trentmkelly/LessWrong-43k huggingface dataset:
Non-axiomatic math reasoning with naive Bayes
by cousin_it — 2011-05-30T10:26:49.552Z

ETA: this post is WRONG. Read on only if you like to find mistakes in other people's reasoning. Sorry.

So I've been thinking how to teach machines to reason about integers without running into Goedelian limitations. A couple days ago I suddenly got this painfully obvious idea that I'm sure others have already developed, but I don't know how to search for prior work, so I'm posting it here.

Here's a simple statement about the integers:

5 > 4

It's easy enough to check by direct calculation if your CPU can do arithmetic. Here's a more complex statement:

For any x, x+1 > x

How do you check this? Let's denote the whole statement as S, and the individual statements "x + 1 > x" as S(x). The statements S(1), S(2) and so on are all directly checkable like the first one. For simplicity we can make the "naive Bayesian" assumption that they're all independent, so P(S) = P(S(1))*P(S(2))*... (ETA: this is shady, though it doesn't break the construction. See AlephNeil's comment and my reply for a slightly better way.) After manually checking that S(1) holds, we update its probability to 1, so our estimate of P(S) increases. Then we try S(2), etc. After a while P(S) will become as close to 1 as we want, if S is true.

We can also use a different quantifier:

There exists x such that x*x = 9

You can check this one by naive Bayesian reasoning too, except "exists" corresponds to probabilistic "or", whereas "for any" corresponded to probabilistic "and". So you check S(1), S(2) and so on, revising P(S) downward at each step, until you stumble on the right x and P(S) jumps to 1.

Here's a still more complex statement:

For any x, there exists y such that x ≥ y*y and x < (y+1)*(y+1)

How do you check this one? If we again denote it as S and the individual parts as S(x), we already know how to approximately check S(x) for each value of x, because it's a statement of the previous "level"! So we know how to check the whole S approximately, too.

Hopefully you get the general idea by now: add as many quantifiers as you wish. The fancy math name for this method of organizing statements is the arithmetical hierarchy. The calculations become expensive, but in the limit of enough computing power the system will get closer and closer to correct reasoning about the integers. As time passes, it will come to believe pretty strongly in all the axioms of PA (or at least t...
Existing Absurd Technologies
by Desrtopa — 2011-05-30T06:12:18.646Z

When attempting to introduce non-rationalists to the ideas of cryonics or Strong AI, it appears that their primary objections tend to be rooted in the absurdity heuristic. They don't believe they inhabit a universe where such weird technologies could actually work. To deal with this, I thought it would be useful to have a cache of examples of technologies that have actually been implemented that did, or ideally, still do, challenge our intuitions about the way the universe works.

The first example that comes to my mind is computers in general; imagine what Ernest Rutherford, let alone Benjamin Franklin, would have thought of a machine that uses electricity to calculate, and do those calculations so fast that they can express nearly anything as calculations. Nothing we know about how the universe works says it shouldn't be possible, indeed it obviously is knowing what we do now, but imagine how weird this would have seemed back when we were just coming to grips with how electricity actually worked.

I suspect there may be better examples to challenge the intuitions of people who've grown up in an age where computers are commonplace though. So does anyone have any to volunteer?
A simple counterexample to deBlanc 2007?
by PhilGoetz — 2011-05-30T05:09:03.725Z

Peter de Blanc submitted a paper to arXiv.org in 2007 called "Convergence of Expected Utilities with Algorithmic Probability Distributions." It claims to show that a computable utility function can have an expected value only if the utility function is bounded.

This is important because it implies that, if a utility function is unbounded, it is useless. The purpose of a utility function is to compare possible actions k by choosing the k for which U(k) is maximal. You can't do this if U(k) is undefined for any k, let alone for every k.

I don't know whether any agent we contemplate can have a truly unbounded utility function, since the universe is finite. (The multiverse, supposing you believe in that, might not be finite; but as the utility function is meant to choose a single universe from the multiverse, I doubt that's relevant.) But it is worth exploring, as computable functions are worth exploring despite not having infinitely long tapes for our Turing machines. I previously objected that the decision process is not computable; but this is not important - we want to know whether the expected value exists, before asking how to compute (or approximate) it.

