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Comment author: lmm 26 November 2015 12:48:10PM 0 points [-]

I would two-box on this problem because of diminishing returns, and one-box on the original problem.

Comment author: gjm 26 November 2015 03:15:12PM 4 points [-]

Your returns must be very rapidly diminishing. If u is your kilobucks-to-utilons function then you need [7920u(1001)+80u(1)]/8000 > [3996u(1000)+4u(0)]/4000, or more simply 990u(1001)+10u(1) > 999u(1000)+u(0). If, e.g., u(x) = log(1+x) (a plausible rate of decrease, assuming your initial net worth is close to zero) then what you need is 6847.6 > 6901.8, which doesn't hold. Even if u(x) = log(1+log(1+x)) the condition doesn't hold.

If we fix our origin by saying that u(0)=0 (i.e., we're looking at utility change as a result of the transaction) and suppose that at any rate u(1001) <= 1001/1000.u(1000), which is certainly true if returns are always diminishing, then "two-boxing is better because of diminishing returns" implies 10u(1) > 8.01u(1000). In other words, gaining $1M has to be no more than about 25% better than gaining $1k.

Are you sure you two-box because of diminishing returns?

Comment author: AstraSequi 26 November 2015 01:53:56AM *  0 points [-]

My intuition is from the six points in Kahan's post. If the next flip is heads, then the flip after is more likely to be tails, relative to if the next flip is tails. If we have an equal number of heads and tails left, P(HT) > P(HH) for the next two flips. After the first heads, the probability for the next two might not give P(TH) > P(TT), but relative to independence it will be biased in that direction because the first T gets used up.

Is there a mistake? I haven't done any probability in a while.

Comment author: gjm 26 November 2015 02:23:18AM 2 points [-]

If the next flip is heads, then the flip after is more likely to be tails, relative to if the next flip is tails.

No, that is not correct. Have a look at my list of 16 length-4 sequences. Exactly half of all flips-after-heads are heads, and the other half tails. Exactly half of all flips-after-tails are heads, and the other half tails.

The result of Miller and Sanjuro is very specifically about "averages of averages". Here's a key quotation:

We demonstrate that in a finite sequence generated by i.i.d. Bernoulli trials with probability of success p, the relative frequency of success on those trials that immediately follow a streak of one, or more, consecutive successes is expected to be strictly less than p

"The relative frequency [average #1] is expected [average #2] to be ...". M&S are not saying that in finite sequences of trials successes are actually rarer after streaks of success. They're saying that if you compute their frequency separately for each of your finite sequences then the average frequency you'll get will be lower. These are not the same thing. If, e.g., you run a large number of those finite sequences and aggregate the counts of streaks and successes-after-streaks, the effect disappears.

Comment author: Lumifer 25 November 2015 07:08:43PM 2 points [-]

amount of exercise is a better predictor of lifespan than weight

First, there is no reason for you to care about ranking ("better"), you should only care whether something is a good predictor of lifespan. Predictors are not exclusive.

Second, weight effect on lifespan is nonlinear. As far as I remember it's basically a U-shaped curve.

Comment author: gjm 26 November 2015 12:08:07AM 2 points [-]

I think it's only U-shaped if you're plotting mortality rather than lifespan on the y-axis...

Comment author: OrphanWilde 25 November 2015 08:20:35PM -1 points [-]

As one gets older and more successful, one gets less status-anxious.

Which is why you're spending time assuring us that you're high-status?

Comment author: gjm 26 November 2015 12:05:17AM *  2 points [-]

Ilya's comments about status could indeed be explained by the hypothesis that he's attempting some kind of sneaky second-order status manoeuvre. They could also be explained by his meaning what he says and genuinely not caring much (consciously or otherwise) about status here on LW. To me, the second looks at least as plausible as the first.

More precisely: I doubt anyone is ever completely 100% unaffected by status considerations; the question is how much; Ilya's claim is that in this context the answer is "negligibly"; and I suggest that that could well be correct.

You may be correct to say it isn't. But if so, it isn't enough just to observe that someone motivated by status might say the things Ilya has, because so might someone who in this context is only negligibly motivated by status. You need either to show us something Ilya's doing that's substantially better explained in status-seeking terms, or else give a reason why we should think him much more likely to be substantially status-seeking than not a priori.

[EDITED to add: I have no very strong opinion on whether and to what degree Ilya's comments here are status manoeuvres.]

