isomorphismes

Lucas’ “rational expectations” revolution in macroeconomics has been tied to the ending of stagflation in the world’s largest economy, and to the reintroduction of “psychology” into finance and economics. However, I never felt like the models of “expectation” I’ve seen in economics seem like my own personal experience of living in ignorance. I’d like to share the sketch of an idea that feels more lifelike to me.

First, let me disambiguate: the unfortunate term-overlap with “statistical expectation” (= mean = average = total over count = ∑ᵢᴺ•/N = a map from N dimensions to 1 dimension) indicates nothing psychological whatever. It doesn’t even correspond to “What you should expect”.

If I find out someone is a white non-Hispanic Estadounidense (somehow not getting any hints of which state, which race, which accent, which social class, which career track…so it’s an artificial scenario), I shouldn’t “expect” the family to be worth $630,000. I “expect” (if indeed my expectation is not a distribution but rather just one number) them to be worth $155,000. (scroll down to green)

Nor, if I go to a casino with 99% chance of losing €10,000 and 1% chance of winning €1,000,000 (remember the break-even point is €990,000). “On average” this is a great bet. But that ignores convergence to the average, which would be slow. I’d need to play this game a lot to get the statistics working in my favour, and I mightn’t stay solvent (I’d need to get tens of millions of AUM—with lockdown conditions—to even consider this game). No, the “statistical expectation” refers to a long-run or wide-space convergence number. Not “what’s typical”.

Not only is the statistical expectation quite reductive, it doesn’t resemble what I’ve introspected about uncertainty, information, disinformation, beliefs, and expectations in my life.

A better idea, I think, comes from the definition of Riemann integration over 2+ dimensions. Imagine covering a surface with a coarse mesh. The mesh partitions the surface. A scalar is assigned to each of the interior regions inscribed by the mesh. The mesh is then refined (no lines taken away, only some more added—so some regions get smaller/more precise and no regions get larger/less precise), new scalars are computed with more precise information about the scalar field on the surface.

NB: The usual Expectation operator 𝔼 is little more than an integral over “possibilities” (whatever that means!).

(In the definitions of Riemann integral I’ve seen the mesh is square, but Voronoi pictures look awesomer & more suggestive of topological generality. Plus I’m not going to be talking about infinitary convergence—no one ever becomes fully knowledgeable of everything—so why do I need the convenience of squares?)

I want to make two changes to the Riemannian-integral mesh.

First I’d like to replace the scalars with some more general kind of fibre. Let’s say a bundle of words and associations.

(You can tell a lot about someone’s perspective fro the words they use. I’ll have to link up “Obverse Words”, which has been in my drafts folder for over a year, once I finish it—but you can imagine examples of people using words with opposite connotation to denote the same thing, indicating their attitude toward the thing.)

Second, I’d like to use the topology or covering maps to encode the ignorance somehow. In my example below: at a certain point I knew “Rails goes with Ruby” and “Django goes with Python” and “Git goes with Github” but didn’t really understand the lay of the land. I didn’t know about git’s competitors, that you can host your own github, that Github has competitors, the more complex relationship between ruby and python (it’s not just two disjoint sets), and so on.

When I didn’t know about Economics or Business or Accounting or Finance, I classed them all together. But now they’re so clearly very very different. I don’t even see Historical Economists or Bayesian Econometricians or Instrumental Econometricians or Dynamical Macroeconomists or Monetary Economists or Development Economists as being very alike. (Which must imply that my perspective has narrowed relative to everyone else! Like tattoo artists and yogi masters and poppy farmers must all be quite different to the entire class of Economists—and look even from my words how much coarse generalisation I use to describe the non-econ’s versus refinement among the econ’s.

These meshes can have a negative curvature (with, perhaps a memory) if you like. You know when you think that property actuaries are nothing at all like health actuaries that your frame-of-reference has become very refined among actuary-distinguishment. Which might mean a coarse partitioning of all the other people! Like Bobby Fischer’s use of the term “weakies” for any non-chess player—they must all be the same! Or at least they’re the same to me.)

Besides the natural embedding of negatively-curved judgment grids, here are some more pluses to the “refinement regions” view of ignorance:

1. You could derive a natural “conservation law” using some combination of e.g. ability, difficulty, how good your teachers are, and time input to learning, how many “refinements” you get to make. No one can know everything.
   
   (Yet somehow we all are supposed to function in a global economy together—how do we figure out how to fit ourselves together efficiently?
   
