventually found, so the notion of an 'insight' has to fundamentally be about feasibility.
• Previously, I suggested a required insight is something like a private key to a trapdoor function. Without the insight you are facing an infeasible large task. With it, you can suddenly easily solve a whole host of new tasks/ problems
• Insight may be combined in (arbitrarily?) complex ways. 

 When are two proofs of essentially different?

Some theorems can be proved in many different ways. That is  different in the informal sense. It isn't immediately clear how to make this more precise.

We could imagine there is a whole 'homotopy' theory of proofs, but before we do so we need to understand when two proofs are essentially the same or essentially different. 

• On one end of the spectrum, proofs can just be syntactically different but we feel they have 'the same content'. 
• We can think type-theoretically, and say two proofs are the same when their denotations (normal forms) are the same. This is obviously better than just asking for syntactical equality or apartness. It does mean we'd like some sort of intuitionistic/type-theoretic foundation since a naive classicial foundations makes all normals forms equivalent. 
• We can also look at what assumptions are made in the proof. I.e. one of the proofs might use the Axiom of Choice, while the other does not. An example is the famous nonconstructive proof of the irrationality of $a^b$ which turns out to have a constructive proof as well. 
• If we consider proofs as functorial algorithms we can use mono-Anabelian transport to distinguish them in some case. [LINK!]
• We can also think homotopy type-theoretically and ask when two terms of a type are equal in the HoTT sense. 

With the exception of the mono-anabelian transport one - all these suggestions of 'don't go deep enough', they're too superficial.

Phase transitions and insights, Hopfield Networks & Ising Models

(See also my shortform on Hopfield Networks/ Ising models as mixtures of causal models [LW(p) · GW(p)])

Modern ML models famously show some sort of phase transitions in understanding. People have been especially fascinated by the phenomenon of 'grokking, see e.g. here and here. It suggests we think of insights in terms of phase transitions, critical points etc.

Dedeo & Viteri have an ingenious variation on this idea. They consider a collection of famous theorems and their proofs formalized in a proof assistant. 

They then imagine these proofs as a giant directed graph and consider a Boltzmann distributions on it. (so we are really dealing with an Ising model/ Hopfield network here). We think of this distribution as a measure of 'trust' both trust in propositions (nodes) and inferences (edges). 

We show that the epistemic relationship between claims in a mathematical proof has a network structure that enables what we refer to as an epistemic phase transition (EPT): informally, while the truth of any particular path of argument connecting two points decays exponentially in force, the number of distinct paths increases. Depending on the network structure, the number of distinct paths may itself increase exponentially, leading to a balance point where influence can propagate
at arbitrary distance (Stanley, 1971). Mathematical proofs have the structure necessary to make this possible.
In the presence of bidirectional inference—i.e., both deductive and abductive reasoning—an EPT enables a proof to produce near-unity levels of certainty even in the presence of skepticism about the validity of any particular step. Deductive and abductive reasoning, as we show, must be well-balanced for this to happen. A relative over-confidence in one over the other can frustrate the effect, a phenomenon we refer to as the abductive paradox

The proofs of these famous theorems break up into 'abductive islands'. They have natural modularity structure into lemmas. 

EPTs are a double-edged sword, however, because disbelief can propagate just as easily as truth. A second prediction of the model is that this difficulty—the explosive spread of skepticism—can be ameliorated when the proof is made of modules: groups of claims that are significantly more tightly linked to each other than to the rest of the network. 

(...)
When modular structure is present, the certainty of any claim within a cluster is reasonably isolated from the failure of nodes outside that cluster.

One could hypothesize that insights might correspond somehow to these islands.

Final thoughts

I like the idea that a mathemathetical insight might be something like an island of deductively & abductively tightly clustered propositions. 

Some questions:

• How does this fit into the 'Natural Abstraction' - especially sufficient statistics?
• How does this interact with Schmidthuber's Powerplay?

EDIT: The separation property of Ludics, see e.g. here, points towards the point of view that proofs can be distinguished exactly by suitable (counter)models. 

Evidence Manipulation and Legal Admissible Evidence

[This was inspired by Kokotaljo's shortform [LW · GW] on comparing strong with weak evidence] 

In the real world the weight of many pieces of weak evidence is not always comparable to a single piece of strong evidence. The important variable here is not strong versus weak per se but the source of the evidence. Some sources of evidence are easier to manipulate in various ways. Evidence manipulation, either consciously or emergently, is common and a large obstactle to truth-finding. 

