tprice7

11 ms·

61.
by tprice7 8y ago
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.
62.
by tprice7 8y ago
"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
63.
by tprice7 8y ago
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 conv
64.
by tprice7 8y ago
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 t
65.
by tprice7 8y ago
Here's a fun puzzle: concretely, what is a group of groups?
66.
by tprice7 8y ago
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
67.
by tprice7 8y ago
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&#
68.
by tprice7 8y ago
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 so
69.
by tprice7 8y ago
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://conte
70.
by tprice7 8y ago
Hence pendulum clocks.
71.
by tprice7 8y ago
In general it's the eigenvectors of the positive-semidefinite (hence symmetric) part of the left polar decomposition.
72.
by tprice7 8y ago
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.
73.
by tprice7 8y ago
Also, every probability mentioned should also include in the list of conditions that the number of samples observed matches our experiment.
74.
by tprice7 8y ago
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
75.
by tprice7 8y ago
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 t
76.
by tprice7 8y ago
“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.
77.
by tprice7 8y ago
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 F
78.
by tprice7 8y ago
“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
79.
by tprice7 8y ago
“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 a
80.
by tprice7 8y ago
"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
81.
by tprice7 8y ago
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 th
82.
by tprice7 8y ago
I've already addressed that. See "in anticipation of the rebuttal" post.
83.
by tprice7 8y ago
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.
84.
by tprice7 8y ago
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
85.
by tprice7 8y ago
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 t
86.
by tprice7 8y ago
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 w
87.
by tprice7 8y ago
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 generall
88.
by tprice7 8y ago
“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 saf
89.
by tprice7 8y ago
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 t
90.
by tprice7 8y ago
'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) &#

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