Isomorphismes

Since people liked my last opinion piece on #big data, here’s another one.

Imagine there was a technology that allowed me to record the position of every atom in a small room, thereby generating some ridiculous amount of data (Avogadro’s number is 𝒪(10²³)ŽŽ so some prefix around that order of magnitude — eg yoctobytes). And also imagine that there was a way for other scientists to decode and view all of that. (Maybe the latency and bandwidth can still be restricted even though neither capacity nor resolution nor fidelity nor coverage of the measurement are restricted — although that won’t be relevant to my thought experiment, it would seem “like today” where MapReduce is required.)

Let’s say I am running some behavioural economics experiment, because I like those. What fraction of the data am I going to make use of in building my model? I submit that the psychometric model might be exactly the same size as it is today. If I’m interested in decision theory then I’m going to be looking to verify/falsify some high-level hypothesis like “Expected utility” or “Hebbian learning”. The evidence for/against that idea is going to be so far above the atomic level, so far above the neuron level, I will basically still be looking at what I look at now:

• Did the decisions they ended up making (measured by maybe 𝒪(100), maybe even 𝒪(1) numbers in a table) correspond to the theory?
• For example if I draw out their assessment of the probability and some utility ranking then did I get them to violate that?

If I’ve recorded every atom in the room, then with some work I can get up to a coarser resolution and make myself an MRI. (Imagine working with tick-level stock data when you really are only interested in monthly price movements—but in 3-D.) (I guess I wrote myself into even more of a corner here, if we have atomic level data then it’s quantum, meaning you really have to do some work to get it to the fMRI scale!) But say I’ve gotten to fMRI level data, then what am I going to do with them? I don’t know how brains work. I could look up some theories of what lighting-up in different areas of the brain means (and what about 16-way dynamical correlations of messages passing between brain areas? I don’t think anatomy books have gotten there yet). So I would have all this fMRI data and basically not know what to do with it. I could start my next research project to look at numerically / mathematically obvious properties of this dataset, but that doesn’t seem like it would yield up a Master Answer of the Experiment because there’s no interplay beween theories of the brain and trying different experiments to test it out — I’m just looking at “one single cross section” which is my one behavioural econ experiment. Might squeeze some juice but who knows.

Then let’s talk about people critiquing my research paper. I would post all the atomic-level data online of course, because that’s what Jesus would do. But would the people arguing against my paper be able to use that granular data effectively?

I don’t really think so. I think they would look at the very high level of 𝒪(100) or 𝒪(1) data that I mentioned before, where I would be looking.

• They might argue about my interpretation of the numbers or statistical methods.
• They might say that what I count as evidence doesn’t really count as evidence because my reasoning was bad.
• They couldn’t argue that the experiment isn’t replicable because I imagined a perfect-fidelity machine here.
• They could go one or two levels deeper and find that my experimental setup was imperfect—the administrator of the questions didn’t speak the questions in exactly the same tone of voice each time; her face was at a slightly different angle; she wore a different coloured shirt on the other day. But in my imaginary world with perfect instruments, those kinds of errors would be so easy to see everywhere that nobody would take such a criticism seriously. (And of course because I am the author of this fantasy, there actually aren’t significant implementation errors in the experiment.)

Now think about either the scientists 100 years after that or if we had such perfect-fidelity recordings of some famous historical experiment. Let’s say it’s Michelson & Morley. Then it would be interesting to just watch the video from all angles (full resolution still not necessary) and learn a bit about the characters we’ve talked so much about.

But even here I don’t think what you would do is run an exploratory algorithm on the atomic level and see what it finds — even if you had a bajillion processing power so it didn’t take so long. There’s just way too much to throw away. If you had a perfect-fidelity-10²⁵-zoom-full-capacity replica of something worth observing, that resolution and fidelity would be useful to make sure you have the one key thing worth observing, not because you want to look at everything and “do an algo” to find what’s going on. Imagine you have a videotape of a murder scene, the benefit is that you’ve recorded every angle and every second, and then you zoom in on the murder weapon or the grisly act being committed or the face of the person or the tiny piece of hair they left and that one little sliver of the data space is what counts.

What would you do with infinite data? I submit that, for analysis, you’d throw most of the 10²⁵ bytes away.

In 20th-century abstract mathematics, one builds up ideas and properties—not assuming anything except what one is told. You think 2+3=5? Well in my space that I just made up, e₂⊕e₃ = e₁, and “5” doesn’t even exist!

