Slideshow presented at the International Symposium on Imprecise Probabilities: Theories and Applications

De Angelis, Marco

Ferson, Scott

Gray, Ander

Rocchetta, Roberto

University of Liverpool Repository

Constructing consonant beliefs from multivariatedata with scenario theoryMarco De Angelis1, Roberto Rocchetta2, Ander Gray1, Scott Ferson11Institute for Risk and Uncertainty, University of Liverpool2Eindhoven University of TechnologyISIPTA 2021, virtually in GranadaAbstractA method for constructing consonant predictivebeliefs for multivariate datasets is presented. Wemake use of recent results in scenario theory toconstruct a family of enclosing sets that areassociated with a predictive lower probability ofnew data falling in each given set. We show thatthe sequence of lower bounds indexed by enclosingset yields a consonant belief function. Thepresented method does not rely on the constructionof a likelihood function, therefore possibilitydistributions can be obtained without the need fornormalization. We present a practical example intwo dimensions for the sake of visualization, todemonstrate the practical procedure of obtainingthe sequence of nested sets.Problem statement and predictive beliefsGiven a bunch of iid samples X1, ...,Xn, with Xi ∈ Rm, what can be learnt about the unknown underlyingdistribution PX? What is the probability of observing a new sample in a given set?I Samples are few We cannot learn PX exactly!I Multivariate case What about Xi interdependence?We want to obtain a belief function BelX so that the inequality BelX ≤ PX holds at least 100 (1− β)% ofthe times.(i) Pn (BelX ≤ PX) ≥ 1− β(ii) ∀A ⊆ Rm, BelX(A)→ PX(A), n→∞Scenario optimizationLet z ∈ Z ⊆ Rd be a vector of (design) parameters. The scenariooptimization consists in minimizing the convex cost function f : Z → R:minz∈Zf (z)subject to: z ∈⋂i=1,...,nZXi, (1)For example, let z = (cx, r), be the centre x-coordinate and the radius of acircle (cy = 0). The area of the circle is minimized, see Figure (1):min πr 2subject to: (cx − X1)2 + (0− Y1)2 ≤ r 2,...(cx − Xn)2 + (0− Yn)2 ≤ r 2(2)DefinitionsEnclosing set of degree k : The optimal set Bk ⊆ Rm, that strictly containsn − k observations.Lower probability of enclosing set Bk: The precise predictive probability of agiven enclosing set of degree k , PX(Bk), has a lower bound pk, with assignedone-sided coverage probability. Bk is determined by the program Eq.(1).Pn(pk≤ PX(Bk))≥ 1− β, (3)TheoremThe lower bound pkis a sequence of coherent predictive beliefs for anyk ∈ Z+ such that 0 = k0 < k1 < · · · < kn = n.Computing the lower bound probability of the enclosing setsϕ(t) =βn + 1n∑j=k(jk)t j−k −(nk)tn−k, t ∈ [0, 1] (4)ϕ(t̂) = 0; pk= t̂(n, k, β);[1] Campi, M.C. and Garatti, S. (2018). Wait-and-judge scenario optimization. Mathematical Programming, 167(1), pp.155-189.More references[2] Garatti, S. and Campi, M.C. (2019). Risk and complexity in scenario optimization. Mathematical Programming, pp.1-37.[3] Denœux, T. (1999). Reasoning with imprecise belief structures. International Journal of Approximate Reasoning, 20(1), 79-111.[4] Ryan Martin (2021). An imprecise-probabilistic characterization of frequentist statistical inference. Researchers.OneMinimum disk cover problemSolving optimization program Eq.(2) with center y-coordinate = 0. This minimumdisk is our first enclosing set Bk1=3.6 4 2 0 2 4 6x6420246yActive scenariosSample space10.0 7.5 5.0 2.5 0.0 2.5 5.0 7.5 10.0Center (x-coordinate)0246810121416RadiusDesign spaceFigure 1: Left side: 50 iid samples (in blue) and smallest enclosing disk (in light green); the orange dots are the samplesdetermining the optimum. Right side: space of acceptable designs and constraint functions corresponding to the three activescenarios in orange. The optimum is located on the deepest point lying on the constraint surface (green round dot).Constructing predictive beliefsFigure 2: 500 iid samples and sequence of enclosing sets constructed by the algorithm. Each enclosing set is associatedwith a unique belief that determines the marginal plausibility contours shown in the side marginal plots.