Consider aggregating many (potentially biased) sources of evidence versus direct observation. These are not directly comparable and in many cases we feel direct observation should prevail.

This is especially poignant in the court of law: the very strict laws arounding presenting evidence are a culturally evolved mechanism to defend against evidence manipulation. Evidence manipulation may be easier for weaker pieces of evidence - see the prohibition against hearsay in legal contexts for instance.

It is occasionally suggested that the court of law should do more probabilistic and Bayesian type of reasoning. One reason courts refuse to do so (apart from more Hansonian reasons around elites cultivating conflict suppression) is that naive Bayesian reasoning is extremely susceptible to evidence manipulation. 

Imagine a data stream 

 $...X_{-3},X_{-2},X_{-1},X_0,X_1,X_2,X_3...$

assumed infinite in both directions for simplicity. Here $X_0$ represents the current state ( the "present") and while $...X_{-3},X_{-2},X_{-1}$ and $X_1,X_2,X_3,...$ represents the future

Predictible Information versus Predictive Information

Predictible information is the maximal information (in bits) that you can derive about the future given the access to the past. Predictive information is the amount of bits that you need from the past to make that optimal prediction.

Suppose you are faced with the question of whether to buy, hold or sell Apple. There are three options so maximally $lo{g}_2\left(3\right)$ bits of information - not all of that information might be in contained in the past, there a certain part of irreductible uncertainty (entropy) about the future no matter how well you can infer the past. Think about a freak event & blacks swans like pandemics, wars, unforeseen technological breakthroughs, just cumulative aggregated noise in consumer preference etc. Suppose that irreducible uncertainty is half of $lo{g}_2\left(3\right)$ leaving us with $\frac{1}{2}lo{g}_2\left(3\right)$ of (theoretically) predictible information.

To a certain degree, it might be predictible in theory to what degree buying Apple stock is a good idea. To do so, you may need to know many things about the past: Apple's earning records, position of competitiors, general trends of the economy, understanding of the underlying technology & supply chains etc. The total sum of this information is far larger than $\frac{1}{2}lo{g}_2\left(3\right)$

To actually do well on the stock market you additionally need to do this better than the competititon - a difficult task! The predictible information is quite small compared to the predictive information

Note that predictive information is always greater than predictible information: you need to at least $k$ bits from the past to predict $k$ bits of the future. Often it is much larger. 

Mathematical details

Predictible information is also called 'apparent stored information' or commonly 'excess entropy'.

It is defined as the mutual information $I(X_{\le 0},X_{\ge 0})$ between the future and the past.

The predictive information is more difficult to define. It is also called the 'statistical complexity' or 'forecasting complexity' and is defined as the entropy of the steady equilibrium state of the 'epsilon machine' of the process. 

What is the Epsilon Machine of the process $\{X_i{\}}_{i\in Z}$? Define the causal states as the process as the partition on the sets of possible pasts $...,x_{-3},x_{-2},x_{-1}$ where two pasts $\overrightarrow{x},{\overrightarrow{x}}^{\prime }$ are in the same part / equivalence class when the future conditioned on $\overrightarrow{x},{\overrightarrow{x}}^{\prime }$ respectively is the same.

That is $P\left(X_{>0}|\overrightarrow{x}\right)=P(X_{>0},{\overrightarrow{x}}^{\prime })$. Without going into too much more detail the forecasting complexity measures the size of this creature. 

Agent Foundations Reading List [Living Document]
This is a stub for a living document on a reading list for Agent Foundations. 

Causality

Book of Why, Causality - Pearl

Probability theory 
Logic of Science - Jaynes

Hopfield Networks = Ising Models = Distributions over Causal models?

Given a joint probability distributions $p(x_1,...,x_n)$ famously there might be many 'Markov' factorizations. Each corresponds with a different causal model.

Instead of choosing a particular one we might have a distribution of beliefs over these different causal models. This feels basically like a Hopfield Network/ Ising Model. 

You have a distribution over nodes and an 'interaction' distribution over edges. 

The distribution over nodes corresponds to the joint probability distribution while the distribution over edges corresponds to a mixture of causal models where a normal DAG graphical causal G model corresponds to the Ising model/ Hopfield network which assigns 1 to an edge $x\to y$ if the edge is in G and 0 otherwise.