Upper and lower probabilities

Key takeaways

Upper and lower probabilities belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Upper and lower probabilities to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Upper and lower probabilities from memory before moving on to harder problems.

Reference excerpt

Upper and lower probabilities are representations of imprecise probability. Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, this method uses two numbers: the event's upper probability and the event's lower probability. Because frequentist statistics disallow metaprobabilities, frequentists have had to propose new solutions. Cedric Smith and Arthur Dempster each developed a theory of upper and lower probabilities. Glenn Shafer developed Dempster's theory further, now known as Dempster–Shafer theory or Choquet (1953). More precisely, in the work of these authors, one considers in a power set, P ( S ) {\displaystyle P(S)\,\!} , a mass function m : P ( S ) → R {\displaystyle m:P(S)\rightarrow R} satisfying the conditions

m ( ∅ ) = 0 ; m ( A ) ≥ 0 ; ∑ A ∈ P ( S ) m ( A ) = 1. {\displaystyle m(\varnothing )=0\,\,\,\,\,\,\!;\,\,\,\,\,\,m(A)\geq 0\,\,\,\,\,\,\!;\,\,\,\,\,\,\sum _{A\in P(S)}m(A)=1.\,\!}

In turn, a mass is associated with two non-additive continuous measures called belief and plausibility, defined as follows:

bel ⁡ ( A ) = ∑ B ∣ B ⊆ A m ( B ) ; pl ⁡ ( A ) = ∑ B ∣ B ∩ A ≠ ∅ m ( B ) {\displaystyle \operatorname {bel} (A)=\sum _{B\mid B\subseteq A}m(B)\,\,\,\,;\,\,\,\,\operatorname {pl} (A)=\sum _{B\mid B\cap A\neq \varnothing }m(B)}

In the case where S {\displaystyle S} is infinite there can be bel {\displaystyle \operatorname {bel} } such that there is no associated mass function. See p. 36 of Halpern (2003). Probability measures are a special case of belief functions in which the mass function only assigns positive mass to the event space's singletons. A different notion of upper and lower probabilities is obtained by the lower and upper envelopes obtained from a class C of probability distributions by setting

e n v 1 ⁡ ( A ) = inf p ∈ C p ( A ) ; e n v 2 ⁡ ( A ) = sup p ∈ C p ( A ) {\displaystyle \operatorname {env_{1}} (A)=\inf _{p\in C}p(A)\,\,\,\,;\,\,\,\,\operatorname {env_{2}} (A)=\sup _{p\in C}p(A)}

The upper and lower probabilities also relate to probabilistic logic: see Gerla (1994). Observe also that a necessity measure can be seen as a lower probability, and a possibility measure as an upper probability.

See also Possibility theory Fuzzy measure theory Interval finite element Probability bounds analysis

References Choquet, G. (1953). "Theory of Capacities". Annales de l'Institut Fourier. 5: 131–295. doi:10.5802/aif.53. Gerla, G. (1994). "Inferences in Probability Logic". Artificial Intelligence. 70 (1–2): 33–52. doi:10.1016/0004-3702(94)90102-3. Halpern, J. Y. (2003). Reasoning about Uncertainty. MIT Press. ISBN 978-0-262-08320-1. Halpern, J. Y.; Fagin, R. (1992). "Two views of belief: Belief as generalized probability and belief as evidence". Artificial Intelligence. 54 (3): 275–317. CiteSeerX 10.1.1.70.6130. doi:10.1016/0004-3702(92)90048-3. S2CID 11339219. {{cite journal}}: Cite uses deprecated parameter |citeseerx= (help) Huber, P. J. (1980). Robust Statistics. New York: Wiley. ISBN 978-0-471-41805-4. Saffiotti, A. (1992). "A Belief-Function Logic". Procs of the 10h AAAI Conference. San Jose, CA. pp. 642–647. ISBN 978-0-262-51063-9.{{cite book}}: CS1 maint: location missing publisher (link) Shafer, G. (1976). A Mathematical Theory of Evidence. Princeton: Princeton University Press. ISBN 978-0-691-08175-5. Walley, P.; Fine, T. L. (1982). "Towards a frequentist theory of upper and lower probability". Annals of Statistics. 10 (3): 741–761. doi:10.1214/aos/1176345868. JSTOR 2240901.

Worked examples

Example 1 — a first encounter with Upper and lower probabilities

Start with the simplest possible case. Write down what Upper and lower probabilities claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Upper and lower probabilities before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Upper and lower probabilities ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Upper and lower probabilities

In research: Upper and lower probabilities appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Upper and lower probabilities in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Upper and lower probabilities is common in secondary-school and first-year university syllabi. It links to neighbouring topics Dempster–Shafer theory, Exotic probabilities, Probability bounds analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Upper and lower probabilities outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Upper and lower probabilities in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Upper and lower probabilities means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Upper and lower probabilities out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Upper and lower probabilities in simple terms?

Upper and lower probabilities are representations of imprecise probability. Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, this method uses two numbers: the event's upper probability and the event's lower probability.

Why does Upper and lower probabilities matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Upper and lower probabilities?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Upper and lower probabilities.