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Validity- preservation properties of rules for Validity-preservation properties of rules for combining inferential models combining inferential models Ryan Martin and Nicholas Syring Ryan Martin and Nicholas Syring Commercials rgmarti3@ncsu.edu, and nasyring@wustl.edu Statistical inference based on be- lief/plausibility Validity North Carolina State University and Washington University Main problem in St. Louis 07/05/2019 1 / 1

Validity- preservation properties of rules for combining inferential models Ryan Martin and Nicholas Syring Commercials Statistical inference based on be- lief/plausibility Validity Main problem 1 / 1

Validity- preservation properties of rules for combining inferential models Ryan Martin and Nicholas Syring Commercials Statistical inference based on be- lief/plausibility Validity Main problem 1 / 1

Validity- preservation properties of rules for combining inferential models Ryan Martin and Nicholas Syring Commercials Statistical inference based on be- lief/plausibility Validity Main problem 1 / 1

Validity- preservation properties of rules for combining inferential models Ryan Martin and Nicholas Syring Commercials Statistical inference based on be- lief/plausibility Validity Main problem 1 / 1

Validity- preservation properties of rules for combining inferential models Ryan Martin and Nicholas Syring Commercials Statistical inference based on be- lief/plausibility Validity Main problem 1 / 1

Validity- preservation properties of rules for combining inferential models Ryan Martin and Nicholas Syring Commercials Statistical inference based on be- lief/plausibility Validity Main problem 1 / 1

Presentation Outline Validity- preservation properties of rules for 1 Commercials combining inferential models Ryan Martin and Nicholas 2 Statistical inference based on belief/plausibility Syring Commercials Statistical 3 Validity inference based on be- lief/plausibility Validity 4 Main problem Main problem 1 / 1

Presentation Outline Validity- preservation properties of rules for 1 Commercials combining inferential models Ryan Martin and Nicholas 2 Statistical inference based on belief/plausibility Syring Commercials Statistical 3 Validity inference based on be- lief/plausibility Validity 4 Main problem Main problem 1 / 1

Researchers.One Validity- preservation properties of rules for combining inferential models Ryan Martin and Nicholas Syring www.researchers.one Commercials Statistical An author-driven publishing platform. inference based on be- lief/plausibility Speak with Ryan Martin, Harry Crane for details. Validity Main problem 1 / 1

The book Validity- preservation properties of rules for combining inferential models Ryan Martin and Nicholas Syring Commercials Statistical inference based on be- lief/plausibility Validity Main problem 1 / 1

Presentation Outline Validity- preservation properties of rules for 1 Commercials combining inferential models Ryan Martin and Nicholas 2 Statistical inference based on belief/plausibility Syring Commercials Statistical 3 Validity inference based on be- lief/plausibility Validity 4 Main problem Main problem 1 / 1

Basic setup Validity- A three step method to evaluate a statistical inference problem: preservation properties of associate, predict, and combine. rules for combining inferential models Ryan Martin and Nicholas Syring Commercials Statistical inference based on be- lief/plausibility Validity Main problem 1 / 1

Basic setup Validity- A three step method to evaluate a statistical inference problem: preservation properties of associate, predict, and combine. rules for combining inferential models We write down an association between data Y , parameter θ , Ryan Martin and a random variable U , describing how the data is sampled and Nicholas Syring Y = a ( θ, U ) . Commercials Statistical inference based on be- lief/plausibility Validity Main problem 1 / 1

Basic setup Validity- A three step method to evaluate a statistical inference problem: preservation properties of associate, predict, and combine. rules for combining inferential models We write down an association between data Y , parameter θ , Ryan Martin and a random variable U , describing how the data is sampled and Nicholas Syring Y = a ( θ, U ) . Commercials Statistical inference Example: if Y ∼ N( θ, 1) and Φ denotes the standard normal based on be- lief/plausibility CDF then Validity Y = θ + U , U ∼ N(0 , 1) . Main problem 1 / 1

