Introduction to the theory of imprecise probability Erik Quaeghebeur - - PowerPoint PPT Presentation

Introduction to the theory of imprecise probability

Erik Quaeghebeur

TU Delft, the Netherlands

UTOPIAE Training School 2018, Durham, England

Why would you want your probability to be imprecise?

2

versus

3

Uncertainty about outcome of. . .

versus

Agents (Gamblers)

Wiske Yoko Tsuno

4

Assessment (gambles accepted)

1 if ⊗ 5 if ⊗4 if + 1 if

Agents (Gamblers)

Wiske Yoko Tsuno

5

Natural extension (λ, µ ⊙ 0)

λ ⊗ 5λ +µ + µ ⊗4λ + λ +µ + µ

Rational agents

Wiske Yoko Tsuno

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Natural extension (λ, µ ⊙ 0)

λ ⊗ 5λ +µ + µ

COHERENCE

⊗4λ + λ +µ + µ

Rational agents

Wiske Yoko Tsuno

7

Assessment (gambles accepted)

⊗ 5 , ⊗4 +

Agents (Gamblers)

Heroine pool

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Natural extension (λ, µ ⊙ 0)

(λW ⊗ 4λY + µ ) + (⊗5λW + λY + µ )

Irrational agents

Heroine pool

9

Natural extension (λ, µ ⊙ 0)

(λW ⊗ 4λY + µ ) + (⊗5λW + λY + µ )

SURE LOSS!

Irrational agents

Heroine pool

10

Assessment (gambles accepted)

∅

Agents (Gamblers)

Heroine pool

11

Natural extension (λ, µ ⊙ 0)

µ + µ

Rational agents

Heroine pool

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Natural extension (λ, µ ⊙ 0)

µ + µ

VACUOUS

Rational agents

Heroine pool

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Basic concepts

◮ Agent reasoning about experiment with uncertain outcome ◮ Possibility space 𝒴 of outcomes ◮ Gambles are real-valued functions of the outcomes;

ℒ = 𝒴 ⊃ R (ℒ is assumed to be a linear space)

◮ Assessment is a description of a set of acceptable gambles ◮ Natural extension of an assessment is

the set of all acceptable gambles implied by the agent’s rationality criteria (and other assumptions)

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Coherence, the classical rationality criteria

Constructive Positive scaling If 𝑔 is acceptable and λ > 0, then λ𝑔 is acceptable. Addition If 𝑔 and 𝑕 are acceptable, then 𝑔 + 𝑕 is acceptable. Background Accepting gain If 𝑔 is nonnegative for all outcomes, then 𝑔 is acceptable. Avoiding sure loss If 𝑕 is negative for all outcomes, then 𝑕 is not acceptable.

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Where are the imprecise probabilities I came here for!!!

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Previsions/Expectations are prices for gambles

◮ ‘Prevision’ and ‘Expectation’ are synomyms ◮ Prices are real values interpreted as constant gambles ◮ Lower prevision 𝑄(𝑔) is the

supremum acceptable buying price of 𝑔: 𝑄(𝑔) = sup¶ν ∈ R : 𝑔 ⊗ ν is acceptable♢ Upper prevision 𝑄(𝑔) is the infimum acceptable selling price of 𝑔

◮ Conjugacy of coherent lower and upper previsions:

𝑄(𝑔) = ⊗𝑄(⊗𝑔)

◮ If 𝑄(𝑔) = 𝑄(𝑔), then 𝑄(𝑔) = 𝑄(𝑔) is the prevision of 𝑔

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Probabilities are previsions of indicator gambles

◮ Event 𝐵 is a subset of 𝒴 ◮ Indicator gamble

1A(𝑦) =

∮︂

1, 𝑦 ∈ 𝐵, 0, 𝑦 ̸∈ 𝐵

◮ Lower probability 𝑄(𝐵) = 𝑄(1A)

Upper probability 𝑄(𝐵) = 𝑄(1A)

◮ Conjugacy of coherent lower and upper probabilities (𝐵c = 𝒴 \ 𝐵):

𝑄(𝐵) = 1 ⊗ 𝑄(𝐵c)