Concepts are added in incrementally, like

• ∥ A ∥ means the “size” of A. size exists
• ∥ A − B ∥ means the “distance” between A and B. plus exists & negative exists; or, comparison exists
   
• (If zero exists, we could say the size of A = the distance between A and 0: ∥ A − 0 ∥ = ∥A∥.)
   
• ⟨ A | B ⟩ means A “times” B. times exists
• arccos ⟨A|B⟩ ∥A∥⁻¹ ∥B∥⁻¹ inverses exist. times exists. so angle exists
• topology adds in neighbourhood relationships—not necessarily in a way that you can infer size or distance (∵¬□∃ metric), but so that you could talk about paths or connectedness
• order or ranking — is it a total order? a transitive order? a partial order? a lattice? Order is subordinate to size, to distance, and to linearity.
• dimensionality — a set containing { ‘a’, ‘b’, the moon, 12, the vector (0 1 1 0 1)∈ℝ⁵, my cat’s hairball } doesn’t inherently have dimensions to it — so structured sets like ℝ² are supposed to explain how their universe breaks down
  
• linearity — possibly the scariest word in mathematics class? I’ve tried and will continue to try to explain it elsewhere, but “linear” is an extremely-restrictive-but-not-that-restrictive-because-so-many-things-are-linear-once-you-allow-calculus-and-maps-across-domains-for-example-fourier-transforms property. Linearity presumes monotonicity (order preservation), size, and a kind of “constancy” that tells you if 2 went to 4, then 13 is going to go to 26. Or “the 26 of the present land”.

Someone GPL’ed this nice (but not comprehensive) chart of two paths through the theory space—starting with a pair (thing, operation) [“magma”—sweet name, right?] and gradually adding more and more axioms until you get to a group.

Mathematical words obtain everyday meaning—sometimes unexpected meaning—in applications. For example

• “angle” might mean “correlation” — the angle between two pulse-trains would be their correlation; and in recommendation engines the matrix “cosine distance” is a basic measure of similarity
• “multiplication” — well what if you want to multiply two functions together? You could convolve them. Convolution doesn’t seem very much at all the same action as 3×8 = three groups of eight. Neither do Photoshop blends seem like multiplication, but some of them are.
• “size” — well maybe I mean “how well the business did” on a slew of different metrics — in which case, are there 20 different conceptions of “size”? I guess so.

Could you multiply two trees together? Could you define the angle between two natural numbers? The angle between two business models? Sure. If you know what you’re doing and why, you might even come up with a conclusion that makes sense. It all depends on (a) your ingenuity, (b) domain knowledge of the real-life situation, and (c) mathematical vocabulary.

Sometimes there is more than one interpretation that works with a given set. For example, {0,1} × {0,1} → {0,1} might be joined to operations that define “logical AND” and “logical OR”, or it might be interpreted just as on/off. Or it might be interpreted as the story of unrequited love.

All of that preface is meant to dislodge any notions you might have that ℝ² is somehow a “default” or “standard” paradigm. Sometimes number×number is an appropriate metaphor and sometimes not.

For example in the movie Rogue Trader, Nick Leeson’s boss is portrayed talking about “synergy” and “the information curve”. “Nick has positioned himself right there on the information curve!” It’s a parody and nobody seems to know quite what “the information curve” is (what’s on the axes? why is it curved?) but because Nick appears to be earning 70% of Barings’ profits, nobody questions the information curve.

Your typical crappy airport “business advice” books—Thomas Friedman kind of crap—will throw around 2-D charts that make no sense as well. Please leave some pics in the comments if you know what I’m talking about and examples come to mind. Here are a few dubious 2-D metaphors:

The “political compass” labels reduce the complexity of the world in particular ways that suit the rhetorical aims of these libertarian authors. For example projecting totalitarianism and populism into the same neighbourhood when one could just as well project them onto opposite ends of some other spectrum.

Here are some dubious scales—where either order, linearity, or 1-dimensionality is suspect.

(Remember: {“heroic”, ”pragmatic”, ”circumspect”, ”brazen”} also comprises or belongs to a scale—in the ggplot sense of the word as well as other senses.)

Wow! You mean that losses are bad and earnings are good? That is some insightful business insight.

Crappy reductions needn’t be 2-D. The MBTI is a crappy reduction of personality in 4-D. And here are some in 1-D and other-D:

I like how step 5 leads to step 2. This should be a list rather than a flow.