   And what if people use your lack of perspective to suggest you should pay them to teach you something which “evaluates to valuable” from your coarse refinement, but upon closer inspection, doesn’t integrate to valuable?)
2. Maybe this can relate to the story of Tony—how we’re always in a state of ignorance even as we choose what to become less ignorant about. It would be nice to be able to model the fact that one can’t escape one’s biases or context or history.
3. And we could get a fairly nice representation of “incompatible perspectives”. If the topology of your covering maps is “very hard” to match up to mine because you speak dialectics and power structures but I speak equilibria and optima, that sounds like an accurate depiction. Or when you talk to someone who’s just so noobish in something you’re so expert in, it can feel like a very blanket statement over so many refinements that you don’t want to generalise over (and from “looking up to” an expert it can also feel like they “see” much more detail of the interesting landscape.)
4. Ignorance of one’s own ignorance is already baked into the pie! As is the beginner’s luck. If I “integrate over the regions” to get my expected value of a certain coarse region, my uninformed answer may have a lot of correctness to it. At the same time, the topological restrictions mean that my information and my perspective on it aren’t “over there” in some L2-distance sense, rather they’re far away in a more appropriately incompatible-with-others sense.

In conclusion, I’m sure everyone on Earth can agree that this is a Really Nifty and Cool Idea.

I’ll try to give a colourful example using computers and internet stuff since that’s an area I’ve learned a lot more about over the past couple years.

First, what does ignorance sound like?

• (someone who has never seen or interacted with a computer—let’s say from a non-technological society or a non-computery elderly rich person. I’ve never personally seen this)
• “Sure, programming, I know a little about that. A little HMTL, sure!”
• “Well, of course any programming you’re going to be doing, whether it’s for mobile or desktop, is going to use HTML. The question is how.

OK, but I wasn’t that bad. In workplaces I’ve been the person to ask about computers. I even briefly worked in I.T. But the distance from “normal people” (no computer knowledge) to me seems very small now compared to the distance between me and people who really know what’s up.

A few years ago, when I started seriously thinking about trying to make some kind of internet company (sorry, I refuse to use the word “startup” because it’s perverted), I considered myself a “power user” of computers. I used keyboard shortcuts, I downloaded and played with lots of programs, I had taken a C++ course in the 90’s, I knew about C:\progra~1 and how to get to the hidden files in the App packages on a Mac.

My knowledge of internet business was a scatty array of:

• Mark Zuckerberg
• “venture capital”
• programer kid internet millionaires
• Kayak.com — very nice interface!
• perl.
  
  
• mIRC
• TechCrunch
• There seem to be way more programming going on to impress other programmers than to make the stuff I wanted!
• I had used Windows, Mac, and Linux (!! Linux! Dang I must be good)
• I knew that “Java and Javascript are alike the way car and carpet are alike”—but didn’t know a bit of either language.
• I used Alpine to check my gmail. That’s a lot of confusing settings to configure! And plus I’m checking email in text mode, which is not only faster but also way more cooly nerdy sexy screeny.
• Object-Oriented, that’s some kind of important thing. Some languages are Object-Oriented and some aren’t.
• “Python is for science; Ruby is for web”
• sudo apt-get install
  
• I had run at least a few programs from the command line.
• I had done a PHP tutorial at W3CSchools … that counts as “knowing a little PHP”, right?

So I knew I didn’t know everything, but it was very hard to quantify how much I did know, how far I had to go.

A mediocre picture of some things I knew about at various levels. It’s supposed to get across a more refined knowledge of, for example, econometrics, than of programming. Programming is lumped in with Linux and rich programmer kids and “that kind of stuff” (a coarse mesh). But statistical things have a much richer set of vocabulary and, if I could draw the topology better, refined “personal categories” those words belong to.

Which is why it’s easier to “quantify” my lack of knowledge by simply listing words from the neighbourhood of my state of knowledge.

Unfortunately, knowing how long a project should take and its chances of success or potential pitfalls, is crucial to making an organised plan to complete it. “If you have no port of destination, there is no favourable wind”. (Then again, no adverse wind either. But in an entropic environment—with more ways to screw up than to succeed—turning the Rubik’s cube randomly won’t help you at all. Your “ship” might run out of supplies, or the backers murder you, etc.)

