Occam alternatives

The most notable problem with Pascal's Goldpan is that when you calculate the utility of believing a particular hypothesis, you'll find that there is a term in that equation for "is this hypothesis actually true?"

That is, suppose you are considering whether or not to believe that you can fly by leaping off a cliff and flapping your arms. What is the expected utility of holding this belief?

Well, if the belief is correct, there's a large utility to be gained: you can fly, and you're a scientific marvel. But if it's false, you may experience a tremendous disutility in the form of a gruesome death.

The point is that deciding you're just going to believe whatever is most useful doesn't even solve the problem of deciding what to believe. You still need a way of evaluating what is true. It may be that there are situations where one can expect a higher utility for believing something falsely, but as EY has touched on before, if you know you believe falsely, then you don't actually believe. Human psychology does seem to contain an element of Pascal's Goldpan, but that doesn't make it rational (at least not in the sense of "optimal"; it does seem to imply that at some point in our evolution such a system tended to win in some sense).

At present the best we can do seems to be keeping our truth-determining and our utility-maximizing systems mostly separate (though there may be room for improvement on this), and Occam's Razor is one of our tried-and-true principles for the truth-determining part.

The illustration that immediately sprung to my mind was of the characters Samantha Carter and Jack O'Neill in the television sci-fi show...

Beware fictional evidence! Main characters in serial TV shows really should follow "Pascal's Goldpan", because that's the way their universe (usually) works! The episode wouldn't have been written if there wasn't a way out of the problem. I suspect that experiencing just two or three such "insoluble" problems get resolved ought to make a proper rationalist wonder if they were living in a fictional universe.

But our universe doesn't seem to have that property. (Or perhaps I'm just not a main character.) What is true seems to be true independent from how much utility I get from believing it.

BTW, that isn't keeping me from loving your coined expressions!

It looks to me like the goldpan is circular, you need a probability to calculate expected utility to calculate prior probability.

• Diax's Rake Never believe a thing simply because you want it to be true

(the non-sentimental prior-- the universe doesn't care about you)

Interesting thought, but I fear it's not practical for the same reason you can't doublethink.

To evaluate how useful your explanation is, you need an accurate model of the world, in which you'll "run" your explanation and evaluate its consequences. Consider the belief in UFOs (to change from the usual belief in God). You can't evaluate how useful that explanation is regardless of whatever it is true or not. If there really are UFOs, the explanation will have a different utility than if there aren't. So the only way to compute the utility of the belief depends of the probability of UFOs existing. And of many other assumptions. Once you have knowledge required to evaluate how useful having the belief will be, it's too late. You can't forget what you know, you can't doublethink.

Are there good reasons to prefer Occam's Razor?

The probability must add to one. There are exponentially more possibilities with linearly more complexity, so the probability would have to decrease exponentially on average. You can base it on something other than complexity, but whatever you use will still correlate to complexity.

Equally weight all hypotheses that explain the data.

There are infinitely many of them. Now what?

The likelihood of a hypothesis is inversely proportional to the number of observations it purports to explain.

How does that work? Saying the coin will land on heads or tails explains two observations. Saying it will land on heads explains one. Is it twice as likely to land on heads than land on heads or tails?

The likelihood of a new hypothesis that explains the data is proportional to the Solomonoff prior for the Kolmogorov complexity of the code that transforms the previously accepted hypothesis into the new hypothesis.

I don't think you could do this without violating the axioms of probability. Also, what's your first hypothesis?

If your first hypothesis is nothing, then until your first observation, you have the Solomonoff prior. If you do manage to obey the axioms of probability, you will have to use Solomonoff induction.

My aesthetics prefers the O'Neill way over the Carter way: the Goldpan over the Razor.

If that's your razor, you're doing it wrong. Everything is possible. The simplest explanation is just most likely. Preparing for your death may be the most likely action to be relevant, but it also doesn't do much to help.

In addition, the show skews the results by making O'Neill right much more often than reasonable. Of course it will look like it's a good idea to assume you have a good chance of survival if they only ever show the characters surviving.

... instead pursued the most practical course of action with a chance of getting them rescued...

One more thing; this is actually rational, and doesn't violate occam's razor. You should take the course of action with the highest P(this is true) * P(I can be rescued|this is true).

(ignorance prior) Equally weight all hypotheses that explain the data.

This prior is impossible to implement for almost all situations, because there is no way to assign uniform probabilities over an infinite number of things - you just end up with 0 everywhere, which isn't normalized, and hence is not a probability distribution.

(utilitarian prior) The likelihood of a hypothesis is proportional to the expected net utility of the agent believing it.

So problem one is that this is circular if someone values the truth.

Then if this is ignored, there's still a problem that it's unstable - someone who things a stock market crash is likely will think that it is high-utility to believe this, and someone who thinks a stock market crash is unlikely will think that their thought is the high-utility one, and so you have a feedback loop, naturally driving people to extremes but not favoring one extreme over another, which is bad since usually either A or ~A is true, but not both.

The third problem is sentences like "it is high utility to believe that this sentence is true and that Santa Claus exists," (or "you will go to heaven if you believe this sentence and God exists") though this can be solved by including a full zeroth-order theory of human utility inside what used to look like a one-sentence maxim.

However, it does have an important thing going for it - it's a first step towards thinking about what a self-modifying agent with a utility function would choose to use as a prior.

My aesthetics prefers the O'Neill way over the Carter way: the Goldpan over the Razor.

Is this because the show repeatedly portrays one as working and the other as not working? This sounds like the common "Hollywood rationality" trope.