The math in the paper was too difficult for me to follow all the way through; so instead, I tried to construct a counterexample. This counterexample does not work; the flaw is explained in one of comments below. Can you find the flaw yourself? This type of error is both subtle and common. (The problem is not that the theorem actually proves that for any unbounded utility function, there is some set of possible worlds for which the expected value does not converge.)

The abstract says:

We consider an agent interacting with an unknown environment. The environment is a function which maps natural numbers to natural numbers; the agent's set of hypotheses about the environment contains all such functions which are computable and compatible with a finite set of known input-output pairs, and the agent assigns a positive probability to each such hypothesis. We do not require that this probability distribution be computable, but it must be bounded below by a positive computable function. The agent has a utility function on outputs from the environment. We show that if this utility function is bounded below in absolute value by an unbounded computable functi

...
Austin, TX LW Meetup, 6/4 1:30PM
by Vaniver — 2011-05-30T04:57:29.780Z

After a brief hiatus, the Austin LW meetup is back together and stepping things up: for the foreseeable future, we plan on meeting every week. The next meeting is:

Saturday 6/4, 1:30 PM, Caffé Medici; right across the street from UT. Parking is available on the street on in one of the garages around it on the street that runs parallel to Guadalupe.

I plan to be there at least until ~4:20, but we may end up staying longer. (We've had a lot to talk about at past meetups.) Hope to see you there!
A puzzle on the ASVAB
by Hul-Gil — 2011-05-30T04:01:00.757Z

I was linked to this on another forum. No instructions were given, apparently - just this picture. What's the deal?

It seems to me the answer is clearly C, not A as the test indicates; and the members in the original thread appear to agree. However, attempted justifications of A have been made, none of which are very convincing to me - mainly because if there are no instructions and an obvious answer, there's not really any benefit for them to reward a different interpretation, which would almost certainly involve arbitrary assumptions regarding the rules they really want you to apply.

Trick questions on exams seem to rely on failure to pay close attention to instructions, or insufficiently rigorously apply rules; when there are no instructions, what justification would anyone have for not choosing the most obvious interpretation? Any could be right!

What do the geniuses here at MoreRight think?
When not to know?
by KatjaGrace — 2011-05-30T00:38:55.000Z

Jeff at Cheap Talk reports on Andrew Caplin’s good point: making tests less informative can make people better off, because often they don’t want that much information, but may still want a bit.

This reminded me of a more basic question: what makes people want to avoid getting information?

That is, when would people prefer to believe P(X)=y% than to have a y% chance of believing X, and a (1-y)% chance of believing not X?

One such time is when thinking about the question at all would be disconcerting. For instance you may prefer whatever probability distribution you already have over the manners in which your parents may make love, than to consider the question.

Another time is when more uncertainty is useful in itself. A big category of this is when it lets you avoid responsibility. As in, ‘I would love to help, but I’m afraid I have no idea how to wash a cat’, or ‘How unfortunate that I had absolutely no idea that my chocolate comes from slaves, or I would have gone to lots of effort to find ethical chocolate’. If you can signal your ignorance, you might also avoid threats this way.

I’m more interested in situations like the one where you could call the doctor to get the results of your test for venereal disease, but you’d just rather not. Knowing would seem to mostly help you do things you would want to do in the case that you do have such a disease, and you are already thinking about the topic. It seems you actually prefer the uncertainty to the knowledge in themselves. The intuitive interpretation seems to be something like ‘you suspect that you do have such a disease, and knowing will make you unhappy, so you prefer not to find out’. But to the extent you suspect that you have the disease, why aren’t you already unhappy? So that doesn’t explain why you would rather definitely be somewhat unhappy than a chance of being unhappier with a chance of relief from your present unhappiness. And it doesn’t distinguish that sort of case from the more common cases where people like to have information.

