Extending Nearly-Linear Models Chiara Corsato, Renato Pelessoni and - - PowerPoint PPT Presentation

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Extending Nearly-Linear Models Chiara Corsato, Renato Pelessoni and - - PowerPoint PPT Presentation

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UNIVERSIT DEGLI STUDI DI TRIESTE Extending Nearly-Linear Models Chiara Corsato, Renato Pelessoni and Paolo Vicig University of Trieste, Italy ISIPTA 2019 Gent July 6, 2019 Outline Motivations Nearly-Linear Models Definitions and

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UNIVERSITÀ DEGLI STUDI DI TRIESTE

Extending Nearly-Linear Models

Chiara Corsato, Renato Pelessoni and Paolo Vicig

University of Trieste, Italy

ISIPTA 2019 Gent July 6, 2019

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Outline

Motivations Nearly-Linear Models

  • Definitions and basic properties
  • Various types of natural extensions

Results postponed to the Poster Session

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Motivations

  • NL Models include several Neighbourhood Models:
  • NL Models may elicit various beliefs, even conflicting ones.
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Nearly-Linear Models

NL Models are simple functions of a given probability P0:

Definition (Corsato, Pelessoni, Vicig, 2019)

µ ∶ A(P) → R is a Nearly-Linear (NL) imprecise probability if

  • µ(∅) = 0, µ(Ω) = 1
  • given P0 on A(P), a ∈ R, b > 0, ∀A ∈ A(P) ∖ {∅,Ω},

µ(A) = min{max{bP0(A) + a,0},1} = max{min{bP0(A) + a,1},0}. We denote µ by NL(a,b).

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Nearly-Linear Models

NL Models are simple functions of a given probability P0:

Definition (Corsato, Pelessoni, Vicig, 2019)

µ ∶ A(P) → R is a Nearly-Linear (NL) imprecise probability if

  • µ(∅) = 0, µ(Ω) = 1
  • given P0 on A(P), a ∈ R, b > 0, ∀A ∈ A(P) ∖ {∅,Ω},

µ(A) = min{max{bP0(A) + a,0},1} = max{min{bP0(A) + a,1},0}. We denote µ by NL(a,b). A NL µ is a linear affine transformation of P0, with barriers.

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Nearly-Linear Models - 2

  • The family of NL imprecise probabilities is self-conjugate: if µ is NL(a,b),

then µc is NL(c,b), with c = 1 − (a + b).

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Nearly-Linear Models - 2

  • The family of NL imprecise probabilities is self-conjugate: if µ is NL(a,b),

then µc is NL(c,b), with c = 1 − (a + b). lower probability P NL(a,b) upper probability P NL(c,b)

Definition

A Nearly-Linear Model is a couple (P,P), where P ∶ A(P) → R is a NL lower probability and P is its conjugate.

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Nearly-Linear Models - 2

  • The family of NL imprecise probabilities is self-conjugate: if µ is NL(a,b),

then µc is NL(c,b), with c = 1 − (a + b). lower probability P NL(a,b) upper probability P NL(c,b)

Definition

A Nearly-Linear Model is a couple (P,P), where P ∶ A(P) → R is a NL lower probability and P is its conjugate.

  • b + 2a ≤ 1

→ P NL(a,b) 2-coherent (minimal consistency property).

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Nearly-Linear Models - 3

We have classified 2-coherent Nearly-Linear Models into 3 subfamilies:

  • 1. Vertical Barrier Model (VBM)
  • 2. Horizontal Barrier Model (HBM)
  • 3. Restricted Range Model (RRM)

and studied their consistency properties (Corsato, Pelessoni, Vicig, 2019).

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Nearly-Linear Models - 3

We have classified 2-coherent Nearly-Linear Models into 3 subfamilies:

  • 1. Vertical Barrier Model (VBM)
  • 2. Horizontal Barrier Model (HBM)
  • 3. Restricted Range Model (RRM)

and studied their consistency properties (Corsato, Pelessoni, Vicig, 2019).

Aim

Find manageable formulae for natural extensions of NL Models.

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Nearly-Linear Models - 3

We have classified 2-coherent Nearly-Linear Models into 3 subfamilies:

  • 1. Vertical Barrier Model (VBM)
  • 2. Horizontal Barrier Model (HBM)
  • 3. Restricted Range Model (RRM)

and studied their consistency properties (Corsato, Pelessoni, Vicig, 2019).

Aim

Find manageable formulae for natural extensions of NL Models.

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Vertical Barrier Model (VBM)

Parameters

a ≤ 0, 0 ≤ a + b ≤ 1, c = 1 − (a + b)(≥ 0)

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Vertical Barrier Model (VBM)

Parameters

a ≤ 0, 0 ≤ a + b ≤ 1, c = 1 − (a + b)(≥ 0) P(A) = max{bP0(A) + a,0}, ∀A ∈ A(P) ∖ {Ω}, P(A) = min{bP0(A) + c,1}, ∀A ∈ A(P) ∖ {∅}, with P(Ω) = 1, P(∅) = 0.