◮ If 𝑄(𝐵) = 𝑄(𝐵), then 𝑄(𝐵) = 𝑄(𝐵) is the probability of 𝐵

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Natural extension (λ, µ ⊙ 0)

(λ + µ ) +(⊗5λ + µ ) (⊗4λ + µ ) (λ + µ ) (λW ⊗ 4λY + µ ) + (⊗5λW + λY + µ ) µ + µ

Agents

Wiske Yoko Tsuno Irrational pool Rational pool

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Wiske’s lower probability that Belgium will win

𝑄( ) = sup

∮︂

ν :

⎟

1 ⊗ ν ⊗ν

⟨

=

⎟

λ + µ ⊗5λ + µ

⟨

, µ ⊙ 0, λ ⊙ 0

⨀︁

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Wiske’s lower probability that Belgium will win

𝑄( ) = sup

∮︂

ν :

⎟

1 ⊗ ν ⊗ν

⟨

=

⎟

λ + µ ⊗5λ + µ

⟨

, µ ⊙ 0, λ ⊙ 0

⨀︁

= sup

{︁5λ + µ

: 1 ⊗ 5λ + µ = λ + µ , µ ⊙ 0, λ ⊙ 0

⟨

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Wiske’s lower probability that Belgium will win

𝑄( ) = sup

∮︂

ν :

⎟

1 ⊗ ν ⊗ν

⟨

=

⎟

λ + µ ⊗5λ + µ

⟨

, µ ⊙ 0, λ ⊙ 0

⨀︁

= sup

{︁5λ + µ

: 1 ⊗ 5λ + µ = λ + µ , µ ⊙ 0, λ ⊙ 0

⟨

= sup

⎭

5λ + µ : λ = 1 6(1 + µ ⊗ µ ), µ ⊙ 0, λ ⊙ 0

⎨

22

Wiske’s lower probability that Belgium will win

𝑄( ) = sup

∮︂

ν :

⎟

1 ⊗ ν ⊗ν

⟨

=

⎟

λ + µ ⊗5λ + µ

⟨

, µ ⊙ 0, λ ⊙ 0

⨀︁

= sup

{︁5λ + µ

: 1 ⊗ 5λ + µ = λ + µ , µ ⊙ 0, λ ⊙ 0

⟨

= sup

⎭

5λ + µ : λ = 1 6(1 + µ ⊗ µ ), µ ⊙ 0, λ ⊙ 0

⎨

= sup

⎭5

6 ⊗ 1 6µ ⊗ 5 6µ : µ ⊙ 0

⎨

= 5 6

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P( ), P( ) P( ), P( )

5 6, 1

0, 1

6

0, 1

5 4 5, 1

+∞, ⊗∞ + ∞, ⊗∞ 0, 1 0, 1

Agents

Wiske Yoko Tsuno Irrational pool Rational pool

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Assessments of lower previsions

◮ Assume a lower prevision 𝑄 with values assessed for a set of

gambles 𝒧

◮ How can we apply the theory we have seen? ◮ Translate the lower prevision 𝑄(𝑔) for a gamble 𝑔 ∈ 𝒧 into a set

¶𝑔 ⊗ 𝑄(𝑔) + 𝜁 : 𝜁 > 0♢

• f acceptable gambles

◮ The gambles 𝑔 ⊗ 𝑄(𝑔) are called marginal gambles

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Expressions for assessments of lower previsions

Avoiding sure loss sup

x∈𝒴 n

∑︂

k=1

(𝑔k(𝑦) ⊗ 𝑄(𝑔k)) ⊙ 0 for all 𝑜 ⊙ 0 and 𝑔k ∈ 𝒧

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Expressions for assessments of lower previsions

Avoiding sure loss sup

x∈𝒴 n

∑︂

k=1

(𝑔k(𝑦) ⊗ 𝑄(𝑔k)) ⊙ 0 for all 𝑜 ⊙ 0 and 𝑔k ∈ 𝒧 Coherence sup

x∈𝒴

⎠ n ∑︂

k=1

(𝑔k(𝑦) ⊗ 𝑄(𝑔k)) ⊗ 𝑛(𝑔0 ⊗ 𝑄(𝑔0))

⎜

⊙ 0 for all 𝑜, 𝑛 ⊙ 0 and 𝑔k ∈ 𝒧

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Expressions for assessments of lower previsions

