After 10 breathing experiments I decided to run some numbers on the results. The raw data is here. I don’t know how a statistician would handle inferences from this data, but I can calculate probabilities for a binomial distribution. The fundamental question was whether there was a detectable difference between wine that had been allowed to sit in a decanter for a few hours (typically 3–6) and wine from the same bottle that had sat under cork and argon gas (or from a different bottle of the same wine in earlier experiments). Subjects were presented with three glasses and had to pick the one that was not like the others, following this protocol. Here are the statistics for the whole group, and for me alone. The last column is the probability of doing as well or better by chance alone (assuming a 1/3 chance of a correct guess in each trial).
Group 
No. of trials 
No. correct 

Everyone 
30 
16 
0.02 
Me 
10 
7 
0.02 
I am sufficiently convinced by this that there is a difference detectable by humans. But feel free to check the data and disagree in the comments.
I also ran the conditional probabilities on correctly identifying the source of the wine given that you had correctly picked the odd one out.
Group 
No. of trials 
No. correct 
Probability 
Everyone 
16 
10 
0.22 
Me 
7 
5 
0.23 
This is less convincing, although of course the sample size is smaller. But it suggests to me the next series of experiments. Having established that decanting makes a small but detectable difference, the next question is: what is the difference? Reading over my notes, I think it is reasonable to make a hypothesis in terms of primary, secondary, and tertiary aromas. There’s a nice description these at Wine Folly. Primary aromas are the aromas coming from the grape itself (fruit, flowers, herbs). Secondary aromas come from the fermentation process (yeast, cream, lees, mushroom, and Brettanomyces aromas such as barnyard, game, and bacon) and tertiary aromas come from the aging process, particularly from aging in oak barrels (vanilla, nuts, baking spices, cedar, smoke, tobacco, leaves). We will adopt the terminology from Wine Folly: “aroma” means the primary aromas, and “bouquet” means the secondary and tertiary aromas.
The hypothesis we will work with in this series of experiments is:
Wine from a freshly opened bottle has a stronger bouquet, which tends to mask the aroma. The aroma is more prominent in wine from the decanter.
I’ll use a new experimental protocol. The question is no longer whether there is a difference, but whether a particular description of the difference is accurate and usable by humans. I’ll present subjects with two glasses blind, one from the bottle and one from the decanter, ask them to read the description of aromas above, and then ask them to apply the hypothesis to identify which is which.
I will also collect data on preference, and for that reason I will organize the experiments by varietal, starting with Bordeaux varietals and blends. Then I will move onto pinot noir. The conventional wisdom is that breathing is good for the former and not for the latter. The breathing experiments so far somewhat support this.