A proposition is true if the use of its content would be beneficial in all relevant utility functions.

There is no such proposition, if we don't allow the Santa Claus sentences.

(social prior) The likelihood of a hypothesis is proportional to how recently and how often it has been proposed and to the social status of its proponents.

Yeah that's just a bad plan. "It is the highest-utility option for you to grovel before me, peons." And it has no reason to exist, since what are the people making the propositions using to guess at what's true?

~~

The others didn't have any really killer (e.g. "the things this outputs aren't probabilities") flaws I could see - they just end you up with different sets of beliefs. That can still be bad, if the belief "that's a tiger about to leap at me" gets rated at low probability when it should have been high, but oh well.

Upvoted for seeking a tiebreaker among tiebreaking heuristics.

comparing and contrasting Marian apparitions with UFO sightings

Any conclusions you care to share?

The likelihood of a hypothesis is proportional to the expected net utility of the agent believing it.

I'm sure there's some way to translate that into expected return from betting on it.

Pascal's Goldpan is an interesting idea. However, it produces all kinds of problems. Over which time period are we defining utility? Surely most pieces of knowledge would have either no effect on utility, regardless of truth, or wildly unpredictable effects, and in some cases negative effects. I don't think it's useful.

Knowledge needs to constrain expectations. Usefulness is a helpful test of whether it does that, but doesn't work as a general definition.

Kudos for originality, though.

The one that initially leapt out at me was what I'll call Pascal's Goldpan. Granted, a human trying to understand the world and using the Goldpan would likely settle on largely the same theories as a human using the Razor since simple theories have practical advantages for our limited mental resources.

It doesn't require much mental effort to postulate the hypothesis "I will certainly become a millionaire tomorrow" and its expected utility is pretty high if I happen to believe it. Usefulness is a good criterion for directing future research, but I can't imagine its sensible use in assigning priors to already stated hypotheses.

All else being equal, the most useful explanation is usually correct.

If living in an overwhelmingly Christian-dominated society, the most useful explanation for your existence is that god in heaven created you - but that's not the correct explanation.

Occam\s razor is famously difficult to justify except by circular appeal to itself. It's interesting to think of alternatives but you should be aware of what you give up when you give up Occam's razor. You can no longer make sensible inferences about the future based on your past experiences. For example, you can no longer have any confidence that the direction of gravity will still point downwards tomorrow, or that the laws of physics won't spontaneously change a minute from now. The experimental method itself no longer makes sense if you have no reason to think that the future will resemble the past.

You should read:

• http://plato.stanford.edu/entries/simplicity/

• http://lesswrong.com/lw/s0/where_recursive_justification_hits_bottom/

None of these are worth anything unless you look for testable predictions, not for simply explaining the existing data. The problem with the UFO explanation is not that it has a fantastically low Solomonoff prior, but that it predicts nothing that would differentiate it from an explanation with a better prior.

In that vein, the Pascal's Goldpan with a very specific utility function is one way to go: construct (independent) testable predictions and estimate the a priori probability of each one, then add (negative logs of) the probabilities for each model, and call it the utility function. Basically, the more predictions and the unlikelier they are, the higher the utility of a given model. Then test the predictions. Among the models where every prediction is confirmed, pick one with the highest utility.

I'm sure this approach has a name, but google failed me...

I always liked what, I think, was Karl Popper's defense of Okham's Razor -- simpler theories are easier to falsify.

Are there good reasons to prefer Occam's Razor?

An aesthetic (with a hint of rigor, but only a hint) one is: if one is enumerating (and running) Turing machines (using a correspondence between them and the naturals), then ones with a short code (i.e. small description number) will be reached first.

(This argument is working on the premise that a Turing machine could, in theory, simulate a universe such as ours (if not exactly, then at least approximate it, arbitrarily accurately)).

And so, if one then has some situation where the enumeration has some chance of being stopped after each machine, then those with shorter codes have a greater likelihood than those with longer codes. (Or some other situation, like (for instance) the enumeration is occurring in a universe with finite lifespan)

Are you sure that wherever you say “likelihood” you don't mean “prior probability” instead?

Are there good reasons to prefer Occam's Razor?

The probability must add to one. There are exponentially more possibilities with linearly more complexity, so the probability would have to decrease exponentially on average. You can base it on something other than complexity, but whatever you use will still correlate to complexity.

Equally weight all hypotheses that explain the data.

There are infinitely many of them. Now what?

The likelihood of a hypothesis is inversely proportional to the number of observations it purports to explain.

How does that work? Saying the coin will land on heads or tails explains two observations. Saying it will land on heads explains one. Is it twice as likely to land on heads than land on heads or tails?

The likelihood of a new hypothesis that explains the data is proportional to the Solomonoff prior for the Kolmogorov complexity of the code that transforms the previously accepted hypothesis into the new hypothesis.

I don't think you could do this without violating the axioms of probability. Also, what's your first hypothesis?

If your first hypothesis is nothing, then until your first observation, you have the Solomonoff prior. If you do manage to obey the axioms of probability, you will have to use Solomonoff induction.

My aesthetics prefers the O'Neill way over the Carter way: the Goldpan over the Razor.

If that's your razor, you're doing it wrong. Everything is possible. The simplest explanation is just most likely. Preparing for your death may be the most likely action to be relevant, but it also doesn't do much to help.

In addition, the show skews the results by making O'Neill right much more often than reasonable. Of course it will look like it's a good idea to assume you have a good chance of survival if they only ever show the characters surviving.