In response to comment by gwern on Against NHST
Comment author: Tem42 25 November 2015 11:29:20PM 0 points [-]

Okay, stupid question :-/

“Almost all of them think it gives some direct information about how likely they are to be wrong, and that’s definitely not what a p-value does...”


"...the technical definition of a p-value — the probability of getting results at least as extreme as the ones you observed, given that the null hypothesis is correct..."

Aren't these basically the same? Can't you paraphrase them both as "the probability that you would get this result if your hypothesis was wrong"? Am I failing to understand what they mean by 'direct information'? Or am I being overly binary in assuming that the hypothesis and the null hypothesis as the only two possibilities?

In response to comment by Tem42 on Against NHST
Comment author: gjm 25 November 2015 11:52:48PM *  2 points [-]

What p-values actually mean:

  • How likely is it that you'd get a result this impressive just by chance if the effect you're looking for isn't actually there?

What they're commonly taken to mean?

  • How likely is it, given the impressiveness of the result, that the effect you're looking for is actually there?

That is, p-values measure Pr(observations | null hypothesis) whereas what you want is more like Pr(alternative hypothesis | observations).

(Actually, what you want is more like a probability distribution for the size of the effect -- that's the "overly binary* thing -- but never mind that for now.)

So what are the relevant differences between these?

  • If your null hypothesis and alternative hypothesis are one another's negations (as they're supposed to be) then you're looking at the relationship between Pr(A|B) and Pr(B|A). These are famously related by Bayes' theorem, but they are certainly not the same thing. We have Pr(A|B) = Pr(A&B)/Pr(B) and Pr(B|A) = Pr(A&B)/Pr(A) so the ratio between the two is the ratio of probabilities of A and B. So, e.g., suppose you are interested in ESP and you do a study on precognition or something whose result has a p-value of 0.05. If your priors are like mine, your estimate of Pr(precognition) will still be extremely small because precognition is (in advance of the experimental evidence) much more unlikely than just randomly getting however many correct guesses it takes to get a p-value of 0.05.

  • In practice, the null hypothesis is usually something like "X =Y" or "X<=Y". Then your alternative is "X /= Y" or "X > Y". But in practice what you actually care about is that X and Y are substantially unequal, or X is substantially bigger than Y, and that's probably the alternative you actually have in mind even if you're doing statistical tests that just accept or reject the null hypothesis. So a small p-value may come from a very carefully measured difference that's too small to care about. E.g., suppose that before you do your precognition study you think (for whatever reason) that precog is about as likely to be real as not. Then after the study results come in, you should in fact think it's probably real. But if you then think "aha, time to book my flight to Las Vegas" you may be making a terrible mistake even if you're right about precognition being real. Because maybe your study looked at someone predicting a million die rolls and they got 500 more right than you'd expect by chance; that would be very exciting scientifically but probably useless for casino gambling because it's not enough to outweigh the house's advantage.

[EDITED to fix a typo and clarify a bit.]

Comment author: Lumifer 25 November 2015 04:27:40PM 0 points [-]

The winning situation is !F(P)

That doesn't have anything to do with the halting problem, it looks like a close relative of the Barber paradox.

Comment author: gjm 25 November 2015 06:08:25PM 1 point [-]

It has something to do with the halting problem. The usual way of demonstrating that no program can solve the halting problem is to suppose you've got one that does and use it to carry out a construction a bit like the one HungryHobo is gesturing towards, where F arranges to halt iff the halting-tester says it doesn't.

Comment author: Lumifer 25 November 2015 05:08:14AM 0 points [-]

Lewandowsky has an... interesting reputation.

Comment author: gjm 25 November 2015 02:34:30PM 2 points [-]

If there's something wrong with the article, it seems like you should be able to say what it is rather than making insinuations about one of its authors.

(Lewandowsky is strongly disliked by those whose position on global warming differs from the mainstream scientific consensus, no doubt. So far as I can tell he doesn't have a reputation for dishonesty or incompetence among groups without a strong motivation to put him down.)

Comment author: Gleb_Tsipursky 25 November 2015 04:44:45AM 0 points [-]

I understand you use posturing and accusations without evidence as part of your Dark Arts arsenal, and accept that. I doubt anyone will come across this comment, since it's so far down and the thread is not new. Just wanted you to know personally, in case you aren't simply posturing, that I don't use sockpuppets. I have a number of Less Wronger friends who support the cause of spreading rational thinking broadly, and whenever I make significant posts or particularly salient comments, I let them know, so that they can give me optimizing suggestions and feedback.

Since they happen to share many of my views, they sometimes upvote my comments. They generally don't participate actively, and this is a rare exception on the part of Raelifin, as they do not want to be caught in the backlash. So FYI for the future. Feel free to continue making these accusations for Dark Arts purposes if you wish, but I just wanted you to know.

Comment author: gjm 25 November 2015 02:26:02PM 3 points [-]

The term for extreme versions of this is "meatpuppet". Of course having friends is not the same thing as having meatpuppets, and I have no way of knowing to what extent your friends are LW participants who just happen to be your friends and would have upvoted your articles anyway, and to what extent they're people who come here only to upvote your articles for you. The nearer the latter, the more meatpuppety.

Comment author: Viliam 25 November 2015 09:33:42AM *  1 point [-]

However, if you're predicting the next flip in a finite series of flips that has already occurred, it's actually more likely that you'll alternate between heads and tails.

...because heads occurring separately are on average balanced by heads occurring in long sequences; but limiting the length of the series puts a limit on the long sequences.

In other words, in infinite sequences, "heads preceeded by heads" and "heads preceeded by tails" would be in balance, but if you cut out a finite subsequence, if the first one was "head preceeded by head", by cutting out the subsequence you have reclassified it.

Am I correct, or is there more?

Comment author: gjm 25 November 2015 02:13:29PM 1 point [-]

I don't think this is correct. See my reply to AstraSequi.

(But I'm not certain I've understood what you're proposing, and if I haven't then of course your analysis and mine could both be right.)

Comment author: AstraSequi 25 November 2015 02:24:34AM *  2 points [-]

I just found out about the “hot hand fallacy fallacy” (Dan Kahan, Andrew Gelman, Miller&Sanjuro paper) as a type of bias that more numerate people are likely more susceptible to, and for whom it's highly counterintuitive. It's described as a specific failure mode of the intuition used to get rid of the gambler's fallacy.

I understand the correct statement like this. Suppose we’re flipping a fair coin.

*If you're predicting future flips of the coin, the next flip is unaffected by the results of your previous flips, because the flips are independent. So far, so good.

*However, if you're predicting the next flip in a finite series of flips that has already occurred, it's actually more likely that you'll alternate between heads and tails.

The discussion is mostly about whether a streak of a given length will end or continue. This is for length of 1 and probability of 0.5. Another example is

...we can offer the following lottery at a $5 ticket price: a fair coin will be flipped 4 times. if the relative frequency of heads on flips that immediately follow a heads is greater than 0.5 then the ticket pays $10; if the relative frequency is less than 0.5 then the ticket pays $0; if the relative frequency is exactly equal to 0.5, or if no flip is immediately preceded by a heads, then a new sequence of 4 flips is generated. While, intuitively, it seems like the expected payout of this ticket is $0, it is actually $-0.71 (see Table 1). Curiously, this betting game may be more attractive to someone who believes in the independence of coin flips, rather than someone who holds the Gambler’s fallacy.

Comment author: gjm 25 November 2015 02:03:18PM 5 points [-]

I think this is not quite right, and it's not-quite-right in an important way. It really isn't true in any sense that "it's more likely that you'll alternate between heads and tails". This is a Simpson's-paradox-y thing where "the average of the averages doesn't equal the average".

Suppose you flip a coin four times, and you do this 16 times, and happen to get each possible outcome once: TTTT TTTH TTHT TTHH THTT THTH THHT THHH HTTT HTTH HTHT HTHH HHTT HHTH HHHT HHHH.

  • Question 1: in this whole sequence of events, what fraction of the time was the flip after a head another head? Answer: there were 24 flips after heads, and of these 12 were heads. So: exactly half the time, as it should be. (Clarification: we don't count the first flip of a group of 4 as "after a head" even if the previous group ended with a head.)
  • Question 2: if you answer that same question for each group of four, and ignore cases where the answer is indeterminate because it involves dividing by zero, what's the average of the results: Answer: it goes 0/0 0/0 0/1 1/1 0/1 0/1 1/2 2/2 0/1 0/1 0/2 1/2 1/2 1/2 2/3 3/3. We have to ignore the first two. The average of the rest is 17/42, or just over 0.4.

What's going on here isn't any kind of tendency for heads and tails to alternate. It's that an individual head or tail "counts for more" when the denominator is smaller, i.e., when there are fewer heads in the sample.

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