Order, 1-dimensionality questionable.

Again, a list. This one has a heading. Apparently headings deserve 4 connecting wires whereas list items only deserve 3?

This is just a list of things. There is no “center” or “flow” or “order” or “cycle” relationship. Maybe “give them” and “get them” could have used a two-way arrow between them.

8-D and I just do not understand what these axis labels mean.

I actually spent hours finding the worst graphics evar. Not gonna tell you my google keywords though.

And, not to be critical all the time, here’s a 2-D metaphor that does work:

Stagepiece one: undermine the conceit that ℝ² is a default. Stagepiece two: cruddy graphics from various domains that force a metaphor that doesn’t really work. And now, the main act.

Today, I want to take aim at a highly suspect 2-D chart from the world of psychology:  the affect × intensity description of feelings.

Right away when I look at this, it seems like an overly limiting and not internally valid picture of emotional range. Like so many taxonomies, it gets deeply under my skin in a way that I can’t explain, except to shout: Bad theory! Bad theory!  I mean — how does it make sense to say

1. that each of these states is a point, as opposed to a spray or splotch or something else
2. that this precise “point” is the same for all individuals
3. “delighted” is slightly to the left of “happy” but happy is directly above “pleased”
4. that “sleepy” is to the right of “tired” instead of the other way around
5. that tired and sleepy are the same distance from each other as “pleased” and “glad”
6. WTF is “droopy”? It sounds like a word to be applied to a plant, not a person. I also don’t think it qualifies as an emotion. “Droopy” sounds like a word Good Housekeeping would use to shame a 1950’s American married woman for not being perky! happy! sexy! listening! rubbing his feet! when her husband returns home from work.
7. Are “sleepy” and “tense” actually moods or emotions? They sound like physical states.
8. All of these emotions are near the perimeter, but some are closer to the origin than others
9. sad minus gloomy = satisfied minus calm

Remember what I was outlining at first. In abstract mathematics and in deciding the shape of a theory, we shouldn’t assume anything that doesn’t have to be assumed to explain the results.

I could attack the valence-intensity model in at least two ways.

1. First would be to exclaim “But you didn’t justify any of that stuff! Linearity? Dimensionality? Order? You skipped it all! Where’s the justification?”
2. Second, perhaps a little stronger than merely asking for backup, would be to point out flaws. For example if I could find a counterexample showing that emotional states don’t have magnitude, can’t be added, don’t break down on dimensions, or aren’t linear across dimensions.

I came up with a list—several years ago—of different feelings which all could contend for “emotional zero”.

• neither happy nor sad
• neutral
• feel blank
• both happy and sad (bittersweet)
• not sure
• ambivalent
• “I feel nothing”
• kinda sort
• middling

That’s just feelings we have the words for. There are lots of nameless emotions (or emotional superpositions) that could contend for the neutral canvas — the origin from which all other emotions are measured.

The fact that so many clearly distinct feelings all contend for the “origin” made me think there is, in fact, no origin. But making the space affine (removing zero) doesn’t fix the problems I had begun to notice with the circumplex view of the emotional spectrum. I think we just have to think of the range of emotions as a totally different kind of space. I don’t know its topology; I do believe there should be some “activation level” (like a scalar) at least sometimes; I do believe that superpositions are possible.

http://isomorphismes.tumblr.com/post/6559201759/graph-tradeoffs-design-pattern

http://isomorphismes.tumblr.com/post/4840897988/logic-emotion

Random thought. If end-of-life health care costs eat up 33% of US health care spending = $850 billion, then that means that if you could make people less afraid of dying and more willing to accept it, you would save = make a colossal amount of money. (In fact $850bn = roughly ten years of revenues of US President Obama’s optimistic projection if he raises taxes on the richest Americans.)

In other words, changing people’s attitudes could add 10% to the GDP of the biggest economy in the world.

Random thought #2. If we’re interested in maximising utility across the economy rather than increasing production levels, then perhaps the most important field of research is not bioengineering but the psychology of satisfaction. If you could figure out how to make people appreciate the things they have and not covet the things others have, then gross utility would shoot way up. How much? Billions? Maybe even on the order of the entire economy itself?

• The shape of the continents depends on the global temperature. (Cold locks ice in polar caps.) Google “Morse theory”.
• The price of housing always rises, until it doesn’t.
   
