Summary: The significance of the Compression Rate Method (CRM) is that it justifies a form of empirical inquiry into aspects of reality that have previously resisted systematic interrogation. Some examples of potential investigations are described. A key hypothesis is discussed, and the link between empirical science and lossless data compression is emphasized.

In my previous post, the protagonist Sophie developed a modified version of the scientific method. It consists of the following steps:

1. Obtain a large database T related to a phenomenon of interest.
2. Develop a theory of the phenomenon, and instantiate the theory as a compression program.
3. Test the theory by invoking the compressor on T and measuring the net codelength achieved (encoded data plus length of compressor).
4. Given two rival theories of the phenomenon, prefer the one that achieves a shorter net codelength.

This modified version preserves two of the essential attributes of the traditional method. First, it employs theoretical speculation, but guides and constrains that speculation using empirical observations. Second, it permits Strong Inference by allowing the field to make decisive comparisons between rival theories.

The key difference between the CRM and the traditional method is that the former does not depend on the use of controlled experiments. For that reason, it justifies inquiries into aspects of empirical reality that have never before been systematically interrogated. The kind of scientific theories that are tested by the CRM depend on the type of measurements in the database target T. If T contains measurements related to physical experiments, the theories of physics will be necessary to compress it. Other types of data lead to other types of science. Consider the following examples:

The paper "Strong Inference" by John R. Platt is a meta-analysis of scientific methodology published in Science in 1964. It starts off with a wonderfully aggressive claim:

Scientists these days tend to keep up a polite fiction that all science is equal.

The paper starts out by observing that some scientific fields progress much more rapidly than others. Why should this be?

Summary: This post provides a brief discussion of the traditional scientific method, and mentions some areas where the method cannot be directly applied. Then, through a series of thought experiments, a set of minor modifications to the traditional method are presented. The result is a refined version of the method, based on data compression.

Related to: Changing the Definition of Science, Einstein's Arrogance, The Dilemma: Science or Bayes?

ETA: For those who are familiar with notions such as Kolmogorov Complexity and MML, this piece may have a low ratio of novelty:words. The basic point is that one can compare scientific theories by instantiating them as compression programs, using them to compress a benchmark database of measurements related to a phenomenon of interest, and comparing the resulting codelengths (taking into account the length of the compressor itself).

The self-indication assumption (SIA) states that

Given the fact that you exist, you should (other things equal) favor hypotheses according to which many observers exist over hypotheses on which few observers exist.

The reason this is a bad assumption might not be obvious at first.  In fact, I think it's very easy to miss.

Argument for SIA posted on Less Wrong

First, let's take a look at a argument for SIA that appeared at Less Wrong (link).  Two situations are considered.

1.  we imagine that there are 99 people in rooms that have a blue door on the outside (1 person per room).  One person is in a room with a red door on the outside.  It was argued that you are in a blue door room with probability 0.99.

2.  Same situation as above, but first a coin is flipped.  If heads, the red door person is never created.  If tails, the blue door people are never created.  You wake up in a room and know these facts.  It was argued that you are in a blue door room with probability 0.99.

So why is 1. correct and 2. incorrect?  The first thing we have to be careful about is not treating yourself as special.  The fact that you woke up just tells you that at least one conscious observer exists. 

In scenario 1 we basically just need to know what proportion of conscious observers are in a blue door room.  The answer is 0.99.

In scenario 2 you never would have woken up in a room if you hadn't been created.  Thus, the fact that you exist is something we have to take into account.  We don't want to estimate P(randomly selected person, regardless of if they exist or not, is in a blue door room).  That would be ignoring the fact that you exist.  Instead, the fact that you exist tells us that at least one conscious observer exists.  Again, we want to know what proportion of conscious observers are in blue door rooms.  Well, there is a 50% chance (if heads landed) that all conscious observers are in blue door rooms, and a 50% chance that all conscious observers are in red door rooms.  Thus, the marginal probability of a conscious observer being in a blue door room is 0.5.

The flaw in the more detailed Less Wrong proof (see the post) is when they go from step C to step D.  The *you* being referred to in step A might not exist to be asked the question in step D.  You have to take that into account.

General argument for SIA and why it's wrong

Let's consider the assumption more formally.

Assume that the number of people to be created, N, is a random draw from a discrete uniform distribution¹ on {1,2,...,N_max}.  Thus, P(N=k)=1/N_max, for k=1,...,N_max.  Assume N_max is large enough so that we can effectively ignore finite sample issues (this is just for simplicity).

Assume M= N_max*(N_max+1)/2 possible people exist, and we arbitrarily label them 1,...,M.  After the size of the world, say N=n, is determined, then we randomly draw n people from the M possible people.

After the data are collected we find out that person x exists.

We can apply Bayes' theorem to get the posterior probability:

P(N=k|x exists)=k/M, for k=1,...,N_max.

The prior probability was uniform, but the posterior favors larger worlds.  QED.

Well, not really.

The flaw here is that we conditioned on person x existing, but person x only became of interest after we saw that they existed (peeked at the data).

What we really know is that at least one conscious observer exists -- there is nothing special about person x.

So, the correct conditional probability is:

P(N=k|someone exists)=1/N_max, for k=1,...,N_max.

Thus, prior=posterior and SIA is wrong.

Egotism

The flaw with SIA that I highlighted here is it treats you as special, as if you were labeled ahead of time.  But the reality is, no matter who was selected, they would think they are the special person.  "But I exist, I'm not just some arbitrary person.  That couldn't happen in small world.  It's too unlikely."  In reality, that fact that I exist just means someone exists. I only became special after I already existed (peeked at the data and used it to construct the conditional probability).

