(no subject)

Zeilberger:

Ewens and Wilf are very right when they claim that P(r, n,m) and Q(r, n,m) are very far apart around the “tail” of the distribution, but who cares about the tail? Definitely not a scientist and even not an applied mathematician. It turns out, empirically (and we did extensive numerical testing, see Procedure HowGoodPA1(R0,N0,Incr,M0,m,eps) in BallsInBoxes), that whenever P(r, n,m) is not extremely small, it is very well approximated by Q(r, n,m), and using the latter (it is so much faster!) gives very good approximations, and enables one to construct the “center” of the probability distribution (i.e. ignoring the tails) very accurately.

Statistical geneticists care about the tails. Imagine a gene association study, where you have some disease, and you want to know, for each of the 20,000 or so genes in the human genome, whether rare variants in those genes increase risk of the disease. You use the Bonferroni correction: instead of using 0.05 as a p-value cutoff, you use 0.05/20,000. So your question is about whether a probability is above or below a number which is around 1 in a million.

The approximation Zeilberger is considering does seem to be good even to probabilities around 10^-7 in the example he looks at, so it seems it can handle another order of magnitude more tests, but this could certainly occur in a study with a larger scope. For example, if instead of one disease, we consider 10, or 100.

Ewens knows this, because he works in statistical genetics. Zeilberger has no reason to think he knows more than Ewens about what matters in scientific practice. Zeilberger knows very little about what matters in practice, and doesn’t need to. He just needs to keep making his solutions more general, so that whatever does come up in practice will turn out to be a special case of something.

Which is what he does in fact do, so I have no problem with how Zelberger manages his research, it’s just ridiculous for him to pepper his papers with baseless value judgments.

(no subject)

Sobel and Frankowski:

The special case in which all p_i are equal and all r_i (i=1, 2, …, b) are equal is the most important application and we have…

The part of the paper by “applied” mathematicians where they tell me that my application isn’t important, as a rhetorical device to transition to a special case that they happen to have results for.

(no subject)

Rick Durrett:

The results for E[η_{n,m}] are useful for population genetics, but are not really relevant to cancer modeling. To investigate genetic diversity in the exponentially growing population of humans, you would sequence the DNA of a sample of individuals from the population. However, in the study of cancer each patient has their own exponentially growing cell population, so it is more interesting to have the information provided by Theorem 1 about the fraction of cells in the population with a given mutation.

The results he seems to think are so useless still seem to be the only results in the paper that has been used in data analysis (in this paper).

What Rick Durrett doesn’t seem to have realized is that, two years prior to the publication of this paper, people started doing DNA sequencing of individual cells from patient’s tumors. So, the study of a single patient became a population genetics problem, and results from a population genetics perspective were exactly what was needed.

Rick Durrett can be forgiven for not noticing that. He’s a mathematician, and can’t be expected to keep up with the latest advances in genomics. But as a consequence he should worry less about whether his results are applicable to the few problems he’s familiar with from his collaborators.

I prefer the spirit he demonstrated in this paper with Foo and Leder, where they provide extensive information about the growth of cancer in a space of 3 or more dimensions.