tprice7

The Stone-Weierstrass theorem is not related to Taylor series, in fact the target function need not have a derivative, let alone have all higher-order derivatives, a prerequisite for the Taylor series to even exist.

"Piecewise continuous functions can be approximated by differentiable ones."

It depends on how you measure the distance between functions. If you are using the uniform norm (as is used in the universal approximation theorem) then this is false.

I gave a quasi-explanation in response to lamename's comment. If you want some search queries, learn what convolution is, then observe the connection between discrete convolution and polynomial multiplication, then learn about the convolution theorem.

I haven't studied the Schönhage–Strassen algorithm closely so take this with a grain of salt. But I think I can give you at least a vague, simplified sense of how the FFT is relevant.

Multiplying polynomials is the same as convolving their coefficients, which is the same as multiplying the Fourier transforms of their coefficients (this is the convolution theorem), which can be done efficiently with the FFT. Now a base-10 number like 971 can be represented as the polynomial 9x^2 + 7x + 1 evaluated at x = 10. So multiplying base-10 numbers (or base-whatever for that matter) can sort of be seen as a special case of multiplying polynomials (plus a step afterward for carrying the digits), which can be done with the FFT.

Here's a fun puzzle: concretely, what is a group of groups?

Sometimes yes, sometimes no. Of course I can go into much more depth in my studies / research while not working. I work the minimum amount necessary to pay my living expenses so I can devote a maximum amount of time to freely pursuing other interests, which includes pure math among other things.

There are plenty of people who devote several years of their life to studying, and must pay not only their living expenses but tuition fees as well. In my opinion, those are the people who you should be asking “how did you manage this”.

Are you referring to the "An Inequality for Gaussians on Lattices" paper? They cite my paper, but it's to give an example of an application of their result (which I use), not because they built on it. But anyway, I think it's very fascinating that the people who discovered a key result that I needed for that paper, which could probably be best classified as arithmetic geometry, are mainly computer scientists!

And I think abnry is right that you need some motivating problem / project. In my case it was to prove the Riemann hypothesis. Obviously I have not succeeded but I've learned a lot in the process and it has indirectly led me to some good research questions. I think choosing an outrageously ambitious pie-in-the-sky problem is ok if you are patient and don't try to approach it too directly.

Not necessarily. I left undergrad after four terms and self-studied to the point of having published research, giving invited talks, and even being a visiting researcher for three months with my expenses paid.

Links:

http://content.algebraicgeometry.nl/2017-2/2017-2-007.pdf

https://projecteuclid.org/download/pdfview_1/euclid.ecp/1508...

Look for my name (Thomas / Tom Price) on these pages:

https://web.archive.org/web/20170121000748/https:/wwwmath.un...

https://www2.math.binghamton.edu/p/seminars/arit/arit_spring...

https://www2.math.binghamton.edu/p/seminars/arit/arit_fall20...

https://www2.math.binghamton.edu/p/seminars/arit/arit_fall20...

Hence pendulum clocks.

In general it's the eigenvectors of the positive-semidefinite (hence symmetric) part of the left polar decomposition.

Also, in case this clears up some confusion, I was using the nonstandard definition in some posts because I inferred that this was the definition you were using.

Also, every probability mentioned should also include in the list of conditions that the number of samples observed matches our experiment.

Also, as an aside, I’m not sure what you mean specifically when you say “model”. The usual definition is a set of probability distributions on a common sample space. But when you say things like “all models are wrong”, I’m assuming you are just using “model” as a synonym for “distribution”, because this claim is vacuously false using the usual definition of model (just take the set of distributions to be every distribution on that sample space; this must contain the true distribution by definition). But then when you say “Looking closer at this you are describing one model but testing different values of one of the model parameters”, this seems to make more sense if you are using the usual definition of model. So I’m not sure what definition you are using.

Ooops, I forgot to include something in the list of conditions. For every probability I mentioned in the post you just quoted and its grandparent post, we should add to the list of conditions that the number of samples used is the same as the number of samples used in our experiment.

