A language for computing under uncertainty

In the late 1800s, mathematicians held an international conference to unify and harmonise mathematical notation to facilitate greater communication across the subdisciplines. A reviewer at the time reported, "As might have been anticipated, they had not the slightest success whatever." The limited unity in notation that been has achieved over the last dozen decades seems to be the result of the innervating mathematical formalism with the notation of set theory, which has been, in Wittgenstein's quip, a cancer on the body of mathematics. Kari Sentz jokes that mathematicians would rather share underwear than adopt a common notation, yet everyone acknowledges the clear and massive advantages to finding consensus where possible.

There have already been several proposed languages to express and compute with uncertainty. And several implementations and software packages have been created and promulgated. Now that some flowers have bloomed, it may be time to start to look for the commonalities and to suggest conventions that can be helpful broadly.

The interval analysis community conscientiously met in committee over many months to create their standards (<<link?>>), and we should be cognizant of their work and its fruit, if not obsequious to them.

The advantages and disadvantages of one choice or another should be expressed in terms of these criteria, roughly ranked in order of importance:

• 1. Ease for end-user

  2. Comprehensiveness

  3. Flexibility and adaptability

  4. Simplicity and straightforwardness

1. Ease and clarity in communicating to colleagues who may be using different software

2. More x-think and less p-think

3. Computational speed

4. Protectedness & good programming practice

5. Independence from implementation language

   1. Ease for coders and developers (methods v. functions, object orientation v. functionality)

Let's meet to argue the issues this fall or winter. Anyone interested is welcome; please let others know about it. We have already emailed

Francis Baumont de Oliveira <francis@farmurban.co.uk>,

Marco De Angelis <mda@liv.ac.uk>,

Scott Ferson <ferson@liverpool.ac.uk>,

Ander Gray <Ander.Gray@liverpool.ac.uk>,

Nick Gray <nickgray@liv.ac.uk>,

Michael Hanss, <michael.hanss@itm.uni-stuttgart.de>,

Dominik Hose, <dominik.hose@itm.uni-stuttgart.de>>,

Josh Kaizer <jsk181@gmail.com>,

Enrique Miralles Dolz <E.Miralles-Dolz@liverpool.ac.uk>,

Jason O'Rawe <jazon33y@gmail.com>,

Kari Sentz <kari.sentz@gmail.com>,

Alexander Wimbush <a.wimbush@liverpool.ac.uk>.

Others to invite include people from DigiTwin, especially Robin Langley, Keith Worden, and David Wagg, as well as Kevin Bevin, Jon Helton, Bill Oberkampf, Ullrika Sahlin, Matthias Troffaes, Gert de Cooman, Willem Roelofs, Vladik Kreinovich, Michael Beer, Edo Patelli, Bruno Sudret, Michael Shields, Bob McMullen, Wolfgang Graf, Michael Kaliske, Andrew Dienstfrey, among others.

The meeting should address what functionality and features an uncertainty computing language should have, and other issues toward the general goal of articulating a useful general standard for such a language. We should first develop a flier-style advertisement for the meeting, with a nice visual and few words, that we can use on Twitter and splash screens on the Events page. Maybe it should have the basic questions we'll address. Perhaps the next thing to do is develop an draft agenda for the meeting. As a possibly self-serving starting point, I have outlined some of the basic functionality in black text below. The green text reflects additions I wanted to make. The orange text marks features that I am not sure I want to support. Toward the bottom of the list, we expect there will be controversy and divergence of opinions, but let us focus on recognising and heralding as much agreement as we can find starting perhaps in the upper parts of the list.