Avoiding sure loss sup

x∈𝒴 n

∑︂

k=1

(𝑔k(𝑦) ⊗ 𝑄(𝑔k)) ⊙ 0 for all 𝑜 ⊙ 0 and 𝑔k ∈ 𝒧 Coherence sup

x∈𝒴

⎠ n ∑︂

k=1

(𝑔k(𝑦) ⊗ 𝑄(𝑔k)) ⊗ 𝑛(𝑔0 ⊗ 𝑄(𝑔0))

⎜

⊙ 0 for all 𝑜, 𝑛 ⊙ 0 and 𝑔k ∈ 𝒧 Natural extension 𝐹(𝑔) = sup

∮︂

inf

x∈𝒴

⎭

𝑔(𝑦)⊗

n

∑︂

k=1

λk

(︁𝑔k(𝑦)⊗𝑄(𝑔k) )︁⎨

: 𝑜 ⊙ 0, 𝑔k ∈ 𝒧, λk > 0

⨀︁

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Does it really have to be so involved?

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Lower previsions on linear spaces

If the lower prevision 𝑄 is defined for all gambles in a linear space ℒ, the coherence criteria simplify: Accepting sure gains 𝑄(𝑔) ⊙ inf 𝑔 for all 𝑔 ∈ ℒ Super-linearity 𝑄(𝑔 + 𝑕) ⊙ 𝑄(𝑔) + 𝑄(𝑕) for all 𝑔, 𝑕 ∈ ℒ Positive homogeneity 𝑄(λ𝑔) = λ𝑄(𝑔) for all 𝑔 ∈ ℒ and λ > 0

30

Upper previsions on linear spaces

If the upper prevision 𝑄 is defined for all gambles in a linear space ℒ, the coherence criteria simplify: Accepting sure gains 𝑄(𝑔) ⊘ sup 𝑔 for all 𝑔 ∈ ℒ Sub-linearity 𝑄(𝑔 + 𝑕) ⊘ 𝑄(𝑔) + 𝑄(𝑕) for all 𝑔, 𝑕 ∈ ℒ Positive homogeneity 𝑄(λ𝑔) = λ𝑄(𝑔) for all 𝑔 ∈ ℒ and λ > 0

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Coherent lower & upper previsions

For a coherent lower prevision 𝑄 and its conjugate upper prevision 𝑄 many useful properties can be derived; we present a few: Upper dominates lower 𝑄(𝑔) ⊙ 𝑄(𝑔) for all 𝑔 ∈ ℒ Constants 𝑄(µ) = µ for all µ ∈ R Constant additivity 𝑄(𝑔 + µ) = 𝑄(𝑔) + µ for all 𝑔 ∈ ℒ and µ ∈ R Gamble dominance if 𝑔 ⊙ 𝑕 + µ then 𝑄(𝑔) ⊙ 𝑄(𝑕) + µ for all 𝑔, 𝑕 ∈ ℒ and µ ∈ R Mixed sub/super-additivity 𝑄(𝑔 + 𝑕) ⊘ 𝑄𝑔 + 𝑄(𝑕) ⊘ 𝑄(𝑔 + 𝑕) for all 𝑔, 𝑕 ∈ ℒ

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I heard that imprecise probabilities are just sets of probabilities?

33

The probability simplex

𝑞 = (𝑞 , 𝑞 )

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The probability simplex

(1

2, 1 2)

(1, 0) (0, 1)

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The probability simplex

1 𝑄p( ) = 0 ≤ 𝑞 + 1 ≤ 𝑞

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The probability simplex

1 𝑄p( ) = 0 ≤ 𝑞 + 1 ≤ 𝑞 𝑄 Y( ) = 4

5

(0, 4

5)

ℳY

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The probability simplex

1 𝑄p( ) = 0 ≤ 𝑞 + 1 ≤ 𝑞 𝑄 Y( ) = 4

5

(0, 4

5)

ℳY

CREDAL SET

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From lower previsions to credal sets

◮ The prevision of a gamble

is a linear function over the probability simplex

◮ The lower prevision of a gamble

can be seen as bounding the prevision of that gamble so constraining the possible probability mass functions

