An entropy paradox

Entropy/diversity is personally one of my favourite brain-wasting topics (I have my reasons for that, good ones :) ). Here's a short line of thoughts that illustrates the tricky nature of these notions. At the end, I'll probably not connect this back to naive everyday thinking which everyone is taking for granted, as it would be either embarrassing for many or worse even (in case I'm wrong), very embarrassing, but only for me alone :) But beware I think I could (make a fool out of myself)...

So the opening thought is that when we are young, we are more similar to both of our parents, and as we turn older, we will exhibit stuff related to the matching gender parent, i.e. specialize.

Now let's describe this by something that behaves like the Herfindahl index (any entropy measure does the job in one way or another) over a pair of similarity metrics. We'll find that from a pair: (mom_similarity, dad_similarity) closer to (0.5, 0.5) the individual's stats move closer to either (1, 0.0) or (0, 1.0), and that this means this index will increase, suggesting a less diverse individual.

Now let's take a look at this on the macro level, multiply up the aforementioned individual so that it becomes a population (of roughly indetically aged, random gender people which is growing up)!

From a series similar to [(0.5, 0.5), ..., (0.5, 0.5)] we observe a transition towards (assuming equal probabilities for the genders) a series that is more like [(1.0, 0.0), (0.0, 1.0), ...] etc.

What we then find is that the diversity on the macro level did exactly the opposite - a population of randomly grown ups is more diverse than that of babies. Actually that was quite trivial without the numbers already, but the joy and the words with the weird spelling ... :)

So yes, almost paradoxically, micro and macro level entropy may work against each other - the level of abstraction does matter a lot!



P.S.: Don't calm down. I look forward to distribute similarly useless thoughts in the future.