Basic setup Validity- A three step method to evaluate a statistical inference problem: preservation properties of associate, predict, and combine. rules for combining inferential models We predict the random variable U whose distribution is fully Ryan Martin known. Specifically, we predict using a (valid) random set S and Nicholas Syring (catching butterflies). Commercials Statistical inference based on be- lief/plausibility Validity Main problem 1 / 1

Basic setup Validity- A three step method to evaluate a statistical inference problem: preservation properties of associate, predict, and combine. rules for combining inferential models We predict the random variable U whose distribution is fully Ryan Martin known. Specifically, we predict using a (valid) random set S and Nicholas Syring (catching butterflies). Commercials Statistical Example: a sort of default random set is inference based on be- lief/plausibility S = { u : | u | < | U | , U ∼ N(0 , 1) } . Validity Main problem 1 / 1

Basic setup Validity- A three step method to evaluate a statistical inference problem: preservation properties of associate, predict, and combine. rules for combining inferential models Let the solutions in ϑ for a given ( y , u ) according to the Ryan Martin association be denoted Θ y ( u ) = { ϑ : y = a ( ϑ, u ) } . Then, and Nicholas Syring combine the solutions over S to obtain the random set on the Commercials parameter space � Statistical Θ y ( S ) = Θ y ( u ) . inference based on be- u ∈S lief/plausibility Validity Main problem 1 / 1

Basic setup Validity- A three step method to evaluate a statistical inference problem: preservation properties of associate, predict, and combine. rules for combining inferential models Let the solutions in ϑ for a given ( y , u ) according to the Ryan Martin association be denoted Θ y ( u ) = { ϑ : y = a ( ϑ, u ) } . Then, and Nicholas Syring combine the solutions over S to obtain the random set on the Commercials parameter space � Statistical Θ y ( S ) = Θ y ( u ) . inference based on be- u ∈S lief/plausibility Example: in the normal example this becomes the set Validity Main problem { ϑ : | y − ϑ | < | U |} , U ∼ N(0 , 1) . 1 / 1

Basic setup Validity- The inferential output for an assertion A about θ is the preservation properties of belief/plausibility pair ( b y ( A ) , p y ( A )) where rules for combining inferential and p y ( A ) = 1 − b y ( A c ) . models b y ( A ) = P S (Θ y ( S ) ⊆ A ) , Ryan Martin and Nicholas Syring Commercials Statistical inference based on be- lief/plausibility Validity Main problem 1 / 1

Basic setup Validity- The inferential output for an assertion A about θ is the preservation properties of belief/plausibility pair ( b y ( A ) , p y ( A )) where rules for combining inferential and p y ( A ) = 1 − b y ( A c ) . models b y ( A ) = P S (Θ y ( S ) ⊆ A ) , Ryan Martin and Nicholas Syring Example: for the normal example the plausibility contour function may be written Commercials Statistical p y ( { ϑ } ) = 2(1 − Φ( | y − ϑ | )) inference based on be- lief/plausibility Validity Main problem 1 / 1

Presentation Outline Validity- preservation properties of rules for 1 Commercials combining inferential models Ryan Martin and Nicholas 2 Statistical inference based on belief/plausibility Syring Commercials Statistical 3 Validity inference based on be- lief/plausibility Validity 4 Main problem Main problem 1 / 1

Validity property Validity- We insist the inferential model output is valid . preservation properties of rules for combining inferential models We rarely place high Ryan Martin belief on false assertions. and Nicholas Syring equivalently Commercials Statistical We rarely place low inference based on be- plausibility on true lief/plausibility assertions. Validity Main problem 1 / 1

Validity property Validity- Precisely, preservation properties of sup P Y | θ ( p Y ( A ) ≤ α ) ≤ α rules for combining θ ∈ A inferential models for every true A and every α ∈ (0 , 1). Ryan Martin and Nicholas By rare we mean calibrated to a uniform distribution. Syring Commercials Statistical inference based on be- lief/plausibility Validity Main problem 1 / 1

Valid inferential models Validity- Our previous construction provides a valid inferential model preservation properties of whenever S is valid, details omitted. Not difficult to find a rules for combining valid S . inferential models Ryan Martin and Nicholas Syring Commercials Statistical inference based on be- lief/plausibility Validity Main problem 1 / 1