Constructing consonant beliefs from multivariate data with scenario theory

Introduction Scenario theory Method for constructing beliefsConstructing consonant beliefs from multivariate data withscenario theoryMarco De Angelis1, Roberto Rocchetta2, Ander Gray1, Scott Ferson11Institute for Risk and Uncertainty, University of Liverpool2Eindhoven University of TechnologyISIPTA 2021, virtually in GranadaJuly 7, 2021https://marcodeangelis.github.ioMarco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 1 / 15Introduction Scenario theory Method for constructing beliefsOutline1 Introduction2 Scenario theory3 Method for constructing beliefsMarco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 2 / 15Introduction Scenario theory Method for constructing beliefsProblem statementGiven a bunch of iid samples X1, ..., Xn, with Xi ∈ Rm, what can be learnt aboutthe unknown underlying distribution PX? What is the probability of observing a newsample in a given set?Samples are fewMultivariate caseMarco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 3 / 15Introduction Scenario theory Method for constructing beliefsProblem statementGiven a bunch of iid samples X1, ..., Xn, with Xi ∈ Rm, what can be learnt aboutthe unknown underlying distribution PX? What is the probability of observing a newsample in a given set?Samples are fewMultivariate caseMarco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 3 / 15Introduction Scenario theory Method for constructing beliefsProblem statementGiven a bunch of iid samples X1, ..., Xn, with Xi ∈ Rm, what can be learnt aboutthe unknown underlying distribution PX? What is the probability of observing a newsample in a given set?Samples are fewMultivariate caseMarco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 3 / 15Introduction Scenario theory Method for constructing beliefsProblem statementGiven a bunch of iid samples X1, ..., Xn, with Xi ∈ Rm, what can be learnt aboutthe unknown underlying distribution PX? What is the probability of observing a newsample in a given set?Samples are few We cannot learn PX exactly!Multivariate caseMarco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 3 / 15Introduction Scenario theory Method for constructing beliefsProblem statementGiven a bunch of iid samples X1, ..., Xn, with Xi ∈ Rm, what can be learnt aboutthe unknown underlying distribution PX? What is the probability of observing a newsample in a given set?Samples are few We cannot learn PX exactly!Multivariate case What about Xi interdependence?Marco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 3 / 15Introduction Scenario theory Method for constructing beliefsPredictive beliefsWe want to obtain a belief function BelX so that the inequality BelX ≤ PX holds atleast 100 (1− β)% of the times.1 ∀A ⊆ Rm, BelX(A)→ PX(A), n→∞2 Pn (BelX ≤ PX) ≥ 1− βMarco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 4 / 15Introduction Scenario theory Method for constructing beliefsPredictive beliefsWe want to obtain a belief function BelX so that the inequality BelX ≤ PX holds atleast 100 (1− β)% of the times.1 ∀A ⊆ Rm, BelX(A)→ PX(A), n→∞2 Pn (BelX ≤ PX) ≥ 1− βMarco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 4 / 15Introduction Scenario theory Method for constructing beliefsPredictive beliefsWe want to obtain a belief function BelX so that the inequality BelX ≤ PX holds atleast 100 (1− β)% of the times.1 ∀A ⊆ Rm, BelX(A)→ PX(A), n→∞2 Pn (BelX ≤ PX) ≥ 1− βMarco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 4 / 15Introduction Scenario theory Method for constructing beliefsCoherent lower probabilitiesBasic mass assignments: (i) m(∅) = 0, (ii)∑A∈2X m(A) = 1Beliefs obtained from basic mass assignments are coherent lower probabilities.The subsets A ⊆ X such that m(A) > 0 are called focal elements. The belief of afocal set A, for all B ∈ 2X , isBelX(A) =∑B:B⊆Am(B). (1)Marco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 5 / 15Introduction Scenario theory Method for constructing beliefsScenario optimizationLet z ∈ Z ⊆ Rd be a vector of (design) parameters and X1, ..., Xn a bunch ofiid samples , with Xi ∈ Rm. The scenario optimization consists in minimizing theconvex cost function f : Z → R:minz∈Zf(z)subject to: z ∈⋂i=1,...,nZXi ,(2)Marco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 6 / 15Introduction Scenario theory Method for constructing beliefsScenario optimizationLet z ∈ Z ⊆ Rd be a vector of (design) parameters and X1, ..., Xn a bunch ofiid samples , with Xi ∈ Rm. The scenario optimization consists in minimizing theconvex cost function f : Z → R:minz∈Zf(z)subject to: z ∈⋂i=1,...,nZXi ,(2)Design parameters can be the center coordinates and the radius of a circle (Rm=2)or sphere (Rm=3), as it will be illustrated in the next slide.Marco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 6 / 15Introduction Scenario theory Method for constructing beliefsScenario optimization on the disk R2For example, let z = (cx, r), be the centre x-coordinate and the radius of a circle(cy = 0). The scenario optimization consists in minimizing the area of the circle:min(cx,r)πr2subject to: (cx −X1)2 + (0− Y1)2 ≤ r2,...(cx −Xn)2 + (0− Yn)2 ≤ r2(3)Marco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 7 / 15Introduction Scenario theory Method for constructing beliefsVisualizing the constraints: m = 2, d = 2Center y-coordinate = 0Marco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 8 / 15Introduction Scenario theory Method for constructing beliefsObtaining the smallest disk O(n)Center y-coordinate = 0Marco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 9 / 15Introduction Scenario theory Method for constructing beliefsActive scenarios6 4 2 0 2 4 6x6420246ySample space10.0 7.5 5.0 2.5 0.0 2.5 5.0 7.5 10.0Center (x-coordinate)0246810121416RadiusDesign spaceMarco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 10 / 15Introduction Scenario theory Method for constructing beliefsActive scenarios6 4 2 0 2 4 6x6420246yActive scenariosSample space10.0 7.5 5.0 2.5 0.0 2.5 5.0 7.5 10.0Center (x-coordinate)0246810121416RadiusDesign spaceMarco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 10 / 15Introduction Scenario theory Method for constructing beliefsDefinitionsEnclosing set of degree k: The optimal set Bk ⊆ Rm, that strictly contains n − kobservations.Lower probability of enclosing set Bk: The precise predictive probability of agiven enclosing set of degree k, PX(Bk), has a lower bound pk, with assignedone-sided coverage probability.Pn(pk≤ PX(Bk))≥ 1− β, (4)Marco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 11 / 15Introduction Scenario theory Method for constructing beliefsComputing the lower boundϕ(t) =βn+ 1n∑j=k(jk)tj−k −(nk)tn−k, t ∈ [0, 1] (5)ϕ(t̂) = 0; pk= t̂(n, k, β);Pn(pk≤ PX(Bk))≥ 1− βCampi, M.C. and Garatti, S., 2018. Wait-and-judge scenario optimization. Mathematical Programming, 167(1), pp.155-189.Garatti, S. and Campi, M.C., 2019. Risk and complexity in scenario optimization. Mathematical Programming, pp.1-37.Marco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 12 / 15Introduction Scenario theory Method for constructing beliefsTheorem 1The lower bounds pkmake a sequence of coherent predictive beliefs for anyk ∈ Z+ such that 0 = k0 < k1 < · · · < kn = n.Marco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 13 / 15Introduction Scenario theory Method for constructing beliefsTheorem 1The lower bounds pkmake a sequence of coherent predictive beliefs for anyk ∈ Z+ such that 0 = k0 < k1 < · · · < kn = n.The proof follows from Eq.(5), noticing that the roots of the polynomial are decreas-ing with k.Marco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 13 / 15Introduction Scenario theory Method for constructing beliefsMarco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 14 / 15Introduction Scenario theory Method for constructing beliefsConclusionsInference on multidimensional datasetsNo need to estimate the likelihoodAdditional constraints can ensure sets are fully inter-nestedStructures can be propagated and retain the confidence interpretationThe interdependence is encoded in the shape of the enclosing setsMarco De Angelis, Roberto Rocchetta, Ander Gray, Scott Ferson Constructing consonant beliefs from multivariate data July 7, 2021 15 / 15

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Constructing consonant beliefs from multivariate data with scenario theory

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