Prior and loss robustness for varoius loss functions Agnieszka Kami - PowerPoint PPT Presentation

Introduction Symmetric loss function Asymmetric loss function ABL Remarks Bibliography Prior and loss robustness for varoius loss functions Agnieszka Kami´ nska and Zdzis� law Porosi´ nski Institute of Mathematics and Computer Science, Wroclaw University of Technology Wybrzeze Wyspianskiego 27 50-370 Wroclaw December 8, 2009 Agnieszka Kami´ nska and Zdzis� law Porosi´ nski Prior and loss robustness for varoius loss functions

Introduction Symmetric loss function Model Asymmetric loss function ABL Bayesian estimation Remarks Robust Bayes estimators Bibliography Model Agnieszka Kami´ nska and Zdzis� law Porosi´ nski Prior and loss robustness for varoius loss functions

Introduction Symmetric loss function Model Asymmetric loss function ABL Bayesian estimation Remarks Robust Bayes estimators Bibliography Model Let X 1 , . . . , X n be i.i.d. random variables with a distribution P ϑ indexed by a real parameter ϑ . We denote X = ( X 1 , . . . , X n ) . Agnieszka Kami´ nska and Zdzis� law Porosi´ nski Prior and loss robustness for varoius loss functions

Introduction Symmetric loss function Model Asymmetric loss function ABL Bayesian estimation Remarks Robust Bayes estimators Bibliography Model Let X 1 , . . . , X n be i.i.d. random variables with a distribution P ϑ indexed by a real parameter ϑ . We denote X = ( X 1 , . . . , X n ) . Let ( X , B , P ) be a statistical space determined by X , where X ⊂ R n , B is σ -field of X and P = { P ϑ : ϑ ∈ Θ = R } . Agnieszka Kami´ nska and Zdzis� law Porosi´ nski Prior and loss robustness for varoius loss functions

Introduction Symmetric loss function Model Asymmetric loss function ABL Bayesian estimation Remarks Robust Bayes estimators Bibliography Model Let X 1 , . . . , X n be i.i.d. random variables with a distribution P ϑ indexed by a real parameter ϑ . We denote X = ( X 1 , . . . , X n ) . Let ( X , B , P ) be a statistical space determined by X , where X ⊂ R n , B is σ -field of X and P = { P ϑ : ϑ ∈ Θ = R } . Let L ( ϑ, d ) be a loss function. Agnieszka Kami´ nska and Zdzis� law Porosi´ nski Prior and loss robustness for varoius loss functions

Introduction Symmetric loss function Model Asymmetric loss function ABL Bayesian estimation Remarks Robust Bayes estimators Bibliography Model Let X 1 , . . . , X n be i.i.d. random variables with a distribution P ϑ indexed by a real parameter ϑ . We denote X = ( X 1 , . . . , X n ) . Let ( X , B , P ) be a statistical space determined by X , where X ⊂ R n , B is σ -field of X and P = { P ϑ : ϑ ∈ Θ = R } . Let L ( ϑ, d ) be a loss function. Let ϑ have a prior distribution π ( ϑ ) , defined on the measurable space (Θ , Ξ) . Agnieszka Kami´ nska and Zdzis� law Porosi´ nski Prior and loss robustness for varoius loss functions

Introduction Symmetric loss function Model Asymmetric loss function ABL Bayesian estimation Remarks Robust Bayes estimators Bibliography Model Let X 1 , . . . , X n be i.i.d. random variables with a distribution P ϑ indexed by a real parameter ϑ . We denote X = ( X 1 , . . . , X n ) . Let ( X , B , P ) be a statistical space determined by X , where X ⊂ R n , B is σ -field of X and P = { P ϑ : ϑ ∈ Θ = R } . Let L ( ϑ, d ) be a loss function. Let ϑ have a prior distribution π ( ϑ ) , defined on the measurable space (Θ , Ξ) . The posterior distribution has a form π ( ϑ | x ) , for X = x . Agnieszka Kami´ nska and Zdzis� law Porosi´ nski Prior and loss robustness for varoius loss functions

Introduction Symmetric loss function Model Asymmetric loss function ABL Bayesian estimation Remarks Robust Bayes estimators Bibliography Model Let X 1 , . . . , X n be i.i.d. random variables with a distribution P ϑ indexed by a real parameter ϑ . We denote X = ( X 1 , . . . , X n ) . Let ( X , B , P ) be a statistical space determined by X , where X ⊂ R n , B is σ -field of X and P = { P ϑ : ϑ ∈ Θ = R } . Let L ( ϑ, d ) be a loss function. Let ϑ have a prior distribution π ( ϑ ) , defined on the measurable space (Θ , Ξ) . The posterior distribution has a form π ( ϑ | x ) , for X = x . We consider the problem of constructing the point Bayes estimator of ϑ under L ( ϑ, d ) . Agnieszka Kami´ nska and Zdzis� law Porosi´ nski Prior and loss robustness for varoius loss functions

