Our universe is “local” - things only interact directly with nearby things, and only so many things can be nearby at once. 

After reading this sentence, I had a short moment of illumination, that this is actually backwards: perhaps what our brains perceive as locality, is the property of "being influenced by/related to". Perhaps childs brain learns which "pixels" of retina are near each other, by observing they often have correlated colors, and similarly which places in space are nearby because you can move things or itself between them etc. So, whatever high-dimensional structure the real universe would have, we would still evolve to notice which nodes in the graph are connected and declare them "local". This doesn't mean, that the observation from the quoted sentence is a tautology: it wouldn't be true in a universe with much higher connectivity - we're lucky to live in a universe with a low [Treewidth](https://en.wikipedia.org/wiki/Treewidth), and thus can hope to grasp it.

I was expecting the central idea of this post to be more similar to/an extension of Everyday Lessons from High-Dimensional Optimization. That in a high-dimensional world, a good scientist can't afford to waste time testing implausible hypotheses. Doing so will get you the right answer eventually, but it is far too slow. In a high-dimensional world, there are just too many variables to tweak. Relevant excerpt from My Wild and Reckless Youth:

The way Traditional Rationality is designed, it would have been acceptable for me to spend thirty years on my silly idea, so long as I succeeded in falsifying it eventually, and was honest with myself about what my theory predicted, and accepted the disproof when it arrived, et cetera. This is enough to let the Ratchet of Science click forward, but it’s a little harsh on the people who waste thirty years of their lives. Traditional Rationality is a walk, not a dance. It’s designed to get you to the truth eventually, and gives you all too much time to smell the flowers along the way.

To what extent is this post making these points?

Your description of the second type of science where you repeatedly control variables to isolate one reminds me a lot of debugging a complex program.

Curated. Although written a while ago, I love posts discussing how to approach figuring out the world. I'd be interested to see more examples of the process advocated here though. Where are the cases of this working?

I detect the ghost of Jaynes in this!

I am not sure exactly why, but this and the optimization post both call to mind the current of thought suggesting we segregate the hypothesis and experimental steps explicitly. I have encountered this in three places:

• An unfinished textbook on Arxiv (which I cannot now locate to my frustration) that described treating machine learning as a science, which proposed gathering data and then the goodness of machine learning algorithm is measured by compression.
• The Report likelihoods, not p-values article on Arbital.
• This is basically how astronomy works by default: no one has a hypothesis for how pulsars interact and then gets a grant from their university department to launch a satellite network to look for pulsars; instead they identify phenomena on which they have little data, and pool resources to build a telescope or satellite or underground neutrino detector to gather the data, and then the publications test their hypotheses against the data gathered from one or more such projects.

I have a vague intuition that dividing up scientific practice in this way chunks the dimensions more tractably, or at least allows for it. Allowing optimization of data gathering and hypothesis formulation independently seems like a clear win for similar reasons.

Maybe the appeal is that it allows hypotheses to come from multiple directions in dimension space. The dimensionality of a body of data is fixed, but if it is generated as a tuple with a single hypothesis then it can only be approached from the perspective of that single hypothesis; if it is independent, then any hypothesis concerned with any of the dimensions of the data can be applied. By analogy, consider convergent evolution: two different paths in phase space arrive at essentially the same thing. Segregating the data step radically compresses this by allowing hypotheses from any other chain of development to be tested against it directly.

This post gave me an idea about how you might approach magic in fiction while keeping it ground in reality: something like magic users are people who learn to pick out relevant variables from the noise to consistently nudge reality in ways that otherwise seem not possible.

Basically placebomancy from Unsong.

Ultimately, this is a picture of “gears-level science”: look for mediation, hunt down sources of randomness, rule out the influence of all the other variables in the universe. 

I'm very doubtful that hunting down sources of randomness is a good way to go about doing science where there's a big solution space. 

There's a lot of human pattern matching involved in coming up with good hypothesis to test.

As a special case of this, we can also handle noise using repeated experiments. If I roll a die, I can’t predict the outcome perfectly, so I can’t rule out influences from all the billions of variables in the universe. But if I roll a die a few thousand times, then I can approximately-perfectly predict the distribution of die-rolls (including the mean, variance, etc). So, even though I don’t know what influences any one particular die roll, I do know that nothing else in the universe is relevant to the overall distribution of repeated rolls (at least to within some small error margin).

I'm not sure I fully understand this, so I wanna try to sketch out an example to see.

Suppose you've got a family of unknown variables $X_0$, $X_1$, $X_2$, ... which each influence the observable variables $Y_0$, $Y_1$, $Y_2$, .... Given some observations for some of the $Y_i$s, you can learn some summary statistics $Y_g$ that you can use to predict others $Y_i$s.

I think the counterintuitive thing about this view then, is that $Y_i$ is not independent of $X_i$ given $Y_g$. So what have we really learned? It doesn't immediately tell us anything about the $X_i$/$Y_i$ relationship. So where's the science?

I think my answer to this question is that while $Y_g$ doesn't tell us anything about the $X_i$s, it does tell us things about the $Y_i$s. (And $Y_g$ would essentially be a measure of the common causes underlying the $Y_i$s, I suppose.) Which is useful if you care about things that are downstream from the $Y_i$s. But I don't really see what determinism buys you here.

My model of you says that you'd mention something about the KPD theorems. But I don't know what.

I think this post would be stronger if it covered at least basic metrology and statistics. 

