← IPML LabsLEARNING UNDER PARTIAL IGNORANCE

POSSIBILITY THEORY / INTERACTIVE LAB

What is possible?
What is supported?

Explore uncertainty when you can rank plausible outcomes, but cannot justify a single probability distribution.

01 / POSSIBILITY DISTRIBUTION

Grade compatibility with your knowledge

Exactly one of A, B, C is the true outcome. The grade π(ω) describes how compatible each outcome is with the available information.

π = 1 means fully possible; π = 0 means ruled out. Several outcomes may be fully possible. These grades need not sum to 1.

Possibility is not probability

Possibility πCandidate probability p
A possibility of 1 does not mean the outcome is certain. It means your information does not rule it out.
Explore nested α-cuts

The sets Cα = {ω : π(ω) ≥ α} shrink as α increases. C₀ = Ω with this ≥ convention, even when some grades are zero. These are compatibility levels, not automatically probability confidence levels.

02 / POSSIBILITY AND NECESSITY

Select an event E

An event is a set of outcomes. Select its members below; the empty event and Ω are allowed.

NECESSITY · LOWER BOUND

POSSIBILITY · UPPER BOUND

CANDIDATE PROBABILITY

Π(E) = maxω∈E π(ω)
Take the most possible member; max ∅ = 0.
N(E) = 1 − Π(Eᶜ)
Support for E comes from ruling out its complement.
Why use max instead of addition?

Possibility is maxitive: Π(E ∪ F) = max(Π(E), Π(F)). Necessity is minitive: N(E ∩ F) = min(N(E), N(F)). Probability adds across disjoint events. In general Π(E ∩ F) ≤ min(Π(E), Π(F)); equality is not an axiom.

03 / CONNECTION TO IMPRECISE PROBABILITY

One possibility model, many probabilities

In the numerical upper-probability interpretation, π encodes a set of probability distributions:

M(π) = {p ∈ Δ₃ : P(E) ≤ Π(E) for every E}
Compatible distributions M(π)Candidate p (red if outside)

Each point is a probability distribution. A normalized possibility model always has a nonempty credal set. The region may be a polygon, a line, or a single point.

Test a candidate probability

Move p(A) or p(B); p(C) is the remainder. The controls keep all probabilities nonnegative and summing to 1.

N(E) = minp∈M(π) P(E)
Π(E) = maxp∈M(π) P(E)

These bounds are tight for each event. The distributions attaining different event bounds can differ. Checking only singleton bounds p(ω) ≤ π(ω) is insufficient.

All eight events, checked together

A candidate is compatible only when it respects every row. Select a row to explore its event.

Event EN(E)P(E)Π(E)Check

04 / WHAT THIS REPRESENTATION BUYS YOU

Leave room for what you do not know

Ignorance is different from a uniform belief

π = (1, 1, 1) leaves every distribution in the simplex possible. Every nonempty proper event has bounds [0, 1]. A uniform probability instead commits to p = (⅓, ⅓, ⅓).

Uniform probability can be a deliberate prior choice. It adds assumptions beyond simply leaving all outcomes open.

Useful when information is incomplete

  • Keep several alternatives fully possible without dividing a fixed probability budget.
  • Encode “A is more plausible than B” before precise frequencies are available.
  • Represent set-valued knowledge and graded exclusions in one model.
  • Compute event measures using maxima and complements; numerical grades also give coherent probability bounds.
A benefit with a tradeoff. Possibility models have a nested (consonant) structure. They are a useful, compact subclass of imprecise probability, but cannot represent every credal set. A precise probability model is useful when its quantitative commitments are warranted.
Ordinal rankings versus numerical probability bounds

Qualitative possibility can use an ordered scale to express relative plausibility. Any strictly increasing rescaling preserves the ranking and max operation, but changes numerical probability bounds. This lab uses numerical grades in [0, 1] and N(E) = 1 − Π(E) throughout. The probabilistic interpretation is an additional numerical interpretation, not a frequency meaning inherent in every possibility ranking.

Connection to belief functions: nested focal sets

Sort the distinct positive grades λ₁ > ⋯ > λₖ > λₖ₊₁ = 0. The nested sets Fⱼ = {ω : π(ω) ≥ λⱼ} receive masses m(Fⱼ) = λⱼ − λⱼ₊₁. Their belief and plausibility functions are N and Π.

This is why the induced credal model is called consonant. Explore ties and zero grades using the presets.

Three experiments for students

  1. Choose Complete ignorance. Select {A} and move the candidate probability. Why can π(A) = 1 coexist with p(A) = 0?
  2. Choose Partial knowledge and select {A, B}. Lower π(C). Explain why necessity increases even though possibility stays at 1.
  3. Choose Singleton trap. All singleton upper bounds pass, yet the candidate is outside M(π). Identify the event that excludes it.