Challenges on the use of Imprecise Prior for Imprecise Inference on - PowerPoint PPT Presentation

Challenges on the use of Imprecise Prior for Imprecise Inference on Poisson Sampling Models Chel Hee Lee 1 , elis Bickis 2 Mik 1 Clinical Research Support Unit Community Health and Epidemiology College of Medicine University of Saskatchewan 2 Department of Mathematics and Statistics University of Saskatchewan 9th WPMSIIP, Durham, England 4 September 2016 Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 1 / 37

Motivation 1 Imprecise Inferential Framework 2 Illustration 3 Scenario I Scenario II Scenario III-1 Scenario III-2 Discussion 4 References 5 Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 2 / 37

Motivation General Framework P. I. N. Count 1 1 1 1 1 1 1 1 X8213 0 2 2 2 2 2 1 1 1 X8222 2 3 3 3 3 3 1 1 1 X8223 1 4 4 4 4 4 1 1 1 X8224 1 5 5 5 5 5 1 1 1 X8225 3 6 6 6 6 6 1 1 1 X8226 2 7 7 7 7 7 1 1 1 X8227 7 8 8 8 8 8 1 1 1 X8227 7 8 8 8 8 8 1 1 1 X8227 7 8 8 8 8 8 1 1 1 Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 3 / 37

Motivation General Framework P. I. N. Count 1 1 1 1 1 1 1 1 X8213 0 2 2 2 2 2 1 1 1 X8222 2 3 3 3 3 3 1 1 1 X8223 1 4 4 4 4 4 1 1 1 X8224 1 5 5 5 5 5 1 1 1 X8225 3 6 6 6 6 6 1 1 1 X8226 2 7 7 7 7 7 1 1 1 X8227 7 8 8 8 8 8 1 1 1 X8227 7 8 8 8 8 8 1 1 1 Sampling Model Y i X8227 7 8 8 8 8 8 1 1 1 Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 3 / 37

Motivation General Framework P. I. N. Count 1 1 1 1 1 1 1 1 X8213 0 2 2 2 2 2 1 1 1 X8222 2 3 3 3 3 3 1 1 1 X8223 1 4 4 4 4 4 1 1 1 X8224 1 5 5 5 5 5 1 1 1 X8225 3 6 6 6 6 6 1 1 1 X8226 2 7 7 7 7 7 1 1 1 θ Prior Distribution X8227 7 8 8 8 8 8 1 1 1 X8227 7 8 8 8 8 8 1 1 1 Sampling Model Y i X8227 7 8 8 8 8 8 1 1 1 Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 3 / 37

Motivation General Framework P. I. N. Count 1 1 1 1 1 1 1 1 X8213 0 2 2 2 2 2 1 1 1 X8222 2 3 3 3 3 3 1 1 1 X8223 1 4 4 4 4 4 1 1 1 X8224 1 5 5 5 5 5 1 1 1 Hyperparameters ξ X8225 3 6 6 6 6 6 1 1 1 X8226 2 7 7 7 7 7 1 1 1 θ Prior Distribution X8227 7 8 8 8 8 8 1 1 1 X8227 7 8 8 8 8 8 1 1 1 Sampling Model Y i X8227 7 8 8 8 8 8 1 1 1 Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 3 / 37

Motivation General Framework P. I. N. Count 1 1 1 1 1 1 1 1 X8213 0 2 2 2 2 2 1 1 1 X8222 2 3 3 3 3 3 1 1 1 X8223 1 4 4 4 4 4 1 1 1 ξ 1 X8224 1 5 5 5 5 5 1 1 1 Hyperparameters ξ X8225 3 6 6 6 6 6 1 1 1 X8226 2 7 7 7 7 7 1 1 1 θ Prior Distribution X8227 7 8 8 8 8 8 1 1 1 X8227 7 8 8 8 8 8 1 1 1 Sampling Model Y i X8227 7 8 8 8 8 8 1 1 1 Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 3 / 37

Motivation General Framework P. I. N. Count 1 1 1 1 1 1 1 1 X8213 0 2 2 2 2 2 1 1 1 X8222 2 3 3 3 3 3 1 1 1 X8223 1 4 4 4 4 4 1 1 1 ξ 2 ξ 1 X8224 1 5 5 5 5 5 1 1 1 Hyperparameters ξ X8225 3 6 6 6 6 6 1 1 1 X8226 2 7 7 7 7 7 1 1 1 θ Prior Distribution X8227 7 8 8 8 8 8 1 1 1 X8227 7 8 8 8 8 8 1 1 1 Sampling Model Y i X8227 7 8 8 8 8 8 1 1 1 Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 3 / 37

Motivation General Framework P. I. N. Count 1 1 1 1 1 1 1 1 X8213 0 2 2 2 2 2 1 1 1 X8222 2 3 3 3 3 3 1 1 1 X8223 1 4 4 4 4 4 1 1 1 ξ 2 ξ 1 ξ 3 X8224 1 5 5 5 5 5 1 1 1 Hyperparameters ξ X8225 3 6 6 6 6 6 1 1 1 X8226 2 7 7 7 7 7 1 1 1 θ Prior Distribution X8227 7 8 8 8 8 8 1 1 1 X8227 7 8 8 8 8 8 1 1 1 Sampling Model Y i X8227 7 8 8 8 8 8 1 1 1 Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 3 / 37