Here are some of the words I learned early on (and many more refinements since then):

• Rails
• Django
• IronPython
• Jython
• JSLint
• MVC
• Agile
• STL
• pointers
• data structures
• frameworks
• SDK’s
• Apache
• /etc/.httpd
• Hadoop
• regex
• nginx
• memcached
• JVM
• RVM
• vi, emacs
• sed, awk
• gdb
• screen
• tcl/tk, cocoa, gtk, ncurses
• GPG keys
• ppa’s
• lspci
• decorators
• virtual functions
• ~/.bashrc, ~/.bash_profile, ~/.profile
• echo $SHELL, echo $PATH
• “scripting languages”
• “automagically”
• sprintf
• xargs
• strptime, strftime
• dynamic allocation
• parser, linker, lexer
• /env, /usr, /dev,/sbin
• GRUB, LILO
• virtual consoles
• Xorg
• cron
• ssh, X forwarding
• UDP
• CNAME, A record
• LLVM
• curl.haxx.se
• the difference between jQuery and JSON (they’re not even the same kind of thing, despite the “J” actually referring to Javascript in both cases)
• OAuth2
• XSALT, XPath, XML

This is only—as they say—“the tip of the iceberg”. I didn’t know a ton of server admin stuff. I didn’t understand that libraries and frameworks are super crucial to real-world programming. (Imagine if you “knew English” but had a vocabulary of 1,000 words. Except libraries and frameworks are even better than a large vocabulary because they actually do work for you. You don’t need to “learn all the vocabulary” to use it—just enough words to call the library’s much larger program that, say, writes to the screen, or scrapes from the web, or does machine learning, for you.)

The path should go something like: at first knowing programming languages ⊃ ruby. Then knowing programming languages ⊃ ruby ⊃ rubinius, groovy, JRuby. At some point uncovering topological connections (neighbourhood relationships) to other things (a comparison to node.js; a comparison to perl; a lack of comparability to machine learning; etc.)

I could make some analogies to maths as well. I think there are some identifiable points across some broad range of individuals’ progress in mathematics, such as:

• when you learn about distributions and realise this is so much better than single numbers!
  
  
  
  
• when you learn about Gaussians and see them everywhere
  
• when you learn that Gaussians are not actually everywhere
  
  
  
• in talking about probability and randomness, you get stuck on discussions of “what is true randomness?” “Does randomness come from quantum mechanics?” and such whilst ignorant of stochastic processes and probability distributions in general.
• (not saying the more refined understanding is the better place to be!)
• A brilliant fellow (who now works for Google) was describing his past ignorance to us one time. He remembered the moment he realised “Space could be discrete! Wait, what if spacetime is discrete?!?!?! I am a genius and the first person who has ever thought of this!!!!” Humility often comes with the refinement.
  
• when you start understanding symbols like ∫ , ‖•‖, {x | p} — there might be a point at which chalkboards full of multiple integrals look like the pinnacle of mathematical smartness—
  
• but then, notice how real mathematicians’ chalkboards in their offices never contain a restatement of Physics 103!
  
  
  A parsimonious statement like “a local ring is regular iff its  global dimension is finite” is so, so much higher on the maths ladder than a tortuous sequence of u-substitutions.
• and so on … I’m sure I’ve tipped my hand well enough all over isomorphismes.tumblr.com that those who have a more refined knowledge can place me on the path. (eg it’s clear that I don’t understand sheaves or topoi but I expect they hold some awesome perspectives.) And it’s no judgment because everyone has to go through some “lower” levels to get to “higher” levels. It’s not a race and no one’s born with the infinite knowledge.

I think you’ll agree with me here: the more one learns, the more one finds out how little one knows. One can’t leave one’s context or have knowledge one doesn’t have. And all choices are embedded in this framework.

The classic red/green colouring scheme for trading screens seems too alarmist.

Conceptually, the red/green distinction makes sense as corresponding to stop/go in traffic signals. But traffic signals need to be neon and striking in a hectic 3-D environment where it’s paramount for everyone to definitely not-miss the stop command.

But in a sheltered 2-D environment where goals commonly include to master emotion, to control passive reactivity, to keep a long-term head in the middle of short-term volatility, and to digest (calmly) massive amounts of information en simultáneo, neon red/green seems too grating.

One theory of the evolution of trichromacy in primates says that

• red/green dichotomy tells us whether meat or fruit is rotten or ripe (especially in dappled light)
• blue/yellow dichotomy tells us how cool/warm something is
• light/dark (value) is the most basic kind of vision.

If we take that as a starting point, a less alarmist colour scheme for trading software could use the blue/yellow dichotomy to indicate whether a security price went up or down. Use a neutral chroma for “small” moves (this depends upon one’s time-frame, but properly the definition of “big move” should be calibrated to an exponential moving average with some width depending on one’s market telescope). Intensity of the move could be signalled with lightness, so that most figures on a screen are a readable lightness of a neutral colour, but “big moves” are tinged with convexly more chroma and very-convexly more lightness.