Here's another way to look at it.  Imagine that a random number between 1 and 1 trillion was drawn.  Suppose 34,441 was selected.  If someone then asked what the probability of selecting that number was, the correct answer is 1 in 1 trillion.  They could then argue, "that's too unlikely of an event.  It couldn't have happened by chance."  However, because they didn't identify the number(s) of interest ahead of time, all we really can conclude is that a number was drawn, and drawing a number was a probability 1 event.

I give more examples of this here.

I think Nick Bostrom is getting at the same thing in his book (page 125):

..your own existence is not in general a ground for thinking that hypotheses are more likely to be true just by virtue of implying that there is a greater total number of observers. The datum of your existence tends to disconfirm hypotheses on which it would be unlikely that any observers (in your reference class) should exist; but that’s as far as it goes. The reason for this is that the sample at hand—you—should not be thought of as randomly selected from the class of all possible observers but only from a class of observers who will actually have existed. It is, so to speak, not a coincidence that the sample you are considering is one that actually exists. Rather, that’s a logical consequence of the fact that only actual observers actually view themselves as samples from anything at all

Related arguments are made in this LessWrong post.  

¹ for simplicity I'm assuming a uniform prior... the prior isn't the issue here

Less Wrong readers are familiar with the idea you can and should put a price on life. Unfortunately the Big Lie that you can't and shouldn't has big consequences in the current health care debate. Here's some articles on it:

Yvain's blog post here (HT: Vladimir Nesov).
Peter Singer's article on rationing health care here.
Wikipedia here.
Experts and policy makers who debate this issue here.

For those new to Less Wrong, here's the crux of Peter Singer's reasoning as to why you can put a price on life:

A tenet of traditional rationality is that you can't learn much about the world from armchair theorizing. Theory must be epiphenomenal to observation-- our theories are functions that tell us what experiences we should anticipate, but we generate the theories from *past* experiences. And of course we update our theories on the basis of new experiences. Our theories respond to our evidence, usually not the other way around. We do it this way because it works better then trying to make predictions on the basis of concepts or abstract reasoning. Philosophy from Plato through Descartes and to Kant is replete with failed examples of theorizing about the natural world on the basis of something other than empirical observation. Socrates thinks he has deduced that souls are immortal, Descartes thinks he has deduced that he is an immaterial mind, that he is immortal, that God exists and that he can have secure knowledge of the external world, Kant thinks he has proven by pure reason the necessity of Newton's laws of motion.

These mistakes aren't just found in philosophy curricula. There is a long list of people who thought they could deduce Euclid's theorems as analytic or a priori knowledge. Epicycles were a response to new evidence but they weren't a response that truly privileged the evidence. Geocentric astronomers changed their theory *just enough* so that it would yield the right predictions instead of letting a new theory flow from the evidence. Same goes for pre-Einsteinian theories of light. Same goes for quantum mechanics. A kludge is a sign someone is privileging the hypothesis. It's the same way many of us think the Italian police changed their hypothesis explaining the murder of Meredith Kercher once it became clear Lumumba had an alibi and Rudy Guede's DNA and hand prints were found all over the crime scene. They just replaced Lumumba with Guede and left the rest of their theory unchanged even though there was no longer reason to include Knox and Sollecito in the explanation of the murder. These theories may make it over the bar of traditional rationality but they sail right under what Bayes theorem requires.

Most people here get this already and many probably understand it better than I do. But I think it needs to be brought up in the context of our ongoing discussion of normative ethics.

Unless we have reason to think about ethics differently, our normative theories should respond to evidence in the same way we expect our theories in other domains to respond to evidence. What are the experiences that we are trying to explain with our ethical theories? Why bother with ethics at all? What is the mystery we are trying to solve? The only answer I can think of is our ethical intuitions. When faced with certain situations in real life or in fiction we get strong impulses to react in certain ways, to praise some parties and condemn others. We feel guilt and sometimes pay amends. There are some actions which we have a visceral abhorrence of.

These reactions are for ethics what measurements of time and distance are for physics -- the evidence.

In January of last year, Nick Bostrom wrote a post on Overcoming Bias about his and Toby Ord’s proposed method of handling moral uncertainty. To abstract away a bit from their specific proposal, the general approach was to convert a problem involving moral uncertainty into a game of negotiation, with each player’s bargaining power determined by one’s confidence in the moral philosophy represented by that player.

Robin Hanson suggested in his comments to Nick’s post that moral uncertainty should be handled the same way we're supposed to handle ordinary uncertainty, by using standard decision theory (i.e., expected utility maximization). Nick’s reply was that many ethical systems don’t fit into the standard decision theory framework, so it’s hard to see how to combine them that way.

In this post, I suggest we look into the seemingly easier problem of value uncertainty, in which we fix a consequentialist ethical system, and just try to deal with uncertainty about values (i.e., utility function). Value uncertainty can be considered a special case of moral uncertainty in which there is no apparent obstacle to applying Robin’s suggestion. I’ll consider a specific example of a decision problem involving value uncertainty, and work out how Nick and Toby’s negotiation approach differs in its treatment of the problem from standard decision theory. Besides showing the difference in the approaches, I think the specific problem is also quite important in its own right.

The problem I want to consider is, suppose we believe that a singleton scenario is very unlikely, but may have very high utility if it were realized, should we focus most of our attention and effort into trying to increase its probability and/or improve its outcome? The main issue here is (putting aside uncertainty about what will happen after a singleton scenario is realized) uncertainty about how much we value what is likely to happen.