• You develop a system of habits to discipline yourself; maxims for self-motivation; then the working world changes on you. Loyalty is no longer rewarded. Hard work is less valued than the ability to make PlentyOfFish.com.
• For years the normal trading range of [insert spread, instrument, or security] is X, until one day sufficiently many (external) parameters shift. The market changes and you see a 20-sigma event. Heroes only.
  
  
• Whoever coded your profile website (chi.mp, flavors.me, tumblr), wrote a route that takes a string as parameter. Entering the name isomorphismes into this function fetches this webdata. Entering your name fetches your webdata. All part of one and the same formula.
• The Lotka-Volterra equations of a large ecosystem, dancing as the sliders shift around in their hypercube. Death and life hang in the balance. And it’s literally a balance. If the fulcrum moves so far that the lever hits the ground, a species will either become extinct or overpopulate the ecosystem (like an algal bloom)—either phase change being irreversible. (Er, at least anti-entropic.)
   
• You think you know yourself, until you step into a new context—new country, new career, new city—and latent aspects of you become dominant.
  
  Who was I before? If I was her then and am this now, what is the underlying me?
  
  Self as a function of circumstance. Perhaps just as constant at root, but reactive; responsive; springy; primed for change.

This week I posted different viewpoints on The Self.

Particularly I’m interested in self as a function of inputs. Just as the size of eyes a fly is born with is a function of the temperature of the eggs, so too, many facets of ourselves are a function of the environment, other people’s behaviour toward us, game-theoretic strategy, incentives, and so on.

Other people’s theories of us can be seen as functions as well. (For example, a hiring manager’s view of employee performance may assume school quality or GPA to be positively related to human capital.)

• Economics: I didn’t get to Jean Tirole’s theory of money-saving as bargains among multiple selves.
• Psychology: Jim Townsend found that self-versus-other dichotomies can be expressed as a negatively curved metric space.
• Personality: I’ve already written that the MBTI is too restrictive a theory of self. It maps from habits to [0,1]^4.
• Douglas Hofstadter’s thoughts on the extension of the pronoun “we”. ‘We’ went to the moon, ‘we’ share a common ancestor with other primates, ‘we’ are overcrowding the planet, ‘we’ have a nice theory of quantum chromodynamics, ‘we’ do not know if ‘we’ are experiencing a simulation or actual reality, ‘we’ don’t really know what makes an economy grow.
• Criminology: My criminal output is a function of the crime level in the neighbourhood I’m raised in. Except when it’s a function of strongly held beliefs.
• Sociology: In contemporary OECD places, ‘we’ are coerced by our cultures to play roles. “There are” certain scripts — modifiable but still requisite or recommended in some sense; at the very least influential, even if only because benefits and rewards are socially tied to role performance.
• The topic of cultural coercion … is something I’ll return to.
• The concept of people-as-functions is one I want to return to later, in discussing history, economics, and a couple different ways of talking about human behaviour mathematically.

I can think of several other mathematics-inspired questions about ourselves. The difference between habit and personality; the yogic metaphor of a river cutting deeper as related to habituation; choice & free will; Markovian and completely-the-opposite-of-Markovian choices (how constrained we are by our past choices); … and a lot more. But you know what, writing is hard. So I do only a little at a time.

What happens if, instead of doing a linear regression with sums of monomial terms, you do the complete opposite? Instead of regressing the phenomenon against  , you regressed the phenomenon against an explanation like  ?

I first thought of this question several years ago whilst living with my sister. She’s a complex person. If I asked her how her day went, and wanted to predict her answer with an equation, I definitely couldn’t use linearly separable terms. That would mean that, if one aspect of her day went well and the other aspect went poorly, the two would even out. Not the case for her. One or two things could totally swing her day all-the-way-to-good or all-the-way-to-bad.

The pattern of her moods and emotional affect has nothing to do with irrationality or moodiness. She’s just an intricate person with a complex utility function.

If you don’t know my sister, you can pick up the point from this well-known stereotype about the difference between men and women:

“Men are simple, women are complex.” Think about a stereotypical teenage girl describing what made her upset. “It’s not any one thing, it’s everything.”

I.e., nonseparable interaction terms.

I wonder if there’s a mapping that sensibly inverts strongly-interdependent polynomials with monomials — interchanging interdependent equations with separable ones. If so, that could invert our notions of a parsimonious model.

Who says that a model that’s short to write in one particular space or parameterisation is the best one? or the simplest? Some things are better understood when you consider everything at once.