◮ A lower prevision corresponds to a set of constraints,

defining a credal set (closed convex set) ℳ = ¶𝑞 : 𝑄p(𝑔) ⊙ 𝑄(𝑔) for all 𝑔 ∈ 𝒧♢

◮ All this generalizes to infinite 𝒴

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A larger probability simplex

P(enalties)

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A larger probability simplex

P(enalties) 𝑄( ) = 𝑄(P) 𝑄( ) = 𝑄(P)

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A larger probability simplex

P(enalties) 𝑄( ⊗ P) = 0 𝑄( ⊗ P) = 0 ℳPool

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A larger probability simplex

P(enalties) (1, 0, 0) (0, 1, 0) (1

3, 1 3, 1 3)

ℳPool

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A larger probability simplex

P(enalties) (1, 0, 0) (0, 1, 0) (1

3, 1 3, 1 3)

ℳPool 𝑄(P) = 1

3

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From credal sets to lower previsions

◮ A credal set is a closed convex set of probability mass functions

(or more generally, previsions)

◮ A credal set is determined completely by its set of extreme points ℳ* ◮ A nonempty credal set is equivalent to a coherent lower prevision

Lower envelope theorem 𝑄(𝑔) = min¶𝑄p(𝑔) : 𝑞 ∈ ℳ♢ = min¶𝑄p(𝑔) : 𝑞 ∈ ℳ*♢

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Where are the conditional models? We need them to learn!

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Basics of conditioning and updating

◮ Conditioning acceptable gambles is done by restriction to the

subspace of gambles that are zero outside the conditioning event

◮ Conditioning lower previsions is a form of natural extension

𝐹(𝑔 ♣ 𝐵) =

∮︂

infx∈A 𝑔(𝑦) if 𝑄(𝐵) = 0, max¶µ ∈ R : 𝑄(1A(𝑔 ⊗ µ))♢ if 𝑄(𝐵) > 0

◮ Conditional credal set =

credal set of conditional probability mass functions ℳ♣𝐵 =

∮︂

whole simplex if ∃𝑞 ∈ ℳ : 𝑄p(𝐵) = 0,

{︁𝑞(≤ ♣ 𝐵) : 𝑞 ∈ ℳ ⟨

if ∀𝑞 ∈ ℳ : 𝑄p(𝐵) > 0

◮ Natural extension often gives vacuous conditionals;

regular extension is a less imprecise updating rule: it removes those 𝑞 such that 𝑄p(𝐵) = 0 from ℳ

47

Conditioning using natural extension

P(enalties) (1, 0, 0) (0, 1, 0) (1

3, 1 3, 1 3)

ℳPool ℳPool♣¶ , 𝑄♢

48

Conditioning using regular extension

P(enalties) (0, 1, 0) (1

3, 1 3, 1 3)

ℳ′

Pool

ℳ′

Pool♣¶

, 𝑄♢

49

What about imprecise probabilities

• n continuous spaces?

50

Too few remarks about imprecise probabilities on continuous spaces

◮ Mostly defined using credal sets

• f parametric distributions

where the parameters vary in a set

◮ Also common:

defined using probability mass assignments to subsets of the space

◮ Examples:

◮ Imprecise Dirichlet model ◮ P-boxes ◮ lower density functions

◮ Calculating lower and upper previsions (natural extension)

can easily become difficult optimization problems

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Nothing in this lecture had anything to do with aerospace engineering!

52

53

𝑌 ⊗⃗ 1Y cos ψ ⊗⃗ 1Z sin ψ Ω𝑠 + 𝑉r⊤ 𝑉r⊥ φ α 𝜄

54

𝑎 𝑍 𝑆 ψ ⊗⃗ 1Y cos ψ ⊗⃗ 1Z sin ψ 𝑠 𝑎 ⃗ 1X cos β ⊗⃗ 1Y sin β ℎ0 𝑉∞ 𝑉 ℎ 𝑌 𝑍 ⃗ 𝑉 β

55

Relationship between turbulence and angle of attack

𝑉r⊥(1 + δ) φ α 𝜄 αδ = α ⊗ α0 = atan ((1 + δ) tan φ0) ⊗ φ0 δ = tan(φ0 + αδ) ⊗ tan φ0 tan φ0

56

Is this all you have?

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