> The results are already not significant before the correction, so there is no point in making a correction step. Read this part again: they are saying their results are insignificant and would be further away from signficance if they had bothered to apply any correction.

Give me a break. Let me go back and quote from the paper:

"Whether the performance of a method is significantly better than random guessing is not so clear-cut. For example, the p-value for ANM-pHSIC under the null hypothesis of random guessing is 0.068 on the CEP benchmark, only slightly larger than the popular α = 5% threshold. However, because we tested many different methods, one should correct for multiple testing. Using a conservative Bonferroni correction, the results on a single data set would be far from being significant. In other words, from this study we cannot conclude that the ANM-pHSIC method significantly outperforms random guessing on the CauseEffectPairs benchmark. However, the Bonferroni correction would be overly conservative for our purposes, as many of the methods that we compare are small variations of each other, and their results are clearly dependent. In addition, good performance across data sets increases the significance of the results. Although a proper quantitative evaluation of the significance of our results is a nontrivial exercise, we believe that when combining the results on the CEP benchmark with those on the simulated data, the hypothesis that the good performance of the methods ANM-pHSIC, ANM-HSIC, ANM-PSD and ANM-MML is only due to chance is implausible."

A p-value of 0.06 is not horrible, and they do not say it is the smallest p-value obtained. And since they test multiple methods, in different conditions, Bonferonni would divide 0.05 by combinatorially many, which is absurd because clearly these are not independent p-values (which is the assumption behind Bonferroni) and it is not irrelevant at all (and you saying it is suggests you don't understand how Bonferroni works).

In fact, you're not even right about "The results are already not significant before the correction" - ANM-pHSIC was almost statistically-significant at the normal threshold, and it didn't perform the best on the full unmodified dataset, that was ANM-HSIC (see figure 11, pg28, left table; or pg27, figure 9).

A more reasonable interpretation would be, rather than shoehorning in a Bonferroni where it has no business, to ask whether there's an overall difference of the various algorithms from baseline chance of 50-50, with something like an ANOVA.

How would that work? Well, we can estimate the fraction right by eye, use the known 88 pairs, and estimate from there. (There's probably a table of exact results somewhere but I didn't see it.) What is the p-value of all the algorithms, lumped together? Does it suggest outperformance? Why yes it does:

R> cep <- read.csv(stdin(),header=TRUE)
"Type","Fraction"
pHSIC,0.62
HSIC,0.65
ent1sp,0.45
entsp1,0.62
entKDP,0.70
endPSD,0.67
end3nn,0.20
entEdE,0.55
FN,0.49
Gauss,0.51
MML,0.57
entME1,0.60
entME2,0.55

R> cep$Correct <- round(cep$Fraction * 88)
R> cep$N <- 88

R> binom.test(sum(cep$Correct), sum(cep$N), p=0.5)

Exact binomial test

data: sum(cep$Correct) and sum(cep$N)
number of successes = 633, number of trials = 1144, p-value = 0.0003429
alternative hypothesis: true probability of success is not equal to 0.5
95 percent confidence interval:
0.523970021 0.582398490
sample estimates:
probability of success
0.553321678

So even including all the crappy algorithms, they still outperform. An ANOVA would show the same thing. As the authors say: "the hypothesis that the good performance of the methods ANM-pHSIC, ANM-HSIC, ANM-PSD and ANM-MML is only due to chance is implausible."