Anyways, if it sounds like nonsense then I will try rephrasing. Right now the threshold for statistical significance is p = 0.05, sometimes lower depending on the field. Let’s say we want to lower this threshold, and we need some way of determining how low it should be. I am suggesting that this could be done by deciding on an acceptably safe threshold, say q, for p(null | T, num samples, normal, iid), and an acceptably safe upper bound, say x_0, for p(null | num samples, normal, iid) / p(T | num samples, normal, iid). Then we use q / x_0 as the threshold for statistical significance.

We would also be using p-values for a null hypothesis corresponding to a range of means rather than a single value (or a range of differences in means, or something else suitably adapted to the type of test we are doing), because with a single value, having an upper bound on the probability of the null hypothesis is vacuous, as you very eagerly point out.

You could argue that this is useless because what we really want to bound is p(null | num samples, T), not p(null | num samples, T, normal, iid). I would respond that, yes this would be the more desirable quantity to know, but we need to make some simplifying assumptions in order to make this problem tractable, and while the normal and iid assumptions probably aren’t true, there are at least situations where we can be confident that they approximate reality reasonably well.

“Looking closer at this you are describing one model but testing different values of one of the model parameters.”

Furthermore, this is even done with the usual 1-sample t-test because the variance can be anything.

You mean for computing the p-value associated with a range of means rather than a single null mean value? That’s the only thing I can think of for which providing code or calculation steps would be applicable (well, there’s also getting a Fermi approximation for the total number of experiments ever performed in a discipline, which I gave as a preliminary suggestion for a cautious estimate of x, but I don’t have time to do that and the main difficulty of that isn’t the math anyways). Anyways I would be happy to provide that if you can confirm that this is what you meant and that you are asking out of genuine curiosity.

“Looking closer at this you are describing one model but testing different values of one of the model parameters.”

I am inferring from context that this is probably supposed to be a criticism but it doesn’t make much sense to me. Of course we have to consider different values of the mean, the whole point is to get a p-value corresponding to a range of potential different means.

But anyways, I do think that an explanation based on the t_0 statistic I defined in my other post is better.

1. We can define a statistic t_0 that is the infimum of the absolute value of all t-statistics for every candidate mean in the interval.

2. Suppose the mean is some value mu’ in the interval. Whenever the t_0 statistic is at least as big as its observed value, the t-statistic corresponding to mu’ is also at least as big in absolute value as the observed value of t_0, by definition.

3. So we can give an upper bound for the probability of the former event by the probability of the latter event. But the probility of the latter event is the same no matter what mu’ is, and can be computed using Student’s T distribution.

4. Therefore, we have an upper bound for the probability of t_0 attaining a value at least as big as the observed value, assuming that the mean is somewhere in the specified interval (plus the other standard assumptions). This is the p-value.

Do you agree with those assertions? If not, specifically where is the problem?

“I'm not sure what you are arguing anymore.”

It’s the claim I make in 3, and then the secondary claim that making our upper bound on p(null | T, normal, iid) small for significant p-values (i.e. p(T | null, normal, id)) could be used as a criterion for whether our threshold for statistical significance is small enough.

“You seemed to be disagreeing with that”

I’m not sure what I said that gave that impression. I didn’t mention anything about the normal / iid assumptions initially not because I thought we weren’t making these assumptions but because I didn’t think these details were essential to my point.

"Let the statistic t_0 be the inf of all these t-statistics. Let T be the event that t_0 is at least as big as the observed value." Oops, I meant the inf of their absolute values, and T is the event that t_0 is at least as big in absolute value as the observed value.

That is just a trivial semantics issue, and yes I am familiar with the calculation.

Where do you think the flaw is specifically? Say we are doing a 1-sample test.

0. (setup) Suppose we have a real number mu and a positive epsilon. Define the interval I as [mu – epsilon, mu + epsilon]. For each “candidate mean” within this interval, we have a corresponding t-statistic. Let the statistic t_0 be the inf of all these t-statistics. Let T be the event that t_0 is at least as big as the observed value.

