# drghirlanda

Computational behavior theory and cultural evolution

## An identity on falling powers

Here is a bit of combinatorics I encountered when preparing a paper on the co-evolution of behavioral repertoire, brain size, and lifespan (I will talk about the paper another time…). Let’s begin with two definitions:

Definition 1: The falling power $n^{\underline{m}}$ is defined as the product $n(n-1)\cdots(n-m+1)$, or:

(1)    ${\displaystyle n^{\underline{m}}=\prod_{i=1}^m (n-i+1)}$

If you are familiar with binomial coefficients, they are related to falling powers by $n^{\underline m} = m! {n \choose m}$.

Definition 2: The Stirling numbers of the second kind are a double series of numbers that tell us how many ways there are to partition $n$ objects into $k$ non-empty subsets. This is written sometimes $S(n,k)$ and sometimes $\left\{n\atop k\right\}$. I will use the fancier notation. For example, there are only 3 ways to partition 3 elements in to 2 non-empty subsets: (12)(3), (13)(2), (1)(23), where $(xy)$ means that $x$ and $y$ have been put together in the same subset. These numbers, beyond the simplest case, are nowhere near intuitive (at least to me). For example, There are 7 ways to partition 4 objects into 2 subsets, hence $\left\{4\atop 2\right\}=7$ (you can figure these ways out for yourself), and 1701 ways to partition 8 objects in 4 subsets, hence $\left\{8\atop 4\right\}=1701$ (I suggest you do not try this on your own).

Now, it happens that falling powers and Stirling numbers of the second kind are related by the following identity:

(2)    ${\displaystyle n^m = \sum_{k=0}^m\left\{m\atop k\right\}n^{\underline k}}$

I have only seen this equation proved by induction, but working on the above-mentioned paper I stumbled upon a direct proof that goes as follows. Note first that $n^m$ is the number of ways to arrange $n$ objects in sequences of length $m$, with repetitions possible (by sequence I mean an ordered selection so that (1,2,2) and (2,1,2) are two different sequences). So we have

(4)    $\mbox{\# different sequences of }m\mbox{ objects chosen with repetition among }n = n^m$

Equation (2) then comes from the fact that its r.h.s. is a different (and more laborious) way to count the same sequences. In other words, we can first count the sequences that we can form using only $k$ out of the $n$ objects, and then sum over $k$:

(5)    ${\displaystyle n^{m} = \sum_{k=0}^n \mbox{\# different sequences of }m\mbox{ objects using any }k\mbox{ objects out of }n}$

Now we have to calculate the expression in the sum. Consider thus constructing a sequence of length $m$ out of $k$ distinct object, which in turn have been selected among $n$. There are $n^{\underline k}$ ways of selecting which of the $k$ objects are going to be part of the sequence, given that the first object out of $n$, the second out of $n-1$, and so on, until the $k$-th object can be selected out of $n-k+1$. Once we have the $k$ objects, in how many ways we can allocate them among the $m$ places of the sequence? This is exactly the number of ways in which a set of size $m$ can be partitioned in $k$ non-empty subsets, or, if you want, the number of ways in which $m$ balls can be placed in $k$ bins without leaving any one bin empty. Thus

(6)    ${\displaystyle \mbox{\# different sequences of }m\mbox{ objects using any }k\mbox{ objects out of }n} = \left\{m\atop k\right\} n^{\underline k}$

which, together with (5), gives (2).

## Empirical support for openness-persuasiveness dynamics

A recent study by Aral & Walker provides support that the openness-persuasiveness dynamics we suggested a few years ago actually goes on in cultural evolution. In short, we had put forward mathematical and simulation models to support the notion that learning from others produces individuals that, over time, become more conservative (less likely to learn from others) and more persuasive (more likely to convince others of one’s own ideas). These predictions have been confirmed by Aral & Walker, who showed that older Facebook users are more difficult to convince do adopt a Facebook app than younger users, and yet are better at convincing others to adopt the app. Up to now, we only had indirect evidence about openness (older people score low on openness in personality tests), and no evidence on persuasion.

We have submitted a comment to the journal relating Aral & Walker’s intriguing findings to our theory. You can find a slightly extended version here, essentially with more references to relevant work.

Watch them here!

## Videos and slides for Understanding Human Cognitive Uniqueness

I am starting to post videos and slides from Understanding Human Cognitive Uniqueness on the conference page. They will be uploaded as they get ready.

The videos have been recorded and edited by Malene Schjoenning.

## Understanding Human Uniqueness: Full Program

The program for Understanding Human Uniqueness has been finalized and is available here. See the conference page or the invitation flyer for more details.

## Cultural variation in face perception!

My cousin alerted me of this paper about the cross-cultural perception of facial expressions, by Rachel Jack and colleagues. The study uses an innovative method to uncover how we perceive facial expressions, and more specifically whether Westerners’ (Europe, North America) and ‘East Asians’ (China, Japan, Korea, Mongolia, Thailand, Taiwan) use the same criteria.

The authors find several differences, such as that Westerners pay more attention to the lower part of the face, while East Asians to the eyes. Also, East Asians attach more significance to the immediate signs of emotion, while Westerners pay more attention to later parts of facial displays. While more work is neededd to understand the reasons for these differences (the authors have some ideas in their Discussion), the data show clear variation in how emotions are perceived (and, presumably, produced) across cultures. The implication is that our emotion recognition mechanism cannot be wholly innate, but it must be open to learning the specifics of each culture.

This conclusion agrees with the fact that other aspects of facial perception, such as the perception of attractiveness, may vary across cultures. Many years ago (on my timescale, it was 2002), some colleagues and I tried to figure out how much nature and how much nurture we should expect in the perception of attractiveness. We concluded, based on what we know about perceptual mechanisms and about the evolution of biological signals, that people’s criteria of attractiveness should be mostly learned, thus leaving space for cross-cultural variation also in this domain. The paper is here.