A few cases where people often seek ignorance:

academic test results which are expected to be bad

medical test results

especially genetic tendencies to disease

whether a partner is cheating

more?

Notice that these all involve emotionally charged situations – can you think of some th...
Nature: Red, in Truth and Qualia
by orthonormal — 2011-05-29T23:50:28.495Z

Previously: Seeing Red: Dissolving Mary's Room and Qualia, A Study of Scarlet: The Conscious Mental Graph

When we left off, we'd introduced a hypothetical organism called Martha whose actions are directed by a mobile graph of simple mental agents. The tip of the iceberg, consisting of the agents that are connected to Martha's language centers, we called the conscious subgraph. Now we're going to place Martha into a situation like Mary's Room: we'll say that a large unconscious agent of hers (like color vision) has never been active, we'll grant her an excellent conscious understanding of that agent, and then we'll see what happens when we activate it for the first time.

But first, there's one more mental agent we need to introduce, one which serves a key purpose in Martha's evolutionary history: a simple agent that identifies learning.

We recall that Martha's species had evolved ever-better fluency with language, and along the way their minds developed a conscious/unconscious distinction, based on the functional difference (in the ancestral environment) between things that might be beneficial to communicate and things that weren't. Any new conscious node or connection has the potential to be communicated, but not all such elements are created equal; some of them (like a key bit of gossip, or a pattern suddenly grasped) have much larger effects on the full graph (and the conscious subgraph) than others.

Since the minds of Martha's species are similar in structure, it's likely that the new conscious ideas that rearrange Martha's graph might have similar effects on her friends' graphs- and therefore, Martha's species has evolved to give such connections high priority for communication.

One way to do this is to have a simple mental agent which tells Martha when she's learned something of significance. That is, it becomes active when the conscious subgraph rearranges on a large scale1. When this happens, it traces the cascade of changes backward until it finds the most basic new node or connection that started the revolution. When it finds the culprit, it forms the conscious idea "I just learned X", where X is the new node or connection; thus, when Martha is thinking or talking about related ideas, the important bit is more likely to occur to her.

In our toy example from yesterday's post, at the moment that the DREAD VILLAIN reveals himself as the LONE STALWART's fathe...
[SEQ RERUN] Priors as Mathematical Objects
by Tyrrell_McAllister — 2011-05-29T15:36:07.552Z

Today's post, Priors as Mathematical Objects, was originally published on 12 April 2007. A summary (taken from the LW wiki):

As a mathematical object, a Bayesian "prior" is a probability distribution over sequences of observations. That is, the prior assigns a probability to every possible sequence of observations. In principle, you could then use the prior to compute the probability of any event by summing the probabilities of all observation-sequences in which that event occurs. Formally, the prior is just a giant look-up table. However, an actual Bayesian reasoner wouldn't literally implement a giant look-up table. Nonetheless, the formal definition of a prior is sometimes convenient. For example, if you are uncertain about which distribution to use, you can just use a weighted sum of distributions, which directly gives another distribution.

Discuss the post here (rather than in the comments to the original post).

This post is part of the Rerunning the Sequences series, where we'll be going through Eliezer Yudkowsky's old posts in order so that people who are interested can (re-)read and discuss them. The previous post was Marginally Zero-Sum Efforts, and you can use the sequence_reruns tag or rss feed to follow the rest of the series.

Sequence reruns are a community-driven effort. You can participate by re-reading the sequence post, discussing it here, posting the next day's sequence reruns post, or summarizing forthcoming articles on the wiki. Go here for more details, or to have meta discussions about the Rerunning the Sequences series.
Overcoming suffering: Emotional acceptance
by Kaj_Sotala — 2011-05-29T10:57:45.362Z

Follow-up to: Suffering as attention-allocational conflict.

In many cases, it may be possible to end an attention-allocational conflict by looking at the content of the conflict and resolving it. However, there are also many cases where this simply won't work. If you're afraid of public speaking, say, the "I don't want to do this" signal is going to keep repeating itself regardless of how you try to resolve the conflict. Instead, you have to treat the conflict in a non-content-focused way.

