Dilation and Asymmetric Relevance Gregory Wheeler A. Paul Pedersen - - PowerPoint PPT Presentation

Dilation and Asymmetric Relevance

Gregory Wheeler

• A. Paul Pedersen

HMI

Human & Machine Intelligence

Frankfurt School of Finance & Management

This paper repairs characterization results in (Pedersen and Wheeler 2014; Pedersen and Wheeler 2015).

Gregory Wheeler ¨ 3

Dilation

Let B be a positive measurable partition of Ω. Say that B dilates A if each B P B: PpA | Bq ă PpAq ď PpAq ă PpA | Bq. In other words, dilates A just in case the closed interval P A P A is contained within the

• peninterval P A

B P A B foreachB .

Gregory Wheeler ¨ 4

Dilation

Let B be a positive measurable partition of Ω. Say that B dilates A if each B P B: PpA | Bq ă PpAq ď PpAq ă PpA | Bq. In other words, B dilates A just in case the closed interval “ PpAq, PpAq ‰ is contained within the

• peninterval

` PpA | Bq, PpA | Bq ˘ foreachB P B.

Gregory Wheeler ¨ 4

Dependence

Given a probability function p on algebra A and events A, B P A, define: SppA, Bq :“ $ ’ & ’ % ppA X Bq ppAqppBq if ppAqppBq ą 0; 1

• therwise.

Thus the quantity Sp is an index of deviation from stochastic independence between events.

Gregory Wheeler ¨ 5

Neighborhoods

Given lower probability space pΩ, A, P, Pq, events A, B P A with PpBq ą 0, and ϵ ą 0, define: PpA | B, ϵq :“ tp P P : |ppA | Bq ´ PpA | Bq| ă ϵu; PpA | B, ϵq :“ tp P P : |ppA | Bq ´ PpA | Bq| ă ϵu. Call the sets A B and A B lower and upper neighborhoods of A conditional on B, respectively, with radius .

Gregory Wheeler ¨ 6

Neighborhoods

Given lower probability space pΩ, A, P, Pq, events A, B P A with PpBq ą 0, and ϵ ą 0, define: PpA | B, ϵq :“ tp P P : |ppA | Bq ´ PpA | Bq| ă ϵu; PpA | B, ϵq :“ tp P P : |ppA | Bq ´ PpA | Bq| ă ϵu. Call the sets PpA | B, ϵq and PpA | B, ϵq lower and upper neighborhoods of A conditional on B, respectively, with radius ϵ.

Gregory Wheeler ¨ 6

Characterization

Example by Michael Nielsen and Rush Stewart

Corollary5.2of(PedersenandWheeler2014)

B dilates A just in case there is pϵBqBPB P RB

` such

that PpA | B, ϵBq Ď S´pA, Bq and PpA | B, ϵBq Ď S`pA, Bq.

Gregory Wheeler ¨ 7

Characterization

. 2 . 4 . 6 . 8 1 0.2 0.4 0.6 0.8 1

Example by Michael Nielsen and Rush Stewart

Corollary5.2of(PedersenandWheeler2014)

B dilates A just in case there is pϵBqBPB P RB

` such

that PpA | B, ϵBq Ď S´pA, Bq and PpA | B, ϵBq Ď S`pA, Bq.

Gregory Wheeler ¨ 7

Relevance

Sp

SppA, Bq :“ ppA X Bq ppAqppBq S A B p Sp A B 1 S A B p Sp A B 1 I A B p Sp A B 1

Sp and Sp

SppA, Bq :“ ppA X Bq PpAqppBq and SppA, Bq :“ ppA X Bq PpAqppBq S A B p Sp A B 1 S A B p Sp A B 1

Gregory Wheeler ¨ 8

Relevance

Sp

SppA, Bq :“ ppA X Bq ppAqppBq S`

P pA, Bq :“ tp P P : SppA, Bq ą 1u;

S´

P pA, Bq :“ tp P P : SppA, Bq ă 1u;

IPpA, Bq :“ tp P P : SppA, Bq “ 1u.

Sp and Sp

SppA, Bq :“ ppA X Bq PpAqppBq and SppA, Bq :“ ppA X Bq PpAqppBq S

` P pA, Bq :“ tp P P : SppA, Bq ą 1u;

S´

P pA, Bq :“ tp P P : SppA, Bq ă 1u.

Gregory Wheeler ¨ 8

Theorem

Let A be an event and B “ pBiqiPI be a positive measurable partition for a given set of probability functions P over an algebra. The following statements are equivalent (i) B dilates A; (ii) There exists ϵ ą 0 such that for every i P I: PpA | Bi, ϵq Ď S´

P pA, Biq and PpH | Bi, ϵq Ď S ` P pA, Biq

Gregory Wheeler ¨ 9

References

Pedersen, A. P. and G. Wheeler (2014). Demystifying dilation. Erkenntnis 79(6), 1305–1342. Pedersen, A. P. and G. Wheeler (2015). Dilation, disintegrations, and delayed decisions. In Proceedings of the 9th Symposium on Imprecise Probabilities and Their Applications (ISIPTA), Pescara, Italy, pp. 227–236. Gregory Wheeler ¨ 10