Motivation General Framework P. I. N. Count 1 1 1 1 1 1 1 1 X8213 0 2 2 2 2 2 1 1 1 X8222 2 3 3 3 3 3 1 1 1 X8223 1 4 4 4 4 4 1 1 1 ξ 2 ξ 1 ξ 3 X8224 1 5 5 5 5 5 1 1 1 ξ X8225 3 6 6 6 6 6 1 1 1 Lower Bound Upper Bound X8226 2 7 7 7 7 7 1 1 1 θ Prior Distribution X8227 7 8 8 8 8 8 1 1 1 X8227 7 8 8 8 8 8 1 1 1 Sampling Model Y i X8227 7 8 8 8 8 8 1 1 1 Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 3 / 37

Motivation Robust Bayesian Analysis (Berger et al., 1994, pp. 24–25) A prior distribution should be easy to elicit and interpret, easy to handle computationally, reasonable to reflect uncertainty, extensible to higher dimensions, and adaptable to incorporate constraints Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 4 / 37

Motivation HOW? ? Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 5 / 37

Imprecise Inferential Framework Imprecise Inferential Framework Canonical Representation Consider a family of probability measures P θ whose density with respect to µ : dP θ ( y ) = exp { θ · t ( y ) − A ( θ ) } d µ ( y ) , where t : R m → R k is a measurable function of y and the cumulant transform � A ( θ ) = ln exp { θ · t ( y ) } d µ ( y ) serves to normalize the measure P θ . Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 6 / 37

Imprecise Inferential Framework Imprecise Inferential Framework Conjugate Prior Formulation We consider the following family of prior measures for P θ with respect to Lebesgue measure: d π ξ 2 ,ξ 1 ,ξ 0 ( θ ) ∝ exp {− ξ 2 θ 2 + θξ 1 − ξ 0 A ( θ ) − M ( ξ 2 , ξ 1 , ξ 0 ) } d θ, where ξ = ( ξ 2 , ξ 1 , ξ 0 ) are hyperparameters and � ∞ exp {− ξ 2 θ 2 + θξ 1 − ξ 0 A ( θ ) } d θ < + ∞ M ( ξ 2 , ξ 1 , ξ 0 ) = ln −∞ is the cumulant transform of ξ producing the densities π ξ ( θ ) . Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 6 / 37

Imprecise Inferential Framework Illustration (based on Poisson samples) Consider the problem of parameter estimation (Scenario 1) when a prior is conjugate to a likelihood using a log-gamma prior distribution (Scenario 2) when a prior is not conjugate to a likelihood using a normal prior distribution (Scenario 3) under the generalized linear model setting having only intercept incorporating a single predictor (with an intercept) Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 7 / 37

Illustration Scenario I Scenario I Using Log-Gamma Priors If µ ∼ Gamma ( α, β ) and θ = log ( µ ) , π α,β ( θ ) ∝ e αθ − β e θ We see the following e ( y θ − e θ ) e ( αθ − β e θ ) p ( θ | y ) ∝ e ( α + y ) θ − ( β + 1 ) e θ = which has the form p ( θ | y ) ∝ exp ( − ξ 2 θ 2 + ξ 1 θ − ξ 0 e θ ) with hyperparameters ξ 2 = 0 , ξ 1 = α 1 + y , ξ 0 = β + 1 , (1) Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 8 / 37

Illustration Scenario I Set Basic Analytical Frame 10 10 9 ξ 0 (number of observations) 8 8 7 6 6 5 4 4 3 2 2 1 0 0 2 4 6 8 10 ξ 1 (sum) Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 9 / 37

Illustration Scenario I Assume Poisson Mean Parameter ( µ = 1 ) 10 10 9 ξ 0 (number of observations) 8 8 7 6 6 5 4 4 3 2 2 1 0 0 2 4 6 8 10 ξ 1 (sum) Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 10 / 37

Illustration Scenario I Before Seeing Data 10 10 9 ξ 0 (number of observations) 8 8 7 6 6 5 4 4 3 2 2 1 0 0 2 4 6 8 10 ξ 1 (sum) Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 11 / 37

Illustration Scenario I After Seeing One Observation 10 10 9 ξ 0 (number of observations) 8 8 7 6 6 5 4 4 3 2 2 1 1 0 0 2 4 6 8 10 ξ 1 (sum) Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 12 / 37

Illustration Scenario I One More Observation 10 10 9 ξ 0 (number of observations) 8 8 7 6 6 5 4 4 3 1 2 2 1 1 0 0 2 4 6 8 10 ξ 1 (sum) Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 13 / 37

Illustration Scenario I See What Data Tell Us 10 10 9 ξ 0 (number of observations) 8 8 7 6 6 5 4 4 1 3 1 2 2 1 1 0 0 2 4 6 8 10 ξ 1 (sum) Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 14 / 37

Illustration Scenario I Continue to Watch! 10 10 9 ξ 0 (number of observations) 8 8 7 6 6 5 0 4 4 1 3 1 2 2 1 1 0 0 2 4 6 8 10 ξ 1 (sum) Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-04 15 / 37

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