It's incorrect to say that billions of variables aren't affecting a sled sliding down a hill - of course they're affecting the speed, even if most are only by a few planck-lengths per hour. But, crucially, they're mostly not affecting it to a detectable amount. The detectability threshold is the key to the argument. 

For detectability, whether you notice the effects of outside variables is going to come down to the precision of the instrument that you're using to measure your output. If you're using a radar gun that gives readings to the nearest MPH, for example, you won't perceive a difference between 10.1 and 10.2 MPH, and so to you the two are equivalent. Nonetheless, outside variables have absolutely influenced the two readings differently. 

Equally critical is the number of measurements that you're taking. For example, if you're taking repeated measurements after controlling a certain set of variables, you may be able to say with a certain confidence/reliability that no other variables are causing enough variations in speed to register an output that's outside of the parameters that you've set. But that is a very different thing than saying that those other variables simply don't exist! One is a statement of probability, another is a statement of certainty. Maybe there's a confluence of variables that only occur once every thousand times, which you won't pick up when doing an initial evaluation. 

A remarkable empirical finding across many scientific fields, at many different scales and levels of abstraction, is that a small set of control variables usually suffices.

I'm skeptical that this is true for most things we care about. It's true in the scientific fields where we have the most accurate models, such as physics, but that's likely because there are so few relevant variables in those fields.

Most new drugs that go into clinical trials fail. Essentially, a pharmaceutical company identifies a variable that appears to be the mediator of a medical outcome, they create a drug that tweaks that variable, and then it turns out not to produce the outcome that they thought it would. There are too many other relevant variables that are poorly understood.

The other thing that makes me skeptical is the effectiveness of machine learning models that use a large number of inputs. It's possible that there's a simple underlying structure to what they're predicting that we just haven't figured out yet, but based on what exists now, it sure looks like there are a large number of relevant variables.

I don't quite think you've solved the problem of induction.

I think there's a fairly serious issue with your claim that being able to predict something accurately means you necessarily fully understand the variables which causes it because determinism.

The first thing to note is that “perfect predictability implies zero mutual information” plays well with approximation: approximately perfect predictability implies approximately zero mutual information. If we can predict the sled’s speed to within 1% error, then any other variables in the universe can only influence that remaining 1% error. Similarly, if we can predict the sled’s speed 99% of the time, then any other variables can only matter 1% of the time. And we can combine those: if 99% of the time we can predict the sled’s speed to within 1% error, then any other variables can only influence the 1% error except for the 1% of sled-runs when they might have a larger effect.

That's not really the cases. E.g: let's say that ice cream melt twice as fast in galaxies without a supermassive black hole at the center. You do experiments to see how fast ice cream melts. After controlling for type of ice cream, temperature, initial temp of the ice cream, airflow and air humidity, you find that you can predict how ice cream melts. You triumphantly claim that you know which things cause ice cream to melt at different rates, having completely missed the black hole's effects.

Essentially, controlling for A & B but not C won't tell you whether C has a causal influence on the thing you're measuring unless

• you intentionally change C between experiments (not practical given googleplexes of potential causal factors)
• C happens to naturally vary quite a bit and so makes your experimental results different, cluing you in to the fact that you're missing something.

I am kind of suprised you didn't reference causal inference here to just gesture at the task in which we "figure out which variables are directly relevant - i.e. which variables mediate the influence of everything else". Are you pointing to a different sort of idea/do you not feel causal inference is adequate for describing this task?

Also, scenario 1 and 2 seem fairly close to the "linear" and "non-linear" models of innovation Jason Crawford described in his talk "The Non-Linear Model of Innovation." To be honest, I prefered his description of the models. Though he didn't cover how miraculous it is that somehow the model can work. That, to a good approximation, the universe is simple and local.

The great challenge is to figure out which variables are directly relevant - i.e. which variables mediate the influence of everything else.

Is this equivalent to identifying the Markov blanket of the phenomenon being studied?

I think this certainly describes a type of gears level work scientists engage in, but not the only type, nor necessarily the most common one in a given field. There's also model building, for example.

Even once you've figured out which dozen variables you need to control to get a sled to move at the same speed every time, you still can't predict what that speed would be if you set these dozen variables to different values. You've got to figure out Newton's laws of motion and friction before you can do that.

Finding out which variables are relevant to a phenomenon in the first place is usually a required initial step for building a predictive model, but it's not the only step, nor necessarily the hardest one.

Another type of widespread scientific work I can think of is facilitating efficient calculation. Even if you have a deterministic model that you're pretty sure could theoretically predict a class of phenomena perfectly, that doesn't mean you have the computing power necessary to actually use it. 

Lattice Quantum Chromodynamics should theoretically be able to predict all of nuclear physics, but employing it in practice requires coming up with all sorts of ingenuous tricks and effective theories to reduce the computing power required for a given calculation. It's enough to have kept a whole scientific field busy for over fifty years, and we're still not close to actually being able to freely simulate every interaction of nucleons at the quark level from scratch.

The key to answering that question is determinism. If the system’s behavior can be predicted perfectly, then there is no mystery left to explain, no information left which some unknown variable could provide

1. What matters is local determinism. You need to show that behaviour is predictable from factors under your control. If local determinism fails, it is hard to tell whether locality or determinism failed individually.

2. And showing that a system's behaviour is predictable when N factors are held constant by the experimenter doesn't show that those are the only ones it is conditionally dependent one. Its behaviour might counterfactually depend on factors which the experimenter did not vary and which did not naturally change over the course of the experiment. In general, you can't exclude mysterious extra variables.