I wasn't aware of your paper, thank you for sharing it. There is nothing I could disagree with, and perhaps what little benefit I can offer is to show how I might alternatively implement the Explanatory Filter for some of the examples you provided wherein you used the ASC method.
For the number heads and tails, especially for large number of fair coins, I'd determine the violation of expectation from the mean. This is done via the binomial distribution, and the resulting probabilities when converted into bits will resemble the shape of your graph. For a system of 500 fair coins here are the associated probabilities for deviation from the mean
P(H=500) = 1/2^500 = 500 bits = 22 to 26 sigma (depending on which sigma approximation is used)
P(H>=499) = 1.53 x 10^-148 = 491 bits
P(H>=400) = 8.43 x 10^-44 = 143 bits = 13.6 sigma
P(H>=300) = 4.47 x 10^-6 = 17 bits = 4.54 sigma
P(H>=250) = 51.8 % ~= 1 bit = 0 sigma
P(T>=250) = 51.8 % ~= 1 bit = 0 sigma
P(T>=300) = 4.47 x 10^-6 = 17 bits = 4.54 sigma
P(T>=400) = 8.43 x 10^-44 = 143 bits = 13.6 sigma
P(T>=499) = 1.53 x 10^-148 = 491 bits
P(T=500) = 1/2^500 = 500 bits = 22 to 26 sigma (depending on which sigma approximation is used)
I got the binomial calculations from:http://stattrek.com/online-calculator/binomial.aspx
I think this has a comparable shape to one side of the graph in your paper. I would suspect we ought to have comparable bit values???
The binomial distribution in the way I used it above has use in finding some biological designs (with some caveats):
2. nucleotide bias
3. codon bias
One issue however for both the ASC method and the alternate method I suggest is the problem of repeats in DNA. Are some designs, or just copying accidents? The telomeric sequence is repeated a lot "TTAGGG", I think it is designed. It will signal "non-random" in ASC and in my alternate method. But what do we do with accidental repeats (they do happen)? I have no good answer.
Royal flush raises the question of why we should find it special above all other hands. My answer is that it's not just our subjective valuing of it, but it violates 3 dimensions expectation in the most extreme possible ways:
1. we statistically expect a mix of suits not the same suit
2. we statistically expect more number cards (2,3,4,5,6,7,8,9,10) than face cards (J,Q,K,A)
3. we statistically expect cards not to be adjacent to each other in collating sequence
Hence, just the way we affix symbols to a deck of cards makes the Royal special, and a natural specification. I did not have to do a Phi_ST analysis to point out the Royal is special given the symbols in use. The Royal Flush specification emerges naturally from the system of card symbols (suit, rank, collating sequence) much like "all fair coins heads" emerges naturally as a specification from the system of coin symbols (heads, tails).
There is the interesting topic of external specification. One hand called the "dead man's hand" (the hand Wild Bill Hickock held when he was shot): A-spade A-club 8-spade 8-club stands out because of subjective significance. External specifications of this sort probably don't have significance in biology. But there are other external specifications which we might be able to use.
In biology what I view as external specifications are duplications. We have duplicated cells, DNA, etc. We have convergence of form not attributable to common descent. Some biological systems have thematic resemblance to human designs like motors, gears. We can mathematically correlate the relationship.
Finally, there are external specifications that are algorithmic: binary digits of pi, binary digits of e, the chapernowne sequence, etc. There are only a finite number of such conceptual sequences a human mind can hold, or for that matter all the libraries of the world can hold. With this knowledge, Bill's Phi_ST calculations gives an approximation for the likelihood an idea in our mind can be realized physically by chance. I gave an alternate take on the influence of human psychology in defining specification (of which algorithmically compressed specifications are a subset):http://www.uncommondescent.com/psycholo ... formation/
There are algorithmic compressible patterns in biology (like repeats). But more sophisticated algorithmic patterns? One I can think of is the Fibonacci sequence in certain flowers. How good of a specification it is remains an open issue.
I don't have a good working procedure for explaining how functioning systems violate expectation.