Frugality and working from finite data

Well done. Since most of the top-level posts by new users last couple of months have IMHO gone badly off the rails, I was tensing for the moment when this one would, but it never did.

I think this question applies in a lot of areas where experiments are possible but extremely expensive; for most of these that I can think of, we are in much greater trouble because we're starting with even less data.

Two examples: personal: Theory 1 and 2 are I (for some value of I) will be happiest as a 1) doctor or 2) physicist. I can sort of experiment with this, rudely and grossly, and I can spend some time gathering supporting data. However, without actually making a career in physics and in medicine, I haven't really experimented; even if I did there would be all sorts of variables I couldn't control.

national: Theory 1 and 2 are we (for some value of we) will better provide for the future of our citizens as 1) Democratic capitalists or 2) social democrats. This is testable in the same way, and just as one can look to other doctors and physicists to gain insight, a nation can look to other nations, but fundamentally, transitioning back and forth between economic systems is extremely costly.

So, in the absence of a superintelligence that can perfectly simulate doctor-you and physicist-you, or a whole nation of us, how do we reach scientifically justifiable conclusions, and how do we refine our theories and models with so little data?

This sounds a lot like the concept of cross-validation in machine learning. Suppose you have a bunch of data -- emails that are either spam or nonspam, for example -- and you want to make a model that will allow you to predict whether an email is spam or not. You split the data into n subsets, and for each subset S, do this test:

1. Create a model from all the data that are not in this set S.

2. Test the model on the data from S. How good are the predictions?

This method lets you test how well your machine learning methods are doing, so you can tweak them or go with a different method. Google recently got some publicity for offering a service that will apply a bunch of machine learning methods to your data and pick the best one; it picks the best one by running cross-validation.

Applying this to scientists, we get a variation on what you proposed in your penultimate paragraph: give part of the data to separate individuals or groups, stick them in a Science Temple in the mountains somewhere without Internet access, and then once they've come up with theories, test them on the data that they weren't given. For extra hilarity, split the data 50/50 among two large research groups, let them think of theories for a year, then have them exchange theories but not data. Then sit back and watch the argument.

You could also combine this sort of thing with the method suggested in VoF, by giving data out slowly to separate groups, in different orders. I'm not sure which of these ideas is best.

The scientific method is wonderfully simple, intuitive, and above all effective. Based on the available evidence, you formulate several hypotheses and assign prior probabilities to each one. Then, you devise an experiment which will produce new evidence to distinguish between the hypotheses. Finally, you perform the experiment, and adjust your probabilities accordingly.

Either there is more than one "scientific method", it isn't really a method, or science doesn't actually follow the scientific method (and therefore, cannot be justified by that method).

Imagine a certain rare galactic event A that occurs with probability 0.5 whenever a certain set of conditions are met

It seems like this is an indirect description of the Black Swan problem. Consider an astrophysical event so unlikely that it has a 50% chance of occurring in the entire light cone. Theory A completely prohibits the event, while theory B assigns the event a very small probability. If the event does not occur in the observation set, there is no way to distinguish between A and B.

This is a really great post, and I'm glad that this post has raised these issues, because I've been interested in them for a while.

Could you clarify this for me?

all the cosmological observations we can possibly make come from a single slice of the universe - a 4-dimensional cone of spacetime. And as there are a finite number of events in this cone, cosmology has only a limited amount of data it can ever gather; in fact, the amount of data that even exists is finite.

Surely we constantly receive new data from the receding boundary of the observable universe? As we move through time our past light-cone follows us, swallowing up more events that will affect us.

Also, if we literally have all possible evidence, then we lose our motivation to care, as the theory can't actually help us predict anything. When we really need these techniques is when we have all the evidence that is available at the moment, but it might make more predictions in the more distant future.

While I tend to agree with your rationale as it stands on its own, I don't think the biggest problem with implementing is politics in the academic world, a bigger problem may be the potential for abuse that would come with the authority to restrict availability of information.

In fact, politically it this concept already shows itself as very valuable as well as attractive, lots of governments and professional groups do restrict access to information sometimes even as true believers in their own justifications but with an outcome that is decidedly unattractive for the losers in that game.

Upvoted; I think cosmology in particular and astronomy in general is a very important case study of how to make theories and generally become less wrong in a situation where you absolutely cannot make any changes to the thing you're studying. (Of course I'm likely biased because it happens to be the field I'm in at the moment.)

Have you seen Vassar's talk on the development of the scientific method? He mentions the Scholarly method, whereby different scholars would work on the same data , and if they came up with the same theory independently, that was strong evidence.

How about this: it doesn't matter.

If we want to build a device that only works if a certain theory is true, we can use it to test the theory. If not, you can do what you want either way, so what does it matter?

There's still similar useful problems. For instance: you can keep getting new data on economics, but there's no way anyone's going to let you do an experiment. In addition, the data you're getting is very bad if you're trying to eliminate bias. It can't be solved in that way, though.

The Virtues of Frugality - Why cosmological observers should release their data slowly

...doesn't seem very convincing to me. Publish your data already, dammit!

Alternative solution: package all the cosmic observations into a big database. Given an astrophysical theory, instantiate it as a specialized compression program, and invoke it on the database. To select between rival theories, measure the sum of encoded file size plus length of compressor; smaller is better.

Of course, we have no way to create new stars or galaxies of our own

Well, we do make simulations.

I think a sensible step would be to treat a human as a black box statistical learning machine which can produce patterns (systems, theories etc.) that simplify the data and can be used to make predictions. A human has no special qualities that distinguishes it from automated approaches such as a support vector machine. These solutions can be seen as necessary approximations to the correct space of possible hypotheses from which predictions may be impractical (particularly as priors are unknown).

One way of having confidence in a particular output from these black boxes is the use of some additional prior over the likelihood of different theories (their elegance if you like), but I'm not sure to what extent such a prior can rationally be determined, i.e. the pattern of likely theories, of which simplicity is a factor.

Another approach is the scientific method, model with a subset and then validate with additional data (a common AI approach to minimise overfitting) I am not sufficiently knowledgeable in statistical learning theory to know how (or if) such approaches can be shown to provably improve predictive accuracy but I think this book covers some of it (other less wrong readers are likely to know more).

Culturally, we also apply a prior on the black box itself, i.e. when Einstein proposed theories, people rationally assumed they were more likely as given the limited data his black box seemed to suffer from less overfitting, of course we have few samples and don't know the variance so he could just have been lucky.

Another perspective is that if we cannot obtain any more information on a subject is it valuable to continue to try and model it. In effect, the data is the answer and predictive power is irrelevant as no more data can be obtained.