In a nutshell, this is just the map-territory distinction as applied to emotions. Your emotions have evolved as a feedback and attention control mechanism: their purpose is to modify your behavior. If you're afraid of a dog, this is a fact about you, not about the dog. Nothing in the world is inherently scary, bad or good. Furthermore, emotions aren't inherently good or bad either, unless we choose to treat them as such.
We all know this, right? But we don't consistently apply it to our thinking of emotions. In particular, this has two major implications:
1. You are not the world: It's always alright to feel good. Whether you're feeling good or bad won't change the state of the world: the world is only changed by the actual actions you take. You're never obligated to feel bad, or guilty, or ashamed. In particular, since you can only influence the world through your actions, you will accomplish more and be happier if your emotions are tied to your actions, not states of the world.
2. Emotional acceptance: At the same time, "negative" emotions are not something to suppress or flinch away from. They're a feedback mechanism which imprints lessons directly into your automatic behavior (your elephant). With your subconsciousness having been trained to act better in the future, your conscious mind is free to concentrate on other things. If the feedback system is broken and teaching you bad lessons, then you should act to correct it. But if the pain is about some real mistake or real loss you suffered, then you should welcome it.
Internalizing these lessons can have some very powerful effects. I've been making very good progress on consistently feeling better after starting to train myself to think like this. But some LW posters are even farther along; witness Will Ryan:

I internalized a number of different conclusions during this period, although piecing together the exact time fra

...
Training for math olympiads
by D_Malik — 2011-05-29T09:34:17.739Z

Lately I've resolved to try harder at teaching myself math so I have a better shot at the international olympiad (IMO). These basically involve getting, say, three really hard math problems and trying your best to solve them within 5 hours.

My current state:

I have worked through a general math problem-solving guide (Art and Craft of Problem-Solving), a general math olympiad guide (A Primer for Mathematics Competitions) and practice problems.

I've added all problems and solutions and theorems and techniques into an Anki deck. When reviewing, I do not re-solve the problem, I only try to remember any key insights and outline the solution method.

I am doing n-back, ~20 sessions (1 hour) daily, in an attempt to increase my general intelligence (my IQ is ~125, sd 15).

I am working almost permanently; akrasia is not much of a problem.

I am not _yet_ at the level of IMO medallists.

What does the intrumental-rationality skill of LWers have to say about this? What recommendations do you guys have for improving problem-solving ability, in general and specifically for olympiad-type environments? Specifically,

How should I spread my time between n-backing, solving problems, and learning more potentially-useful math?

Should I take any nootropics? I am currently looking to procure some fish oil (I don't consume any normally) and perhaps a racetam. I have been experimenting with cycling caffeine weekends on, weekdays off (to prevent tolerance being developed), with moderate success (Monday withdrawal really sucks, but Saturday is awesome).

Should I add the problems to Anki? It takes time to create the cards and review them; is that time better spent doing more problems?
Advice request: Buying a car
by Maelin — 2011-05-29T07:33:36.218Z

So I'm looking at buying a car. At the moment I am using my parents' old car, however when I move out of home I will not be able to take it with me, and it also lacks some of the features I would like (cruise control in particular).

I'm looking at buying a small car, probably a hatchback. At the moment I have saved around AU$12,000. My parents are willing to lend me some amount, probably up to $5000, and I work for a bank which can give me a loan at a very favourable staff rate. I earn a bit over $325 per week at the moment and I have few living expenses beyond luxuries, that income is bolstered semifrequently when I can work extra shifts. At the beginning of 2013 I will be seeking a full time job as a high school teacher.

Looking at a few car websites (in particular carsales.com.au) it appears I can buy a fairly good second hand car for around $15,000, or a brand new car that seems quite good for around $24,000. This is a very rough guide to local prices.

I am aware that my decision process in this judgement is very fallible because I don't know much about buying cars, e.g. the potential pitfalls of a second hand one, and any other things I need to take into account.