First-class citizenship of uncertain numbers

Printable, readable, writeable, debuggable

Passable as function arguments and results

Changeable of the fly

Standard functions generalised for uncertain numbers

Unary: abs, exp, log = ln, log10, sqrt, sqr, int, tan, sin, cos, atan, asin, acos, not, image, inverse, Gamma, Lambert w, ceiling, diGamma, downround, floor, logit, Logistic,

Binary: atan, power, Beta, betafunction, bico, choose, erf, factorial, fallingfactorial, risingfactorial

N-ary: print, display, show, min, max, list, and, or, xor, nand, nor

New functions introduced for uncertain numbers

Unary: left, right, leftside, rightside, smallest, largest, smaller, larger, centroid, core, mean, median, mode, harmean, geomean, midcore, midrange, stddev, variance, cv, iqrange, width, breadth, range, skewness, widen, wider, specific, spanning, dispersive, alpha, prob, ascloseas, closest, dyadic, edgy, endpt, farthest, fatten, fatter, fattenpositive, fatterpositive, gefgw (good enough for government work), odds, thicken, thicker, above, below, between, censor, highest, lowest, rescale

Binary: convolve, decon, factor, inside, contains, straddles, overlap, cond, equiv, imply, modus ponens, modus tollens

N-ary: envelope, imposition, mixture , concensus, histogram

Infix operators: +, -, *, /, ^, **, unary minus, //, %, ±, ×, ÷, •, ∙, <, <=, >, >=, ≥, ≤, ==, ===, ≈, ≡, !=, <>, ≠, &, |, ¬, ∩, ∪, ⋂, ⋃, ∈, ∉, ⊂, ⊆

Assignment operators: =, ~, <-, is

Communication (notation, file-compatibility)

Special symbols: #, @, ∞, Ø, ?, ℜ, ℝ, ℕ, ℤ

Special constants: dunno, empty, infinity, positives, nonnegatives, negatives, nonpositives, scientific and engineering constants with CODATA uncertainties

Types: reals, intervals, distributions, p-boxes, fuzzy numbers, hybrid numbers, possibility distributions, metadistributions, metaboxes (list of intervals or p-boxes)

Output summaries

One-number (scalar with the null hedge, i.e., a naked scalar)

Simple (the most appropriate hedged scalar value)

Range (left and right)

95%-range (2.5th and 97.5th percentiles)

Five-number (min, first quartile, median, third quartile, max)

Risk Calc summary (distribution family if known, range, mean, var)

Verbose (short paragraph describing the uncertain value)

Constructors

Intervals: [a, b], [a ± b], [s], zero-variance intervals, "sound" dual (Kaucher) intervals, open and partially open intervals

Nonparametric: minmax, box, minmaxmean, minmaxmeanismedian, minmaxmedian, minmaxmeanismode, minmaxmode, minmaxmedianismode, minmaxquantile, minmaxpercentile, meanmin, minmean, meanmax, maxmean, meanvariance, meanvar, meanstddev, meancv, minmeanstddev, maxmeanstddev, mmmms, mmmmv, unimmmms, unimmmmv, fivenumbers, minmaxmean95th, symmeanvariance, symmeanstddev, symmeancv, posmeanstddev, histogram, unimodal

Named distributions: 150 named distributions, supporting interval parameters

Hedges: about, around, approximately, up to, at most, at least, exactly, precisely, etc.

Counts

Measurements

Histograms

Estimators of parameters and distributions from data: canonical Bayesian posteriors, confidence boxes, consonant confident structures

Models of distributions from constraints: maximum entropy, PERT, maximum relative entropy, etc.

Fitted structures: method of matching moments, maximum likelihood, etc.

Dependence specification

Correlation measures

Pairwise dependence

Matrices of pairwise dependence

Multivariate joint structures

Special copulas

Parameterised and general copulas

Empirical copulas

Resolution of questions about the two levels of dependence

Dependence tracking

Repeated uncertainty detection

Other stuff

Special cases: logicals

Structures: stems, trees, lists, arrays

Inference engines à la Stan, Jugs, etc.

Intervals statistics

Finite element code

ODE code

Autodifferentiation

Units checking and propagation

Probabilistic programming (https://en.wikipedia.org/wiki/Probabilistic_programming)

Vocabulary

There have been several 'taxonomies' of uncertainty proffered by various researchers. The Sandia reports offer glossaries about many basic terms. Lots of technical terms are bandied about. Can we agree about the context and meaning for some basic ones?