Enrique Miranda, Ignacio Montes, Sbastien Destercke Univ. of Oviedo - PowerPoint PPT Presentation

A Unifying Frame for Neighbourhood and Distortion Models Enrique Miranda, Ignacio Montes, Sébastien Destercke Univ. of Oviedo (Spain) and Univ. Téchnologique de Compiègne (France) (mirandaenrique,imontes)@uniovi.es; sebastien.destercke@hds.utc.fr ISIPTA’2019, Gent

Comparison of distortion What people expect to find when they models Miranda, Montes, Destercke come to Oviedo... Introduction The models Comparison Conclusions 1/14 Forward

Comparison of distortion ...and what they get (to their horror) models Miranda, Montes, Destercke Introduction The models Comparison Conclusions 2/14 Back Forward

Comparison of distortion Summary models Miranda, Montes, Destercke Introduction The models Comparison Conclusions Introduction 1 The models 2 Comparison 3 Conclusions 4 3/14 Back Forward

Comparison of distortion Summary of the paper models Miranda, Montes, Destercke Introduction The models One family of imprecise probability models are those that Comparison arise from distorting somehow a fixed probability measure Conclusions P 0 by some factor δ > 0, representing: the amount of contaminated data; a taxation from the house; the distance from the original model we are sensitive to; . . . ◮ The goal of the paper is to compare a number of possi- ble distortion models. Here, we consider a finite space X and assume that ∀ x ∈ X P 0 ( { x } ) > 0 and that δ > 0 is small enough (but this can be generalised). 4/14 Back Forward

Comparison of distortion Does anybody care? models Miranda, Montes, Destercke Introduction The models Comparison Conclusions 5/14 Back Forward

Comparison of distortion Does anybody care? models Miranda, Montes, Destercke Introduction The models Comparison Utkin Conclusions Bronevich Walley Distortion Chateauneuf Seidenfeld Huber Pelessoni Vicig 5/14 Back Forward

Comparison of distortion Examples of distortion models (I) models Miranda, Montes, Destercke Introduction Pari-mutuel model 1 The models Comparison P PMM ( A ) = max { 0 , ( 1 + δ ) P 0 ( A ) − δ } . Conclusions Linear-vacuous mixture 2 P LV ( X ) = 1 , P LV ( A ) = ( 1 − δ ) P 0 ( A ) ∀ A � = X . Constant odds ratio on gambles 3 P COR ( f ) is the unique solution of ( 1 − δ ) P 0 (( f − P COR ( f )) + ) = P 0 (( f − P COR ( f )) − ) . Constant odds ratio on events 4 Q COR ( A ) = ( 1 − δ ) P 0 ( A ) 1 − δ P 0 ( A ) . 6/14 Back Forward

Comparison of distortion Examples of distortion models (II) models Miranda, Montes, Destercke Introduction Given a distance d , we can consider the credal set The models Comparison M ( P 0 , d , δ ) = { P ∈ P ( X ) | d ( P , P 0 ) ≤ δ } Conclusions and its lower envelope P d . In this way we can consider: Total variation 5 d TV ( P , Q ) = sup | P ( A ) − Q ( A ) | . A ⊂X Kolmogorov 6 d K ( P , Q ) = sup | F P ( x ) − F Q ( x ) | , x ∈X assuming X totally ordered. 7/14 Back Forward

Comparison of distortion Distortion in terms of P or from a ball? models Miranda, Montes, Destercke Introduction The first examples can also be obtained as envelopes of The models neighbourhoods induced by some d : Comparison Conclusions Q ( A ) − P ( A ) P PMM : d PMM ( P , Q ) = max 1 − Q ( A ) . 1 A � = X Q ( A ) − P ( A ) P LV : d LV ( P , Q ) = max . 2 Q ( A ) A � = ∅ � � 1 − P ( A ) · Q ( B ) P COR : d COR ( P , Q ) = max . 3 P ( B ) · Q ( A ) A , B � = ∅ Q ( A c ) � � 1 − P ( A ) Q COR : d ′ COR ( P , Q ) = max . 4 P ( A c ) Q ( A ) A � = X , ∅ In fact, something similar applies to what are called distor- tion models, given by f ( P 0 ) for some f . 8/14 8/14 Back Forward

Comparison of distortion Comparison between the models models Miranda, Montes, Destercke Introduction The models Comparison We have compared the different examples according to the Conclusions following criteria: ◮ How large is the credal set obtained when distorting P 0 by some fixed factor δ . ◮ The number of extreme points of this credal set. ◮ The properties of the associated coherent lower proba- bility. ◮ The properties of the distorting function d . 9/14 9/14 Back Forward

Comparison of distortion Amount of imprecision models Miranda, Montes, Destercke Introduction If we fix P 0 and δ , we can compare the amount of impreci- The models sion between the different neighbourhood models: Comparison Conclusions M ( P K ) M ( P TV ) M ( P PMM ) M ( P LV ) M ( Q COR ) M ( P COR ) Here, an arrow between two nodes means that parent in- cludes the child. 10/14 10/14 Back Forward

Comparison of distortion Properties of the lower probabilities models Miranda, Montes, Destercke Introduction The models Comparison Model 2-monotone ∞ -monotone Prob. interval Conclusions P PMM YES NO YES P LV YES YES YES P TV YES NO NO P COR NO NO NO Q COR YES YES NO P K YES YES NO Thus, the most precise model, that was the constant odds ratio (on gambles), is the one with worse properties, while the best is the linear vacuous. 11/14 11/14 Back Forward

Comparison of distortion Number of extreme points of M ( P ) models Miranda, Montes, Destercke Introduction In terms of the maximum number of extreme points of the The models neighbourhood model, we have the following: Comparison Conclusions Model Maximal number of extreme points n ! P PMM 2 ⌋ ( ⌊ n 2 ⌋− 1 ) ! ( n −⌊ n 2 ⌋− 1 ) ! ⌊ n P LV n n ! P TV ( ⌊ n 2 ⌋− 1 ) ! ( n −⌊ n 2 ⌋− 1 ) ! 2 n − 2 P COR Q COR n ! P K P n where P n denotes the n -th Pell number. The best is the linear vacuous and the worst is the constant odds ratio on events. 12/14 12/14 Back Forward

Comparison of distortion Conclusions and further work models Miranda, Montes, Destercke Introduction Distortion models can be seen as neighbourhood mod- The models els induced by some d . Comparison Conclusions The linear vacuous seems to be the best overall model, although this depends on the property of interest. The analysis of the imprecision can be done by means of other measures. Additional results, to be reported elsewhere: Study of the model induced by the L 1 distance. Combination of distortion models. Study when the model is preserved by conditioning. 13/14 13/14 Back Forward

Comparison of distortion We eagerly look forward to your models Miranda, Montes, Destercke questions in the poster Introduction The models Comparison Conclusions 14/14 14/14 Back