Extensions of Sets of Markov Operators Under Epistemic Irrelevance - - PowerPoint PPT Presentation

Extensions of Sets of Markov Operators Under Epistemic Irrelevance

Damjan Škulj University of Ljubljana ISIPTA 2019 Ghent, 6 July 2019

Probabilistic models via sets of Markov operators

Markov operators

Definition Let H and K be sets of gambles. A Markov operator is a linear operator T : H → K, such that:

1 if f ≤ g then Tf ≤ Tg; 2 T1X = 1X .

A set of Markov operators T can specify a probabilistic model by requiring:

1 if f is a desirable gamble, then Tf is also desirable for every T ∈ T ; 2 if f is undesirable, then T ∈ T exists such that Tf is undesirable.

A set of (almost) desirable gambles D satisfying the above requirements is said to be generated by T . In general, multiple such sets exist.

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Probabilistic models via sets of Markov operators

Examples

Sets of Markov operators are can be combined to achieve certain properties

• f probabilistic models.

Some motivating examples:

1 Conditional expectation Ep(f |B) of a desirable gambles f is usually

also considered desirable. (If p ∈ M and B = {∅, X}, this model is equivalent to the model given by the credal set M.)

2 Symmetry with respect to elements of Y for the gambles

f : X × Y → R can be modelled by requiring that the set of desirable gambles is generated by the set P(Y) of all operators Tσ of the form [Pσf ](x, y) = f (x, σ(y)), where σ is a permutation on Y.

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Extension of sets of Markov operators

Extension of imprecise probabilistic models

We consider specific question of extending a probabilistic model on X to a lager space, such as X × Y (or X × Y × Z). set gambles desirable gambles

• perators

space X G(X) D(X) T extended space X × Y G(X × Y) D(X × Y) ˜ T =? How can we construct an extension ˜ T of T that generates D(X × Y)? The extension ˜ T depends on the type of extension D(X × Y) of D(X).

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Extension of sets of Markov operators

Example – strong products

Take for example the strong product of credal sets M and N on separate spaces X and Y: set credal sets

• perators

space X, Y M, N T , S extended space X × Y M × N T ⊗ S where T ⊗ S = {T ⊗ S : T ∈ T , S ∈ S} and ⊗ denotes the tensor product of operators. Note: The structure of the model, i.e. independent product, must be pre-specified – not every T ⊗ S-generated credal set is an independent product of its marginals.

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Extensions satisfying epistemic irrelevance

Epistemic irrelevance

A set of desirable gambles D ⊂ G(X × Y) satisfies epistemic irrelevance Y → X if for every X-measurable gamble f the following conditions are equivalent:

f ∈ D; Iyf ∈ D for every y ∈ Y.

Remark A gamble f is X-measurable if f (x, y) = f (x, y′) for every y, y′ ∈ Y. Epistemic irrelevance combines two properties:

symmetry w.r.t. Y; the sum f =

y∈Y Iyf is only desirable if all Iyf are desirable.

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Extensions satisfying epistemic irrelevance

Example – symmetry alone does not imply irrelevance

Let X = Y = {0, 1}. Consider the following gambles: f : Y = 0 Y = 1 X = 0 −1 −1 X = 1 1 1 , which is X-measurable; I(Y =0)f : Y = 0 Y = 1 X = 0 −1 X = 1 1 , I(Y =1)f : Y = 0 Y = 1 X = 0 −1 X = 1 1

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Extensions satisfying epistemic irrelevance

Take two probability mass functions: p1 : Y = 0 Y = 1 X = 0 3/16 3/16 X = 1 2/16 8/16 , p2 : Y = 0 Y = 1 X = 0 3/16 3/16 X = 1 8/16 2/16 , Let M = {p1, p2} and then PM(f ) = 4/16, PM(I(Y =0)f ) = −1/16, PM(I(Y =1)f ) = −1/16. So we have that I(Y =0)f and I(Y =1)f are undesirable, while f = I(Y =0)f + I(Y =1)f is desirable. Thus, even if there is clear symmetry with respect to Y, we do not have epistemic irrelevance.

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Extensions satisfying epistemic irrelevance

Additive independent extension

In addition to symmetry we need a new property to guarantee epistemic irrelevance: Definition (Additive independent extension) A set D is an additive independent extension of {Di ⊂ Gi}i∈I if f ∈ D; f =

i∈I fi, fi ∈ Gi

imply that ∃i ∈ I : fi ∈ Di. A sum of undesirable gambles cannot be desirable.

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Extensions satisfying epistemic irrelevance

Epistemic irrelevant additive independent extension

Let T be a set of Markov operators on G(X). We construct the following set of Markov operators: ˜ T =    ˜ T : ˜ Tf =

• y∈Y

IyTyf (·, y), Ty ∈ T ∀y ∈ Y    . Restricted to G(X|y) that is isomorphic to G(X), ˜ T acts as T . Hence, the generated sets of desirable gambles D(X|y) are all generated by T , yet are not necessarily equal, as T can have multiple generated models. So, we additionally need to require symmetry, which we can do by adding the set of permutation operators.

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Extensions satisfying epistemic irrelevance

Main result

Let D be a set of desirable gambles that is an additive independent extension of {D ∩ G(X|y): y ∈ Y}. Then the following are equivalent:

1 D satisfies epistemic irrelevance Y → X; 2 D is generated by ˜

T ∪ P(Y);

3 two sets of desirable gambles

D(X) : generated by T and D(Y) : generated by P(Y) (only required to be symmetric w.r.t. Y)

exist so that every f ∈ D can be written in the form: f = fY +

• y∈Y

Iyfy, where fY ∈ D(Y) and fy ∈ D(X) for every y ∈ Y.

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Extensions satisfying epistemic irrelevance

Further work

The advantage of operators approach is that the probabilistic model (even imprecise) does not need to be fully specified – we can only have a set of (local) conditional models, which we can easily extend to larger (global) probability spaces. Moreover, additional requirements, such as symmetry, can easily be given in terms of additional sets of operators. This can be useful in models where compatible probabilistic models need to be constructed based on conditional models only. Such examples include stochastic processes, credal networks, probability trees and others. In particular, it would be interesting to impose requirements such as time homogeneity or Markov property in terms of Markov operators.

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