The definition of “up/down” might be refigured as whether the trader is short/long the security in question, or perhaps redness/greenness could be used in conjunction with the “market view” of cold/hot, to indicate whether a security is moving for/against one’s strategy. That too could be seen as overly alarming, but a (pseudo)convex coding of red-ness might again solve the problem again, only invoking the “panic mode” when there’s really something to worry about.

(Source: twitter.com)

If you’ve read Stats 101 at your local institution of schooling and refinement, you know the difference between false positives and false negatives.

• False positive. Oncologist, to patient: Oh my God. This is terrible. Just terrible. Patient: What? Is it bad, Doc? Oncologist: Oh, not you. My son’s handwriting. It’s terrible! Practically illegible.”
  
• False negative: Pregnancy test: − Eight months later: Waaaah!

False positive is when the canary has spent several years building up an immunity to iocaine mine gas; you stroll in and die. False negative is when the canary dies of canary-pneumonia in a gas-free mine; you scurry away and miss out on $500bn worth of shale coal.

Signals

For algorithmic traders, a “signal” is the switch that tells your software “Buy! Buy!” or “Sell! Sell!”

• Computer: Just give me the signal, boss, and I’ll buy 10,000 shares of the company that makes IcyHot.
• Trader: Let’s see … gold is up 50% on the year … the underlying is retracing between the third and fourth Fibonacci levels  … the volatility of the DOW is below 30 … it’s raining in Moscow … and my Alabama state government newsfeed just flashed the word “indubitably” … throw down the iron condor! Hard!!!

If I think about looking for a signal, I think about: when should I do this trade?

Anti-Signals

The idea behind this exercise is to have a computer search through data streams for you and tell you what’s a good time to trade.

If you take the perspective that the only thing you can control is your bet size (and not what the market will do), then it becomes clear that the choice is not only about {yes, no} but also about [£0, £100bn].

Accordingly, something that tells you when not to trade can be just as valuable as something that tells you when to trade.

The most obvious non-trading scenario is the Federal Open Market Committee. Say you normally trade forex intraday, close out all your positions when you leave the screen, and that’s your game. Right after the FOMC announcement, market movements may be drastic and will have little to do with what you normally bet on — unless you trade FOMC announcements specifically. But the point is it’s a separate modelling problem.

In theory at least, any strategy should be improvable if you can accurately identify conditions when the strategy fails. Removing losers will add to your PnL just as surely as adding winners.

I’ll make up a fake example with fake data (aka, lies). Say your strategy is to trade in the direction of momentum of S&P 500 E-mini’s iff the directionality has been sustained for at least 70% of the last five minutes, and to pull out the trade iff the fraction of price movements in your direction falls below 70%.

Looking at each hour of the past year, the least profitable hour for this trade, statistically, has been 12:00-1:00 New York time. So if you had followed the exact same instructions but closed out all positions and never traded during lunchtime, your PnL during 2011 would have been higher than the strategy as originally stated. (Of course, this example works only because one hour had to be the least profitable. But the same difficulty—distinguishing real patterns from numerical mirages—inheres in signal identification as well as anti-signal identification. If you identify a real cause & effect then the anti-signal should work.)

Any Statistical Model

Say you are trying to calculate when, where, and wieviel an advertiser should bid in a DMP for internet ad space. You take as inputs known or presumed data about site visitors, indexed by cookie, and produce as (eventual) output a list of which ads to buy, when, and how much to bid for them.

Here the same anti-signal concept could apply.

Instead of thinking, What are some damned good characteristics in this space? or Should I try another algorithm? This other paper says RandomForests aren’t as good as Breiman says. , think What data is the AI really failing on? You can remove those data from the training set and decline to make recommendations about cookies within that hull.

Say you are scanning a number of text resumes on a site like Indeed <aff link> and trying to figure out whose application you should invite for a geomodelling job. Just as much as searching for positive keywords like “Petrel”, you might want to filter out negative keywords like “definately”.

Say you are training your machine to learn when tweets will be effective and when not. Instead of shoving every tweet through the lingpipe, first filter out the non-English-language tweets.

OK, that last example is really obvious. I am not claiming that anti-signals are novel. It’s just a word I made up for something that’s common sense. But coining the word reminds me when I look at a modelling problem, to turn the problem upside-down and ask if there’s any low-hanging fruit on the other side.