Introduction Symmetric loss function Model Asymmetric loss function ABL Bayesian estimation Remarks Robust Bayes estimators Bibliography Bayesian estimation If X = x , then the posterior risk of d can be expressed as E π | x [ L ( ϑ, d )] , R x ( π, d ) = where E π | x [ · ] denotes the expected value when ϑ ∼ π ( ϑ | x ) . Agnieszka Kami´ nska and Zdzis� law Porosi´ nski Prior and loss robustness for varoius loss functions

Introduction Symmetric loss function Model Asymmetric loss function ABL Bayesian estimation Remarks Robust Bayes estimators Bibliography Bayesian estimation If X = x , then the posterior risk of d can be expressed as E π | x [ L ( ϑ, d )] , R x ( π, d ) = where E π | x [ · ] denotes the expected value when ϑ ∼ π ( ϑ | x ) . ϑ π satisfies The Bayes estimator � R x ( π, � ϑ π ) = inf d ∈ D R x ( π, d ) . Agnieszka Kami´ nska and Zdzis� law Porosi´ nski Prior and loss robustness for varoius loss functions

Introduction Symmetric loss function Model Asymmetric loss function ABL Bayesian estimation Remarks Robust Bayes estimators Bibliography Prior robustness Information on the appropriate prior is often too inadequate to specify a prior distribution unambiguously. Agnieszka Kami´ nska and Zdzis� law Porosi´ nski Prior and loss robustness for varoius loss functions

Introduction Symmetric loss function Model Asymmetric loss function ABL Bayesian estimation Remarks Robust Bayes estimators Bibliography Prior robustness Information on the appropriate prior is often too inadequate to specify a prior distribution unambiguously. The problem of expressing uncertainty regarding prior information can be solved by using a class Γ of prior distributions. Agnieszka Kami´ nska and Zdzis� law Porosi´ nski Prior and loss robustness for varoius loss functions

Introduction Symmetric loss function Model Asymmetric loss function ABL Bayesian estimation Remarks Robust Bayes estimators Bibliography Prior robustness Information on the appropriate prior is often too inadequate to specify a prior distribution unambiguously. The problem of expressing uncertainty regarding prior information can be solved by using a class Γ of prior distributions. Assume that the prior π ( ϑ ) belongs to the class Γ . Agnieszka Kami´ nska and Zdzis� law Porosi´ nski Prior and loss robustness for varoius loss functions

Introduction Symmetric loss function Model Asymmetric loss function ABL Bayesian estimation Remarks Robust Bayes estimators Bibliography Γ-minimax estimators Let F x ( π, d ) be a posterior functional. The optimal decision � ϑ satisfies F x ( π, � sup ϑ ) = inf d ∈ D sup F x ( π, d ) . π ∈ Γ π ∈ Γ Agnieszka Kami´ nska and Zdzis� law Porosi´ nski Prior and loss robustness for varoius loss functions

Introduction Symmetric loss function Model Asymmetric loss function ABL Bayesian estimation Remarks Robust Bayes estimators Bibliography Γ-minimax estimators Let F x ( π, d ) be a posterior functional. The optimal decision � ϑ satisfies F x ( π, � sup ϑ ) = inf d ∈ D sup F x ( π, d ) . π ∈ Γ π ∈ Γ the conditional Γ-minimax estimator F x ( π, d )= R x ( π, d ) , Agnieszka Kami´ nska and Zdzis� law Porosi´ nski Prior and loss robustness for varoius loss functions

Introduction Symmetric loss function Model Asymmetric loss function ABL Bayesian estimation Remarks Robust Bayes estimators Bibliography Γ-minimax estimators Let F x ( π, d ) be a posterior functional. The optimal decision � ϑ satisfies F x ( π, � sup ϑ ) = inf d ∈ D sup F x ( π, d ) . π ∈ Γ π ∈ Γ the conditional Γ-minimax estimator F x ( π, d )= R x ( π, d ) , the posterior regret Γ-minimax estimator F x ( π, d )= R x ( π, d ) − R x ( π, � ϑ π ) , Agnieszka Kami´ nska and Zdzis� law Porosi´ nski Prior and loss robustness for varoius loss functions

Introduction Symmetric loss function Model Asymmetric loss function ABL Bayesian estimation Remarks Robust Bayes estimators Bibliography Γ-minimax estimators Let F x ( π, d ) be a posterior functional. The optimal decision � ϑ satisfies F x ( π, � sup ϑ ) = inf d ∈ D sup F x ( π, d ) . π ∈ Γ π ∈ Γ the conditional Γ-minimax estimator F x ( π, d )= R x ( π, d ) , the posterior regret Γ-minimax estimator F x ( π, d )= R x ( π, d ) − R x ( π, � ϑ π ) , the most stable estimator F x ( π, d )=sup π ∈ Γ R x ( π, d ) − inf π ∈ Γ R x ( π, d ) . Agnieszka Kami´ nska and Zdzis� law Porosi´ nski Prior and loss robustness for varoius loss functions

Introduction Symmetric loss function Model Asymmetric loss function ABL Bayesian estimation Remarks Robust Bayes estimators Bibliography Prior and loss robustness Agnieszka Kami´ nska and Zdzis� law Porosi´ nski Prior and loss robustness for varoius loss functions