1. You can use Student’s t-distribution to compute an upper bound for the probability of T under the assumptions that the observations are iid normal and the mean lies in I. I will call this probability p(T | null, normal, iid), where “null” is the event that the mean exists and is in I. It makes no difference that it is more typical to lump these assumptions together as “null” because in math you can define things however you want as long as you are consistent.

2. We have that p(null | T, normal, iid) = p(T | null, normal, iid) * p(null | normal, iid) / p(T | normal, iid).

3. Therefore, if we have an upper bound for x = p(null | normal, iid) / p(T | normal, iid) then we can get an upper bound for p(null | T, normal, iid). That is my main claim.

Which of the above statements do you object to?

I've already addressed that. See "in anticipation of the rebuttal" post.

By the way, it's not normally done using one either. With a 1-sample t test for example, the null hypothesis is that the underlying distribution has a certain prescribed mean, but the variance can be anything.

Ok for simplicity let’s assume this is a 1-sample test. So there is a certain “default” mean, say mu, and we are concerned with whether the mean of some random variable on whatever population we are sampling from is within, say, epsilon of this default value. For every number in [mu – epsilon, mu + epsilon] we can get a p-value giving the probability that we would have observed the data if this was the true mean. In order to get a probability that we would have observed the data given that the mean was somewhere in this interval, we need some prior distribution on the means in this interval, which we don’t have. However, we can just take the sup of all the p-values for each mean in this range to get an upper bound. (I think this is similar to how confidence intervals work also but take that with a grain of salt)

And in anticipation of the rebuttal that the probability is still 0 because it's never exactly normal, what I really meant originally but didn't write out explicitly for brevity and because I assumed it would be implicit: when I talk about giving an upper bound on p(null | data) from p(data | null), what I really mean is giving an upper bound on p(null | data, normal) from p(data | null, normal) where normal is the assumption that the distribution of whatever parameter we are looking at is normally distributed and null is the event that the difference in means between the two groups we are looking at it is less than some predetermined positive threshold. Or, for a 1-sample test, that the mean of a single group is within that threshold of some default value.

Ok, I will assume that by "model" here you mean a probability distribution on the parameters relevant to your experiment. In that case I agree with what you just said: knowing exactly the correct model is impossible in a similar way that knowing someone's height to an infinite degree of precision is impossible. But I never said anything to the contrary. The H_0 I gave corresponds to an infinite set of models and not a single one (note that I said the difference is less than a certain threshold, not that the difference is 0 (although it still would be an infinite set in that case, but the probability would be 0)).

Yes, the t-test does assume normality and you can never be sure of perfect normality if that's what you are getting at (although I believe that simulation tests of the robustness of the t-test against deviations from normality generally show that this isn't too much of a practical concern). I wasn't trying to address every potential weakness with the t-test (or p-values in general); I was addressing the one stated in the article.

“how safe is it to conclude that H_0 is not true”

What I meant here by H_0 was the hypothesis that the difference between groups is less than some particular threshold. I think if you made the threshold large enough then it would not be safe to conclude that H_0 is not true.

"You are ignoring a lot of the model being tested here (eg, normality, independence of the measurements, etc) and only considering one parameter. "

I said enough so that if you were arguing in good faith you could fill in the gaps yourself.

I’m assuming the question we are trying to answer is not “which H_n is most probable”, but rather “how safe is it to conclude that H_0 is not true”. For example, say we are concerned with whether the difference between two groups is less than a certain amount or greater than a certain amount.

'The fundamental problem is that p values don't mean what we "need" them to mean, that is p(null | significant effect).'

From Bayes' theorem, this more useful probability is given by p * x, where x = p(null) / p(significant effect). Maybe we could just lower the accepted threshold for statistical significance by several orders of magnitude so that, for statistically significant p, p * x is still small even for careful (i.e. big) estimates of x (e.g. maybe a Fermi approximation of the total number of experiments ever performed in the field in question). This doesn't necessarily imply impractically big sample sizes, although obviously this depends on the specifics (I believe the p value for a given value of the t-statistic decays exponentially with sample size).