[PPT] - Validity-preservation properties of rules for combining inferential PowerPoint Presentation

SLIDE 1

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

Validity-preservation properties of rules for combining inferential models

Ryan Martin and Nicholas Syring rgmarti3@ncsu.edu, and nasyring@wustl.edu North Carolina State University and Washington University in St. Louis 07/05/2019

1 / 1

SLIDE 2

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

SLIDE 3

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

SLIDE 4

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

SLIDE 5

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

SLIDE 6

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

SLIDE 7

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

SLIDE 8

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

Presentation Outline

1 Commercials 2 Statistical inference based on belief/plausibility 3 Validity 4 Main problem

1 / 1

SLIDE 9

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

Presentation Outline

1 Commercials 2 Statistical inference based on belief/plausibility 3 Validity 4 Main problem

1 / 1

SLIDE 10

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

Researchers.One

www.researchers.one An author-driven publishing platform. Speak with Ryan Martin, Harry Crane for details.

1 / 1

SLIDE 11

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

The book

1 / 1

SLIDE 12

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

Presentation Outline

1 Commercials 2 Statistical inference based on belief/plausibility 3 Validity 4 Main problem

1 / 1

SLIDE 13

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

Basic setup

A three step method to evaluate a statistical inference problem: associate, predict, and combine.

1 / 1

SLIDE 14

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

Basic setup

A three step method to evaluate a statistical inference problem: associate, predict, and combine. We write down an association between data Y , parameter θ, and a random variable U, describing how the data is sampled Y = a(θ, U).

1 / 1

SLIDE 15

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

Basic setup

A three step method to evaluate a statistical inference problem: associate, predict, and combine. We write down an association between data Y , parameter θ, and a random variable U, describing how the data is sampled Y = a(θ, U). Example: if Y ∼ N(θ, 1) and Φ denotes the standard normal CDF then Y = θ + U, U ∼ N(0, 1).

1 / 1

SLIDE 16

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

Basic setup

A three step method to evaluate a statistical inference problem: associate, predict, and combine. We predict the random variable U whose distribution is fully

known. Specifically, we predict using a (valid) random set S

(catching butterflies).

1 / 1

SLIDE 17

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

Basic setup

A three step method to evaluate a statistical inference problem: associate, predict, and combine. We predict the random variable U whose distribution is fully

known. Specifically, we predict using a (valid) random set S

(catching butterflies). Example: a sort of default random set is S = {u : |u| < |U|, U ∼ N(0, 1)}.

1 / 1

SLIDE 18

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

Basic setup

A three step method to evaluate a statistical inference problem: associate, predict, and combine. Let the solutions in ϑ for a given (y, u) according to the association be denoted Θy(u) = {ϑ : y = a(ϑ, u)}. Then, combine the solutions over S to obtain the random set on the parameter space Θy(S) =

u∈S

Θy(u).

1 / 1

SLIDE 19

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

Basic setup

A three step method to evaluate a statistical inference problem: associate, predict, and combine. Let the solutions in ϑ for a given (y, u) according to the association be denoted Θy(u) = {ϑ : y = a(ϑ, u)}. Then, combine the solutions over S to obtain the random set on the parameter space Θy(S) =

u∈S

Θy(u). Example: in the normal example this becomes the set {ϑ : |y − ϑ| < |U|}, U ