spider-mario
4 days ago
> Second, besides noise (bars are Wilson 95% confidence intervals, very conservative for run-to-run noise), there is little difference down to 4-bit; only the 2-bit scores a bit lower.
Confidence intervals have nothing to do with run-to-run variation. They have little to do with anything people usually ascribe to them (https://link.springer.com/article/10.3758/s13423-015-0947-8 ), but even less with run-to-run variation (https://link.springer.com/article/10.1007/s10654-016-0149-3 misconception 22).
stared
4 days ago
Point taken, but there is a much more fundamental issue with it - and precisely why I wrote "very conservative".
It is a different problem if we pick two sets from the same data distribution, A and B, and first we have a score on A, then on B. Here we re-run on precisely the same set of Terminal Bench 2.1 problems. It may be that results are so random between runs that each single task has the same probability in a Bernoulli distribution. But more likely, many problems are easy (i.e. each run will solve them consistently), many are too hard (i.e. no run is going to solve them) and only a fraction is somehow in between.
Maybe there is some good trick to find a proper distribution, but to my knowledge, we would need to run it at least two times on TB2.1 to get any more educated estimates. That said, I am open to new ideas.
That said, I consider frequentist probability a dirty trick, and that Bayesian is the proper way of doing things (vide David J.C. MacKay" Information Theory, Inference, and Learning Algorithms" and Cam Davidson-Pilon "Probabilistic Programming & Bayesian Methods for Hackers" https://www.inference.org.uk/itprnn/book.pdf, https://dataorigami.net/Probabilistic-Programming-and-Bayesi...).
tomrod
4 days ago
I'm working on a practical review implementation on this! Great to hear others are thinking along the same way.
ricardobeat
4 days ago
The main problem here is that a model that wildly fluctuates with 60% - 100% - 80% results will have the same wilson score as one that repeatedly scores 80% - 80% - 80%. So the 'confidence interval' bar is meaningless.
I'm not that well versed in statistics, but a standard box plot is probably the best alternative
stared
4 days ago
A single result is binary. All we get from a run is which tasks were solved, which weren’t.
breezybottom
4 days ago
Both ways involve sophistry. If you don't like dirty tricks, statistics isn't for you.
mitxela
3 days ago
> The key confusion underlying the FCF is the confusion of what is known before observing the data — that the CI, whatever it will be, has a fixed chance of containing the true value — with what is known after observing the data. Frequentist CI theory says nothing at all about the probability that a particular, observed confidence interval contains the true value; it is either 0 (if the interval does not contain the parameter) or 1 (if the interval does contain the true value).
They're really reaching here. There's a 95% chance it's 1 and 5% it's 0, and we don't know which, so it's 0.95. Like saying the probability your uncle died of a shark attack is either 0 or 1. But in reality I don't know anything about your uncle so it's the average probability that someone picked randomly died of a shark attack, until I know it about your uncle.
spider-mario
3 days ago
It’s not reaching. See my response here: https://news.ycombinator.com/item?id=49622516
> Like saying the probability your uncle died of a shark attack is either 0 or 1.
For a frequentist, it is. If you want the Bayesian interpretation of probability, you must calculate the probability that the interval you ended up getting contains the true value, and you can easily end up with a value different from 95%. You can’t just stop at “it was drawn from a pool of intervals of which 95% contain the true value” since you have more information than that (the numerical bounds).
mitxela
2 days ago
If I pick an interval uniformly from a pool of intervals of which 95% contain the true value, it has a 95% chance of containing the true value.
Just like if I pick a ball from a bag of balls of which 50% are red, it has a 50% chance of being red.
spider-mario
2 days ago
If you pick a ball from a bag of balls of which half are fuzzy and half are smooth, then before picking the ball, there is indeed a 50% chance that you will pick a fuzzy one. But once you pick it and have it in your hand and can feel the texture, are you still going to say “there’s a 50% chance that this specific ball I am holding is fuzzy because half of the balls in there were”?
diseasedyak
4 days ago
Yah, prediction interval instead, right? (I'm still learning statistics)
Saying there's a confidence interval for run-to-run makes no sense, from the way I understand it, because each run could have a result that's all over the place.
spider-mario
4 days ago
Yes. It’s maybe easier to reason about by imagining that we are trying to estimate the parameter of a Gaussian distribution.
Let’s say that the “true” distribution of the data has mean μ=100 and standard deviation σ=15, but we don’t know that.
95% confidence interval for μ = “if we repeatedly draw N samples from the true distribution and compute a confidence interval every time, 95% of those intervals will contain μ.” That’s all that the definition of a confidence interval guarantees. It does not follow that if we take one of those intervals, it, specifically, has a 95% chance of containing μ. For a frequentist, that’s a meaningless statement (both the interval and μ are fixed so there’s no frequentist probability about it); for a Bayesian, there is no guarantee that that probability is 95%. 95% is instead the probability of “sampling data that will happen to generate an interval that contains μ”.
95% Bayesian credible interval for μ = interval that can be interpreted as having a 95% probability of containing μ, generally obtained by computing the posterior probability density distribution for μ and finding an interval that encompasses 95% of the probability mass. Conventions include highest-density intervals (HDIs), which are obtained by making sure that the PDF is equal at both bounds, and equal-tailed intervals (equal probability mass before and after the interval). With enough samples, it may become arbitrarily narrow (“we are very sure of the mean”), despite the standard deviation of 15 that is built into the “true” distribution that we are estimating, and a Jeffreys prior will happen to make it satisfy the definition of a confidence interval as well (https://sami.boo/jaynes/confidence-intervals-vs-bayesian-int... ).
Posterior predictive distribution = taking into account the uncertainty on both μ and σ, distribution of samples that would be obtained by sampling from N(μ, σ) (which, because of that uncertainty, is a https://en.wikipedia.org/wiki/Compound_probability_distribut... but may have a convenient closed form https://en.wikipedia.org/wiki/Conjugate_prior#Table_of_conju... ), from which we can likewise extract a 95% interval.
jmalicki
4 days ago
> a Jeffreys prior will happen to make it satisfy the definition of a confidence interval as well
Correct, which is why it's crazy to consider a Jeffrey's credible interval as valid, but not a confidence interval.
Perhaps the confidence interval might not be valid under a frequentist interpretation, but the same math can be valid under a Bayesian interpretation even if the person publishing didn't mean it that way.
Does that mean it's valid? Well, the confidence interval is as equally valid as a lot of credible intervals, which is why the credible interval is not some magical better alternative - it's mostly just a choice of how you interpret it, it is not a different procedure that produces different numbers.
spider-mario
3 days ago
> Correct, which is why it's crazy to consider a Jeffrey's credible interval as valid, but not a confidence interval.
The confidence interval is “valid” for what its definition is. But:
- saying “there is a 95% chance that this interval contains the true value because it’s a 95% credible interval” is valid;
- saying “there is a 95% chance that this interval contains the true value because it’s a 95% confidence interval” is not.
It’s very much possible to reach a correct conclusion with invalid reasoning. As Jaynes put it:
“As we stress repeatedly, from a false proposition – or from a fallacious argument that leads to a false proposition – all propositions, true and false, may be deduced. But this is just the danger; if fallacious reasoning always led to absurd conclusions, it would be found out at once and corrected. But once an easy, shortcut mode of reasoning has led to a few correct results, almost everybody accepts it; those who try to warn against it are not listened to.”
> Perhaps the confidence interval might not be valid under a frequentist interpretation, but the same math can be valid under a Bayesian interpretation even if the person publishing didn't mean it that way.
Right, but if you get that by happenstance and not from the fact that it’s a confidence interval, the latter isn’t really getting you anything.
Per my first link: “One of the misconceptions regarding the relationship between Bayesian inference and frequentist inference is that they will lead to the same inferences, and hence all confidence intervals can simply be interpreted in a Bayesian way. In the case where data are normally distributed, for instance, there is a particular prior that will lead to a confidence interval that is numerically identical to Bayesian credible intervals computed using the Bayesian posterior (Jeffreys 1961; Lindley 1965). This might lead one to suspect that it does not matter whether one uses confidence procedures or Bayesian procedures. We showed, however, that confidence intervals and credible intervals can disagree markedly. The only way to know that a confidence interval is numerically identical to some credible interval is to prove it. The correspondence cannot — and should not — be assumed. More broadly, the defense of confidence procedures by noting that, in some restricted cases, they numerically correspond to Bayesian procedures is actually no defense at all. One must first choose which confidence procedure, of many, to use; if one is committed to the procedure that allows a Bayesian interpretation, then one’s time is much better spent simply applying Bayesian theory. If the benefits of Bayesian theory are desired — and they clearly are, by proponents of confidence intervals — then there is no reason why Bayesian inference should not be applied in its full generality, rather than using the occasional correspondence with credible intervals as a hand-waving defense of confidence intervals.”
> which is why the credible interval is not some magical better alternative - it's mostly just a choice of how you interpret it
You don’t get to choose to interpret a confidence interval as if it had the meaning of a Bayesian one. If you want to be able to interpret a computed interval as “there is a 95% chance that this interval contains the true parameter value”, only the Bayesian procedure guarantees it, and it can very well produce different numbers. So it is a better alternative in that sense.
Dylan16807
3 days ago
If 95% of the intervals in your set of intervals include μ, and you randomly pick one of them, in what way is that interval not 95% likely to contain μ? Ignoring the frequentist pedantry that "likelyhood is the wrong word", is there a way for a different number to be the correct number?
spider-mario
3 days ago
https://link.springer.com/article/10.3758/s13423-015-0947-8#...
https://sami.boo/jaynes/confidence-intervals-vs-bayesian-int... (+ Cauchy example below)
It’s like with test accuracy. Test accuracy is the pre-test probability that the test will give a correct result. But once you have a positive or negative result, which way it turned out plays a part in computing the predictive value. Likewise, once you have computed the interval, the specific bounds you ended up getting can affect the plausibility that they contain the true value.
Dylan16807
3 days ago
Your first link is just the pedantry. The second link... I don't see any issue for the question I asked? The confidence interval is supposed to be wrong a certain percent of the time. It being so wrong it disproves itself is funny but the 90% is still 90% isn't it? They just landed in a particularly harsh part of the 10%.
The conclusion they come to is "it is possible to do better in the individual case by taking into account evidence from the sample that the confidence interval method throws away". That means the confidence interval is inefficient, not incorrect.
spider-mario
3 days ago
If you just meant it in the sense that “if I pick one interval at random and don’t look at it, it will (future) have an n% chance of containing the true parameter” then sure. It’s a pre-data statement (“I have an n% chance of sampling data that will happen to generate an interval that contains the parameter”).
But once you have picked one, and you know its bounds (say, [12.1471, 13.8264]), then it’s fallacious to make the post-data reasoning that “because it was picked at random from the set of 90% confidence intervals, it has a 90% chance of containing the true parameter”.
Again, it’s like with medical tests. If a test has 90% sensitivity and 90% specificity, it has 90% accuracy (it will, in 90% of cases, produce a result that matches disease status) – a pre-data statement on the test result (/ on the confidence interval that we will compute). But it does not follow that, if you screen an asymptomatic patient with low prior probability of disease and get a positive result, they have a 90% chance of having the disease – a post-data statement on disease status, given the test result (/ on where the parameter lies, given the interval).
> The conclusion they come to is "it is possible to do better in the individual case by taking into account evidence from the sample that the confidence interval method throws away". That means the confidence interval is inefficient, not incorrect.