Method. In case your are still reading, the method employed to reconstruct the criteria of Westerners and East Asians is as follows (simplifying a bit). A computer software generates short (1.25 s) animation of faces by combining randomly selected “facial action units” (AUs), which represent how different muscles move parts of the face. These movies (lots of them) were rated by observers for intensity and quality (sadness, happiness, anger, etc.). The idea is that, although the movies show random combinations of AUs, statistical analysis of people’s responses allow the reconstruction of each AU’s role. For example, the analysis might pick up that expressions including AU 6 (raising the cheeks) are typically rated as happier than expressions without AU6.

## Talking about yourself feels better if others are listening: Why?

Diana Tamir and Jason Mitchell of the Social Cognitive and Affective Neuroscience Lab at Harvard have just published a paper showing that people find it rewarding to talk about themselves, especially if others are listening (summarized here). Although, put it that way, you may or may not find the result  astonishing, it touches upon an important issue in our understanding of ourselves: the difference between proximate and ultimate causes. Konrad Lorenz explained this difference in the fewest words when he said: the ultimate cause of a car is to travel, the proximate cause is the engine. That is, the ultimate cause is the function, and the proximate is the mechanism that achieves it.

Tamir and Mitchell show that brain areas that respond to reward (food, sex, money, etc.) are also activated when answering questions about oneself, more than when answering questions about Barack Obama (chosen perhaps for his interesting opinions, perhaps because he is familiar to everyone) or about dry facts. And knowing that a friend or relative would read your answer activated the reward areas even more. This, they argue, is the proximate cause of our obsession with talking about ourselves: it activates the reward areas of our brain.

The authors have been careful in validating their results conducting not one, but four distinct experiments. I will just mention that the participants were sure to know the answer to questions about themselves, but not to the other questions. So the reward they felt could reflect the anticipation of knowing the answer rather than the self-referential aspect of the question (we know the same brain areas respond to anticipated reward). After all, we are rewarded all our lives for knowing the answer to questions. But this is not my main point.

My main point is about the ultimate reason why we feel rewarding to talk to others (especially if they listen). In genetic evolution the only ultimate cause is natural selection. Things happen because they make organisms survive and reproduce. It is not hard to imagine potential benefits of sharing your thoughts with others: exchanging knowledge, strengthening social bonds, and so on. But human behavior has another ultimate cause: cultural evolution. What drives cultural evolution is imperfectly understood, but one way to think about it is to ask what are the magical ingredients’ that make ideas popular. One such ingredient is, rather obviously, that the idea should be able to spread. Other things being equal, ideas that spread faster, convincing person after person to adopt them, will become more popular than slow-spreading ideas. And what is the best way to spread ideas? To talk about them! If you like talking to others about your ideas, these will have a good chance of spreading, and among the ideas you spread there will be those that make you like talking to others. Simplifying a bit, if you think talking to others is cool,’ then you will say, among other things, talking to others is cool,’ and others may be convinced of it and start talking to others, furthering the spread of the `talking to others is cool’ idea. If this sounds like a tongue twister, it is because cultural evolution is full of self-referential loops in the dynamics of ideas (one example, and another).

Thus we may like to talk about ourselves because of the dynamics of ideas, rather than because this tendency has been built into us by genetic evolution. Can we distinguish between the two hypotheses? Not yet, I believe, and the main reason is that neither evolutionary psychology nor cultural evolutionary theory (I don’t even have a Wikipedia link for that, but you can look here) have formulated precise predictions about how and when ideas should or should not be shared. But adapting Tamir and Mitchell’s experimental setup to test such hypotheses should be easy. So come on, theoreticians, give us a hypothesis to test!

## If baboons can read, can pigeons, too?

“Can pigeons read?” is the question asked at the beginning of this old video, aimed at illustrating techniques to teach animals complex discriminations by rewarding them for correct choices but not for incorrect ones.

These techniques, developed around 1930, have been used in a study teaching baboons to recognize English words from non-words. Soberly entitled “Orthographic processing in baboons,” the study has been often headlined “Baboons can read,” even by the very journal who published it. My colleague Johan Lind was delighted to hear the news: “If they can read, then I can write to them and ask about animal intelligence.” Unfortunately, the only thing the baboons would be able to tell Johan is which combinations of letters are more likely to appear in English words, which is what they learned by receiving food anytime they correctly identified four-letter sequences as an English word or a non-word.

The study actually demonstrates that you do not need to know language to tell words from non-words. All languages have a statistical signature, whereby some combinations of sounds (and, therefore, letters) are common, and others are rare. Baboons are smart enough, and see well enough, to learn this. I would not be surprised if pigeons could do it too, given that they can, for example, discriminate paintings by different artists, presumably learning something about the artists’ “visual grammar.” Pigeons can also associate different written words with different actions, as the video above shows. All this suggests that the evolutionary origin of our ability to read is even more ancient than “reading” baboons suggest, pigeons being separated from humans by some 150 million years of independent evolution. Analyzing the structure of visual stimuli is a natural task for many animals, and I do not think the key to understanding human uniqueness lies here.

## Understanding Human Uniqueness Flyer

We have prepared a flyer to advertise the Conference on Human Cognitive Uniqueness that will take place at Brooklyn College on May 29-30. Feel free to use it to advertise the Conference yourself!

## The Logic of Fashion Cycles

As announced a few weeks ago, our paper “The Logic of Fashion Cycles” has been published, and is freely available on the PLoS ONE website. You can find a good summary at The National Post.