Being as this is by far the most money I will ever have spent on a single thing and it is likely to last me for most of a decade at least, I am strongly motivated to make this decision rationally.

Does anybody have any advice they can give on how I can decide how much money to spend, or things I should consider when comparing potential cars?
Calculus textbook recommendation?
by Tesseract — 2011-05-29T06:20:02.382Z

I'm seeking suggestions for a calculus textbook that I could use to teach myself the subject.

Details:

[Personal details removed.] I know that most calculus textbooks are designed to be taught to classes, so I was wondering if anyone knew of a textbook specifically designed for autodidacts, or one that would be particularly useful for the purpose. (If you just know of a good general textbook, I'd be grateful to hear that as well.)

Thanks to anyone who gives a suggestion.

EDIT: Chose a Marvin-Gardner-edited version of Calculus Made Easy, accompanied by Khan Academy lessons.
A discussion of an applictation of Bayes' theorem to everyday life
by AdeleneDawner — 2011-05-29T06:04:47.121Z

[12:49:29 AM] Conversational Partner: actually, even if the praise is honest, it makes me uncomfortable if it seems excessive. that is, repeated too often, or made a big deal about.

[12:49:58 AM] Adelene Dawner: 'Seems excessive' can actually be a cue for 'is insincere'.

[12:50:05 AM] Conversational Partner: oh

[12:50:25 AM] Adelene Dawner: That kind of praise tends to parse to me as someone trying to push my buttons.

[12:51:53 AM | Edited 12:52:09 AM] Conversational Partner: is it at least theoretically possible that the praise is honest, and the other person just happens to think that the thing is more praiseworthy than I do? or if the other person has a different opinion than I do about how much praise is appropriate in general?

[12:52:59 AM] Adelene Dawner: Of course.

[12:53:13 AM] Adelene Dawner: This is a situation where looking at Bayes' theorem is useful.

[12:54:03 AM] Conversational Partner: ooh, that might be the first situation I've ever encountered where applying Bayes' theorem directly is useful... or did you not mean applying it directly?

[12:55:18 AM] Adelene Dawner: You might not be able to apply it directly in terms of plugging in numbers and getting a result, but you can use it in terms of 'these things make the probability that this signal is strong evidence higher, and these things make it lower', which then makes the whole situation make more sense.

[12:55:50 AM] Conversational Partner: oh, thanks for explaining that

[12:55:56 AM] Conversational Partner: also, what were you saying before about "costly signal"?

[12:56:16 AM] Adelene Dawner: That actually ties into the Bayes' theorem thing. I'll expand.

[12:56:52 AM] Adelene Dawner: The relevant thing here is the probability that you did something right given that you were praised for it by a particular person or in a particular way, right?

[12:57:38 AM] Conversational Partner: yes

[12:57:44 AM] Conversational Partner: or at least, I think so

[12:59:02 AM] Adelene Dawner: Okay. So A is 'I did the thing right' and B is 'I was praised for A in a particular way'. Bayes' theorem says that P(A given B) is P(B given A) times P(A) over P(B). P(B given A) and P(A) make it go up, P(B) makes it go down.

[1:02:29 AM] Adelene Dawner: So, if I praise you in a particular way every time you do a particular thing right, without fail, P(B given A) is 1 and the chance that you did the thing right given th...
"Thank you for updating"
by rysade — 2011-05-28T23:41:49.958Z

Would it be supercilious to thank someone for updating? I know I would feel uncomfortable doing it, but I often feel the urge to do so anyway.

There seems to be something vicious about thanking them. My own estimation of my own belief has not changed in these situations. I feel fairly satisfied that the other has considered my view and has shakily come to agree with it. I worry it would be a little like saying 'screw your opinion, now you see I'm right.'

However:

It is both rational and polite to thank them. The gentlemen's agreement of rationality allows for one person to be wrong and not lose face at all. When someone concedes something to me (a point) I typically feel the need to thank them.

What do you think?