It means you know in the individual case that the specific confidence interval does not in fact have a 90% chance of containing the parameter.
user
3 days ago
Dylan16807
3 days ago
I can see the analogy here but a test like that being binary throws things off and that's also super asymmetrical error. Weren't we sampling a gaussian?
spider-mario
3 days ago
It’s the same principle. The “trivial interval” from the first link (the one you called pedantry), which is applicable to a Gaussian and tweakable to have different coverage than 50%, clearly shows that being an n% confidence interval, on its own, doesn’t guarantee the ability to directly translate that n% into post-data inferences. Likewise, in the truncated exponential example, when you compute that confidence interval of [12.1471, 13.8264], you know that the probability that it contains θ is 0%. In the Cauchy example, when you compute [-2.31, 10.31] as your 90% CI from the two samples 3 and 5, you can then compute that it actually has a >99% chance of containing θ (table I).
Dylan16807
3 days ago
The trivial example demonstrates that once you pick your confidence interval you don't truly have odds anymore. But if we're judging whether we have the right odds number, what matters is if a different number is correct. And the only coherent way I can think of to interpret "odds" says the odds are 50%. The trivial example survives that challenge.
The examples that actually ruin the number so far have been asymmetrical. Is there a way to do it with a gaussian? Especially if you're trying to make a reasonable internal?
spider-mario
2 days ago
> The trivial example demonstrates that once you pick your confidence interval you don't truly have odds anymore. But if we're judging whether we have the right odds number, what matters is if a different number is correct. And the only coherent way I can think of to interpret "odds" says the odds are 50%. The trivial example survives that challenge.
Sorry, I don’t follow. Let’s say I want to compute a 50% confidence interval for the unknown mean of a Gaussian distribution. I sample two numbers from the distribution, get 9 and 7, compute the interval according to the trivial procedure and get (-∞, ∞). Does the interval (-∞, ∞) have a 50% probability of containing the mean of that Gaussian distribution? I would think it’s closer to 100%.
If what you are saying is “it’s meaningless to talk about the probability of that specific interval containing the unknown-but-fixed parameter” then that’s the purely frequentist view and then you also agree that it’s meaningless to say that [-2.31, 10.31] has a 90% chance of containing the location parameter of the Cauchy distribution that happened to yield the samples 3 and 5. Incidentally, what asymmetry are you referring to in the Cauchy example?
Dylan16807
2 days ago
I misread the Cauchy example since I was going too fast, nevermind the symmetry part.
> then that’s the purely frequentist view and then you also agree that it’s meaningless
I'm saying that when you hit "meaningless" you can back up a step to where you actually had randomness and look at that distribution, which gets rid of a lot of these issues.
But after looking at these examples I think it only makes sense in limited circumstances to do that. Like in the trivial example: your final distribution isn't based on the probability of the mean being any particular number. The only probability was back a step and that was 50%.
At this point I still don't think it's objectively wrong to say a particular interval above is 95% likely, but there's too many ways to interpret the statement so nobody should say it is.
The way we're calculating that these intervals are "wrong" is by looking at all the possible parameters that could have given us the samples we got, and checking how often the range contains the parameter. That's a useful calculation but is it the one people expect? I think that depends on the situation. Treating the parameter as being the thing we sample over is misleading, but treating it as fixed is also misleading.
Snowfield9571
a day ago
just read a book on Bayesian stats bro. Probably just the intro of most of them will get you to your answer.
Dylan16807
a day ago
I already have my answer.
Y_Shalev
3 days ago
[flagged]
jnwatson
4 days ago
Mind blown. The more I read about statistics, the less I know.
exogenousdata
4 days ago
“There are three kinds of lies: Lies, damned lies and statistics.” - Mark Twain (attributed but unsubstantiated to Benjamin Disraeli)
maCDzP
4 days ago
Thank you for these, coz I learned a lot! Great that they are open access.
soltanov
3 days ago
Task success is not enough; reliability and cost matter.
fr2029
4 days ago
the 2nd derivate of shannon covariance of noise begs to differ
user
4 days ago