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Vertical Barrier Model (VBM)

Parameters

a ≤ 0, 0 ≤ a + b ≤ 1, c = 1 − (a + b)(≥ 0) P(A) = max{bP0(A) + a,0}, ∀A ∈ A(P) ∖ {Ω}, P(A) = min{bP0(A) + c,1}, ∀A ∈ A(P) ∖ {∅}, with P(Ω) = 1, P(∅) = 0. P is coherent and 2-monotone (P is coherent and 2-alternating).

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VBM and natural extensions - 1

Proposition (VBM as a natural extension)

The lower probability in the VBM expression for P, Q(A) = bP0(A) + a, ∀A ∈ A(P),

  • avoids sure loss;
  • is convex iff b = 1.
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VBM and natural extensions - 1

Proposition (VBM as a natural extension)

The lower probability in the VBM expression for P, Q(A) = bP0(A) + a, ∀A ∈ A(P),

  • avoids sure loss;
  • is convex iff b = 1.

Its natural extension on A(P) is precisely the lower probability P of the VBM itself.

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VBM and natural extensions - 1

Proposition (VBM as a natural extension)

The lower probability in the VBM expression for P, Q(A) = bP0(A) + a, ∀A ∈ A(P),

  • avoids sure loss;
  • is convex iff b = 1.

Its natural extension on A(P) is precisely the lower probability P of the VBM itself. A VBM is a correction of Q via natural extension.

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VBM and natural extensions - 2

P ∶ A(P) → R lower probability of a VBM is coherent and 2-monotone.

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VBM and natural extensions - 2

P ∶ A(P) → R lower probability of a VBM is coherent and 2-monotone.

Proposition (Natural extension of a VBM)

E P0 natural extension of P0 on L(P).

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VBM and natural extensions - 2

P ∶ A(P) → R lower probability of a VBM is coherent and 2-monotone.

Proposition (Natural extension of a VBM)

E P0 natural extension of P0 on L(P).

  • If a < 0, for any X ∈ L(P) define

˜ x = sup {x ∈ R ∶ P0(X > x) ≥ − a b }. Then E(X) = (a + b)˜ x + (1 − (a + b)) inf X − bE P0((˜ x − X)+).

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VBM and natural extensions - 2

P ∶ A(P) → R lower probability of a VBM is coherent and 2-monotone.

Proposition (Natural extension of a VBM)

E P0 natural extension of P0 on L(P).

  • If a < 0, for any X ∈ L(P) define

˜ x = sup {x ∈ R ∶ P0(X > x) ≥ − a b }. Then E(X) = (a + b)˜ x + (1 − (a + b)) inf X − bE P0((˜ x − X)+).

  • If a = 0,

E(X) = (1 − b) inf X + bE P0(X).

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VBM and natural extensions - 2

P ∶ A(P) → R lower probability of a VBM is coherent and 2-monotone.

Proposition (Natural extension of a VBM)

E P0 natural extension of P0 on L(P).

  • If a < 0, for any X ∈ L(P) define

˜ x = sup {x ∈ R ∶ P0(X > x) ≥ − a b }. Then E(X) = (a + b)˜ x + (1 − (a + b)) inf X − bE P0((˜ x − X)+).

  • If a = 0,

E(X) = (1 − b) inf X + bE P0(X). Remark:

  • a < 0 and a + b = 1 → PMM (Walley, 1991)
  • a = 0 → ε-contamination Model (Walley, 1991)
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Horizontal Barrier Model (HBM)

Parameters

b + 2a ≤ 1, a + b > 1, c = 1 − (a + b)(< 0)

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Horizontal Barrier Model (HBM)

Parameters

b + 2a ≤ 1, a + b > 1, c = 1 − (a + b)(< 0) P(A) = max{min{bP0(A) + c,1},0}. A HBM is generally only 2-coherent. It may avoid sure loss or be even coherent.

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A selection of results for HBMs

  • If P is finite, P in a HBM avoids sure loss iff ∑

ω∈P

P(ω) ≥ 1. Then its natural extension on A(P) is E(A) = min{ ∑

ω∈P

P(ω),1}.

  • E is also the natural extension of the probability interval [0,P(ω)]ω∈P
  • → E is 2-monotone
  • → a HBM and a lower-vacuous probability interval avoiding sure loss are

equivalent (Troffaes, de Cooman, 2014).

  • If P is arbitrary, P avoids sure loss iff, for any finite partition P′ coarser than

P, ∑

ω′∈P′ P(ω′) ≥ 1.

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Further (Poster Session) results

  • Natural extensions of coherent HBMs
  • Natural extensions of RRMs avoiding sure loss (further relationships

with probability intervals)

  • Interpretation of a VBM natural extension as a risk measure
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Thank you... ...and see you at the Poster Session!