Posts tagged with uncertainty

Some people think of postmodernism as the rejection of the existence of objective facts. Another take is that po-mo comprises broader methods of getting one’s point across than didaction. For example: joking, insinuating, or ending sentences with question marks.

For example this sarcastic remark:

A theory is something which nobody literally believes except the person who invented it. An experiment is something which everybody literally believes except the person who invented it.

pokes fun at what a different conversational mode might wax about in general terms such as “human frailty” or “fallibility”—or sound like a stronger attack on the scientific method than it intends to be.

It’s natural to express scepticism when an expert or supposed expert disagrees with something that makes complete sense to you. (I owe ya a post called “The rigid rod of modus tollens & modus ponens”.) ”Says who?” is a sentence anyone can utter. You could view “the scientific method” as one way to respond to that criticism. But is it the only way?

Some (postmodern?) anthropologists and ethnographers begin their essays on people who are foreign to them by discussing their biases and where generally they’re coming from. Which may be a more appropriate response to scepticism with non-experimental data—a different way of addressing the same problem that repeatable double-blind experiments are supposed to, namely errors in judgment by the observer/researcher.

Economists have field-specific ways of addressing problems inherent to what they study. These include models, stylised facts, stating own biases, statistics, and rebuttals against the statistical analysis. But also self-questioning sarcasm. For example

The questions in economics never change. Only the answers do.

or 

When we leave our closet, and engage in the common affairs of life, [reason’s] conclusions seem to vanish, like the phantoms of the night on the appearance of the morning; and ‘tis difficult for us to retain even that conviction, which we had attain’d with difficulty.

or

The Economics Nobel confers upon the laureate an appearance of expertise which in economics no one ought to possess.

I don’t think “a postmodern economics” needs to be “post-autistic” or revolutionary or hip in the ways I’ve seen suggested by heterodoxists. It could simply be the recognition that informal speech like sarcasm can be on the same level of importance as speeches, lectures, claims, statements, and pontifications.




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.

http://www.olivierlanglois.net/images/voro2.jpg

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.

File:Coloured Voronoi 3D slice.svg

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.
a scalar field

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.

image
image

 

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.)

http://i780.photobucket.com/albums/yy90/AlexMLeo/felixsbrain.jpg

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.
image
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.)

image

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.

File:ApproximateVoronoiDiagram.png

 

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.

A tiny portion of Doug Hofstadter’s “semantic network”.  via jewcrew728, structure of entropy

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.
    Regular Expressions
    11th Grade
  • 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
    Sandwich
  • 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.

image

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.)

File:2Ddim-L2norm-10site.png

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

http://www.financialiceberg.com/uploads/iceberg340.jpg
http://www.emeraldinsight.com/content_images/fig/1100190504002.png


http://www.preventa.ca/images/im_risk_anatomy.jpg

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!

    a rug plot or carpet plot is like a barcode on the bottom of your plot to show the marginal (one-dimension only) distribution of data

    who is faster, men or women?
  • when you learn about Gaussians and see them everywhere
    Central Limit Theorem  A nice illustration of the Central Limit Theorem by convolution.in R:  Heaviside <- function(x) {      ifelse(x>0,1,0) }HH <- convolve( Heaviside(x), rev(Heaviside(x)),        type = "open"   )HHHH <- convolve(HH, rev(HH),   type = "open"   )HHHHHHHH <- convolve(HHHH, rev(HHHH),   type = "open"   )etc.  What I really like about this dimostrazione is that it’s not a proof, rather an experiment carried out on a computer.  This empiricism is especially cool since the Bell Curve, 80/20 Rule, etc, have become such a religion.NERD NOTE:  Which weapon is better, a 1d10 longsword, or a 2d4 oaken staff? Sometimes the damage is written as 1-10 longsword and 2-8 quarterstaff. However, these ranges disregard the greater likelihood of the quarterstaff scoring 4,5,6 damage than 1,2,7,8. The longsword’s distribution 1d10 ~Uniform[1,10], while 2d4 looks like a Λ.  (To see this another way, think of the combinatorics.)
  • when you learn that Gaussians are not actually everywhere
    kernel density plot of Oxford boys' heights.

    histogram of Oxford boys' heights, drawn with ggplot.A (bimodal) probability distribution with distinct mean, median, and mode.
  • 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—
    http://www.niemanlab.org/images/math-formula-chalkboard.jpg
  • but then, notice how real mathematicians’ chalkboards in their offices never contain a restatement of Physics 103!
    Kirby topology 2012
    http://whatsonmyblackboard.files.wordpress.com/2011/06/21june2011.jpg
    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.




Apparently the “extra” dimensions of string theory are only supposed to be a few millimetres thick.

If that’s the case, could you dodge a bullet by moving a millimetre in the 10th dimension?

I guess it would depend on how wide the bullet and your liver are in the 10th dimension. Could lead to an interesting superpower: move in hidden dimensions.

  • The hero wouldn’t be invulnerable but would be less vulnerable. Still get an exit wound but maybe she’d only be grazed through the interior rather than completely ripped to shreds.
  • Still worth dodging/blocking a fist in the normal-sized 3 dimensions, but even a “direct” uppercut or body blow could become more of a “glancing blow” if she dodged in the thin directions. (NB: If ∃ 7 extra thin dimensions, each 1mm wide, and she dodged at once to “the other side” of all 7 at once—assuming, as well, that we’re “all the way to one side” of each of the extra dimensions—then she’d have made a total distance of √7mm between her and us.)
  • Joint locks—could she put someone in a joint lock they couldn’t get out of? Couldn’t she also get out of joint locks that no-one else could?
  • Couldn’t become invisible but become less visible.
  • Couldn’t pass through walls but could reach into crevices easier.
  • Could swim faster (twist her torso in the 10th dimension so the hands & feet still pull water, but less resistance on the mass of the body).

Am I thinking about this right?







I don’t feel that it is necessary to know exactly what I am. The main interest in life and work is to become someone else that you were not in the beginning. If you knew when you began a book what you would say at the end, do you think that you would have the courage to write it? What is true for writing and for a love relationship is true also for life. The game is worthwhile insofar as we don’t know what will be the end.

Michel Foucault

in an interview titled Truth, Power, Self printed 25 October 1982. via matryoshhkathe se




The actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on.

Therefore the true logic for this world is the calculus of Probabilities, which takes account of the magnitude of the probability which is, or ought to be, in a reasonable man’s mind.
James Clerk Maxwell (1850), quoted in Plausible Reasoning (1994)