# Taxonomies of chance, risk and unpredictability.

Is it possible to give any sensible taxonomy of "probability in the real world"? Here we show some ways in which people have tried to do so.

## 1. Intrinsic mathematical structure.

The Mathematics Subject Classification (MSC) splits Mathematics, meaning theorem-proof mathematics and some nearby areas, into about 80 fields, one of which (section 60) is Probability. (The MSC is a complement to the journal Mathematical Reviews which seeks to give brief reviews of all research papers in Mathematics, assigning each paper to one or more of its classifications. There are about 2,500 papers a year in Probability.) Then Probability is split into about 100 topics, illustrated by the 12 below.
• 60G42 Martingales with discrete parameter
• 60G44 Martingales with continuous parameter
• 60G45 Martingale theory
• 60G46 Martingales and classical analysis
• 60G48 Generalizations of martingales
• 60G50 Sums of independent random variables; random walks
• 60G51 Processes with independent increments
• 60G52 Stable processes
• 60G55 Point processes
• 60G57 Random measures
• 60G60 Random fields
• 60G70 Extreme value theory; extremal processes
This classification is based upon the ``intrinsic structure of the mathematical objects being studied", and serves quite well to group together papers which use similar mathematical methods or which tackle mathematically similar problems. But it is not directed at real-world uses of mathematical models.

Here is an analogous list from Statistics, where the items represent methodologies.

• 62H12 Estimation
• 62H15 Hypothesis testing
• 62H17 Contingency tables
• 62H20 Measures of association (correlation, canonical correlation, etc.)
• 62H25 Factor analysis and principal components; correspondence analysis
• 62H35 Image analysis
• 62H40 Projection pursuit
• 62H86 Multivariate analysis and fuzziness

## 2. Uses of probability across different academic disciplines

An hour before writing the first draft of this section, a graduate student come into my office, said he was interested in doing research in some area of applied probability, and asked me what were the academic areas to which probability is applied. I get asked this question sufficiently often that I have a slightly flippant, stock response:
Anything called "applied probability" is just a particular style of math theory. If you're really interested in using mathematics in science/engineering/economics ....., find a professor who's studying interesting science/engineering/economics ..... questions.
But returning to the student's question, it's easy to give a list of 10 or 20 very broad areas in which probability is used. For instance (in alphabetical order)
• Bioinformatics
• Cognitive Science
• Finance Theory
• Information (Coding) Theory
• Mathematical Statistics
• Network Theory
• Population genetics
• Queueing Theory
• Reliability Theory
• Risk Theory
• Statistical Physics
• Theory of Algorithms
and haphazard other topics. But listing such broad topics doesn't seem very helpful. It would be very useful to refine to a list of 100 more specific topics in what I call the Wikipedia zone. What does this mean? The zone is the interval within the broad-narrow spectrum of academic topics for which one can write a good Wikipedia article. For instance, Stochastic process is too broad whereas Markov chain mixing time is too narrow; neither fits the goal of providing an informative concise, non-technical account of an interesting topic.

Creating such a refined list -- and writing the 100 Wikipedia articles! -- would constitute an extremely useful resource. Incidently, the existing Wikipedia article on applied probability is rather similar to what I have written above, but very few existing "topic articles" hit the style I am envisaging.

It's curious that nothing like the desired "list of 100" exists already. Within the research community, one can find lists of the topics of special sessions at academic conferences (a 2009 example) which reflect current research-level activity, but these tend to be methodology-based and relate to some rather narrow portion of an application topic. Occasional panel reports, such as a 2002 report Current and Emerging Research Opportunities in Probability, attempt a bigger picture, but ultimately give short lists of very broad areas like our 1-7 above, or (see e.g. an overview talk associated with report above) revert to talking about intrinsic mathematical structure.

## 3. Philosophical interpretations of probability

There are two useful online accounts of the standard philosophical views of what probability is; the less technical one is What is Probability? from the Understanding Uncertainty site, and the more technical one is Interpretations of Probability from the Stanford Encyclopedia of Philosophy. Both give the same top-level five-way split:
• Classical probability (equally likely outcomes)
• Logical probability (probability as objective degree of belief)
• Frequency interpretations
• Propensity interpretations (probability inherent in the experimental set up)
• Subjective probability
xxx I find this unhelpful, mostly because it is based upon "iconic" rather than real examples (xxx cross-ref rant against iconic examples). xxx can't fit our 100 list into these categories.

