Long streaks, not runners’ long runs (which are also surprisingly predictable).

One of the major breakthroughs in Bioinformatics was the recognition that local similarity scores (which can be thought of as runs of positive sequence similarity) are extreme-value distributed.[0] The logic of that discovery uses almost exactly the same mathematical argument as this paper [1], indeed I recognized some of the same equations.

It is difficult to overstate the importance of this discovery for biology, as today, the vast vast majority of protein functional inferences for newly sequenced genomes are based on the statistics of long runs of sequence similarity.

[0] https://www.ncbi.nlm.nih.gov/BLAST/tutorial/Altschul-1.html [1] https://www.pnas.org/doi/epdf/10.1073/pnas.87.6.2264

This is an interesting finding. There are two takeaways from the paper.

1. The length of streaks L for an independent Bernoulli process with success probability p (with q = 1-p) over n trials can easily be calculated.

L = log_{1/p} (n*q)

2. This estimate becomes more accurate as p decreases. Because the distribution of L is an extreme value distribution which gets more concentrated as p decreases.

This means for low values of p, L becomes more predictable and accurate.

I don’t know how this result will change my life, but at least now I know that I can predict streaks if I know p.

I once saw on some website a chart with distribution of flat tire events. Often one does not encounter it in 10 years and suddenly 2 or 3 times in a year. Mathematically, chances of such distribution are quite high.