In the post alluded to a nice self-contained tricky math inequality problem that I am hoping someone will be nerd-sniped by. (I am rusty on my linear algebra inequalities and I don’t care enough to spend more time on it.) Here’s what I wrote:
2025-01-18: I mentioned in a couple places that it might be possible to have non-additive genetic effects that are barely noticeable in rDZ-vs-12rMZ comparisons, but still sufficient to cause substantial Missing Heritability. The Zuk et al. 2012 paper and its supplementary information have some calculations relevant to this, I think? I only skimmed it. I’m not really sure about this one. If we assume that there’s no assortative mating, no shared environment effects, etc., then is there some formula (or maybe inequality) relating rDZ-vs-½rMZ to a numerical quantity of PGS Missing Heritability? I haven’t seen any such formula. This seems like a fun math problem—someone should figure it out or look it up, and tell me the answer!
More details: Basically, when rDZ is less than 12rMZ, then there has to be nonlinearity in the map from genomes to outcomes (leaving aside other possible causes). And if there’s nonlinearity, then the polygenic scores can’t be perfectly predictive. But I’m trying to relate those quantitatively.
Like, intuitively, if rMZ=1.000 and rDZ=0.499, then OK yes there’s nonlinearity, but probably not very much, so probably the polygenic score will work almost perfectly (again assuming infinite sample size etc).
…Conversely, if rMZ=1.000 and rDZ=0.001, then intuitively we would expect “extreme nonlinearity” and the polygenic scores should have very bad predictive power.
But are those always true, or are there pathological cases where they aren’t? That’s the math problem.
I tried this with reasoning LLMs a few months ago with the following prompt (not sure if I got it totally right!):
I have a linear algebra puzzle.
There’s a high-dimensional vector space G of genotypes.
There’s a probability distribution P within that space G, for the population.
There’s a function F : G → Real numbers, mapping genotypes to a phenotype.
There’s an “r” where we randomly and independently sample two points from P, call them X and Y, and find the (Pearson) correlation between F(X) and F((X+Y)/2).
If F is linear, I believe that r^2=0.5. But F is not necessarily linear.
Separately, we try to find a linear function G which approximates F as well as possible—i.e., the G that minimizes the average (F(X) - G(X))^2 for X sampled from P.
Let s^2 be the percent of variance in F explained by G, sampled over the P.
I’m looking for inequalities relating s^2 to r^2, ideally in both directions (one where r is related to an upper bound on s, the other a lower bound).
Commentary on that:
The X vs (X+Y)/2 is not exactly what happens with siblings. It’s similar to comparing a parent to their child—i.e., X is the genotype of the mother, Y the father, (X+Y)/2 the kid. But parent-child should be mathematically similar to sibling-sibling, since both are 50% relatedness. …Except it’s not really parent-child either, because if X has a SNP but Y doesn’t, then the child has the SNP with 50% probability, rather than having “half of that SNP”. But I figured it might amount to the same thing? But I do think you need to be randomizing over individual SNPs to formulate the problem for the actual sibling case we care about. (So really, the individuals are all binary / indicator vectors (entries are 1s and 0s), as opposed to arbitrary elements of the vector space. I’m just guessing that’s not too important to the problem.)
I’m assuming rMZ=1, and then “r” is rDZ, “G” is the polygenic score, and “s” quantifies the predictive power of the polygenic score.
I did this very quickly, there might be other mistakes in this problem formulation, and I wasn’t motivated enough to keep exploring it.
(Btw, I sent that prompt to a few AIs around January 2025, and they gave answers but I don’t think the answers were right.)
In the post alluded to a nice self-contained tricky math inequality problem that I am hoping someone will be nerd-sniped by. (I am rusty on my linear algebra inequalities and I don’t care enough to spend more time on it.) Here’s what I wrote:
More details: Basically, when rDZ is less than 12rMZ, then there has to be nonlinearity in the map from genomes to outcomes (leaving aside other possible causes). And if there’s nonlinearity, then the polygenic scores can’t be perfectly predictive. But I’m trying to relate those quantitatively.
Like, intuitively, if rMZ=1.000 and rDZ=0.499, then OK yes there’s nonlinearity, but probably not very much, so probably the polygenic score will work almost perfectly (again assuming infinite sample size etc).
…Conversely, if rMZ=1.000 and rDZ=0.001, then intuitively we would expect “extreme nonlinearity” and the polygenic scores should have very bad predictive power.
But are those always true, or are there pathological cases where they aren’t? That’s the math problem.
I tried this with reasoning LLMs a few months ago with the following prompt (not sure if I got it totally right!):
Commentary on that:
The X vs (X+Y)/2 is not exactly what happens with siblings. It’s similar to comparing a parent to their child—i.e., X is the genotype of the mother, Y the father, (X+Y)/2 the kid. But parent-child should be mathematically similar to sibling-sibling, since both are 50% relatedness. …Except it’s not really parent-child either, because if X has a SNP but Y doesn’t, then the child has the SNP with 50% probability, rather than having “half of that SNP”. But I figured it might amount to the same thing? But I do think you need to be randomizing over individual SNPs to formulate the problem for the actual sibling case we care about. (So really, the individuals are all binary / indicator vectors (entries are 1s and 0s), as opposed to arbitrary elements of the vector space. I’m just guessing that’s not too important to the problem.)
I’m assuming rMZ=1, and then “r” is rDZ, “G” is the polygenic score, and “s” quantifies the predictive power of the polygenic score.
I did this very quickly, there might be other mistakes in this problem formulation, and I wasn’t motivated enough to keep exploring it.
(Btw, I sent that prompt to a few AIs around January 2025, and they gave answers but I don’t think the answers were right.)