Brain mass vs body mass
An application of the ensemble approach to information equilibrium
In my paper, I showed that if you have a complex ensemble of processes {Aᵢ} in information equilibrium via multiple channels with some controlling variable B with information transfer indices {kᵢ} measuring the amount of information per channel i
where the angle brackets represent an ensemble average using a partition function. One application to macroeconomics (e.g. here or here) takes B to be some kind of ubiquitous aggregate input (e.g. labor or “aggregate supply”) and the Aᵢ to be all the products that make up an economy (i.e. GDP) so that the macroeconomy grows with that ubiquitous input. The {kᵢ} represent growth states where there is some underlying growth rate, say γ, and the different growth rates are kᵢ γ.
What’s really neat about this is that the equation is formally similar to the information equilibrium condition for a single, simple market — you just take off those angle brackets. That’s how the AD/AS model for an entire economy can basically work the same as a supply and demand diagram for a single good. At least that’s the information equilibrium explanation.
While I think it’s the coolest equation I’ve ever personally created, I’ve struggled to find a really good application for its nuances. Its use in the information equilibrium model has mostly been to justify that the simple model should often work just as well as a complex ensemble of markets because ⟨k⟩ changes slowly — therefore all the usual results can be interpreted as happening in a much more complex system just by replacing k with ⟨k⟩.
How good of an approximation that is depends largely on how ⟨k⟩ changes with B. As B gets large, the lowest values of {kᵢ} become the most important and the B-dependence of ⟨k⟩ tends to B^ε where ε could be arbitrarily small — the lowest possible k-value. Since this all has an analogy with thermodynamics (k is energy, B is ~ 1/T), we can say that a cold system is dominated by its lowest energy states with the higher energy states becoming less and less occupied. Empirically, the justification that ⟨k⟩ is approximately constant with respect to B works well for e.g. labor markets. In fact, I've never seen a deviation in an information equilibrium application that could be reasonably attributed to a changing ⟨k⟩. And the applications have ranged from transistors to COVID!
In general, information equilibrium is supposed to address complex systems — the kinds of systems where MaxEnt is a good first guess because it’s basically MaxIgnorance. So biological systems should be a good candidate ...
I ran across a new paper that points out that the usual (log-)linear relationship between body mass and brain size appears to have accumulating issues with a variety of different explanations. The authors suggest a curvilinear (i.e. log quadratic) relationship and say that this relationship
... simultaneously accounts for several phenomena for which diverse biological explanations have been proposed, notably variability in scaling coefficients across clades, low encephalization in larger species and the so-called taxon-level problem.
Much follows from the quadratic relationship along with Cope’s rule: body mass increases with time. Therefore e.g. the taxon-level problem is just that more distantly related species have undergone evolution longer so are more likely to have had body mass go through more orders of magnitude so that the slope should fall (if it is curvilinear).
Now in the ensemble approach to information equilibrium, the (approximate) slope on the log-log plot is given by
where Z is the partition function. Per the argument above, ⟨k⟩ tends to fall with B and that creates a curvilinear relationship. Now I don’t know what Z actually is — the distribution of possible evolutionary states {kᵢ} is not theoretically known. Therefore I did a quick Monte Carlo using 100 different contributing factors (“markets” in econ, but “processes” more generally) with {kᵢ} determined via a normal distribution centered around 0.75 — per the paper, the typical value based on “scaling of metabolic costs of the brain”. A random throw looks like this:
I then computed the average relationship between brain mass B and body mass ⟨A⟩ based on these k-states. I ran this calculation 20 times to produce several possible functions ⟨A⟩ = f(B) (i.e. log ⟨A⟩ ~ ⟨k⟩ log B) and plotted those:
The (brain, body) mass data is from the supplementary information in the paper (mammals only). It’s a pretty decent estimate. I’d imagine some collection of kᵢ is heritable in different species and the observation that primates have seen a rapid expansion in brain size (i.e. higher than typical k) could be something like finding a new evolutionary niche. To create an analogy with markets — primates found a new business model that allowed for a period of higher than average S&P 500 rates of growth.
This isn’t the first time there’s been an analogy between economics and evolution on in my blog posts (see e.g. here, which, incidentally contains the progenitor of what would become the diagram on the cover of my first book1). Both are complex systems with agents exploring a massive state space. However, this is the first time I’ve had a good candidate for an application that uses the details of the ensemble approach — the possibility of a more complex relationship between ⟨k⟩ and B than ⟨k⟩ ~ constant.
I couldn’t have a post without at least one footnote. I discovered Amazon put the Goodreads score on the page and it gets a 3.9, which per here, is “3.7 – 3.9 stars: Genuinely a good book.” (Though they are looking at literature, so not sure how the rating translates to non-fiction.) Additional side note: I discovered that since no one buys books, my book is likely in the top quintile of sales.