## 5. A Taxonomy of Luck

In Luck: The Brilliant Randomness Of Everyday Life Rescher proposes an explicit taxonomy of luck, as the first 8 items below. I have added 2 more.
• Windfalls or wind thefts
• Unforeseeable lost or gained opportunities
• Accidents
• Narrow escapes or flukish victimizations
• Coincidences (e.g. "being in the wrong place at the wrong time")
• Consequence-laden mistakes in identification or classification
• Fortuitous encounters
• Welcome or unwelcome anomalies (in generally predictable matters)
• Other people's actions having (un)favorable consequences for you
• Conscious risk-taking that works out well or badly
One can criticize these categories as vague and partly overlapping; and as I discusss here Rescher derives these categories by considering hypothetical examples. xxx but work out well on examples from Wiseman (xxx similar paragraph in my The Four Faces of Luck page -- edit). (xxx show data).

## 6. A Taxonomy of Serendipity

In Anatomy of the Unsought Finding. Serendipity: Origin, History, Domains, Traditions, Appearances, Patterns and Programmability Pek Van Andel writes
I collected seventeen ways in which unsought findings have been made.
and gives real historical examples of each, in the contexts of science and invention. Interestingly, he interprets seredipity as unsought discoveries rather than lucky ones. He gives a nicely written conclusion on the theme that the actual process of scientific discovery is less rational than it appears in print, though I suspect working scientists realize this perfectly well.
• Analogy
• One surprising observation
• Repetition of a surprising observation
• Successful error
• From side-effect to main effect
• From by-product to main product ('spin-off')
• Wrong hypothesis
• No hypothesis
• Inversion
• Testing of a popular 'belief'
• Child, student or outsider
• Disturbance
• Scarcity
• Interruption of work
• Playing
• Joke
• Dream or "forgetting-hypothesis"

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

In the areas above, people don't seem to pay much attention to taxonomies. In contrast, everyone who thinks abstractly about uncertainty and risk seems to devise thier own taxonomy ..... Here are a few examples.

## 7. A taxonomy of uncertainty

(xxx identical copy at phil_uncertainty.html).

The Intergovernmental Panel on Climate Change (IPCC) issues periodic reports, widely regarded as the most authoritative analysis of scientific understanding of climate change caused by human activity. Future predictions involve uncertainty, and they want their many authors to be consistent in how they write about uncertainty, so provide a technical document Guidance Notes for Lead Authors of the IPCC Fourth Assessment Report on Addressing Uncertainties from which I have extracted the table below, there labelled "A simple typology of uncertainties".

Type Indicative examples of sources Typical approaches or considerations
Unpredictability Projections of human behaviour not easily amenable to prediction (e.g. evolution of political systems). Chaotic components of complex systems. Use of scenarios spanning a plausible range, clearly stating assumptions, limits considered, and subjective judgments. Ranges from ensembles of model runs.
Structural uncertainty Inadequate models, incomplete or competing conceptual frameworks, lack of agreement on model structure, ambiguous system boundaries or definitions, significant processes or relationships wrongly specified or not considered. Specify assumptions and system definitions clearly, compare models with observations for a range of conditions, assess maturity of the underlying science and degree to which understanding is based on fundamental concepts tested in other areas.
Value uncertainty Missing, inaccurate or non-representative data, inappropriate spatial or temporal resolution, poorly known or changing model parameters. Analysis of statistical properties of sets of values (observations, model ensemble results, etc); bootstrap and hierarchical statistical tests; comparison of models with observations.

## (8) Another taxonomy of uncertainty

Given via an igloo of uncertainty graphic and a taxonomy of uncertainties and decisions graphic from a 2007 paper The ethics of uncertainty: In the light of possible dangers, research becomes a moral duty.

## (9) A taxonomy of information security risk

From a document Risk Taxonomy Technical Standard .
 Loss event frequency Threat event frequency Contact Action Vulnerability Control strength Threat capability Probable loss magnitude Primary loss factors Asset loss factors Threat loss factors Secondary loss factors Operational loss factors External loss factors
xxx cross-ref "risk" as "EV of loss". and use of usual graphic.

## (10) A taxonomy of near- or medium-term global risks

This (very interesting) list of 36 "global risks" is divided into 5 (less interesting?) categories:
• Economic
• Geopolitical
• Environmental
• Societal
• Technological.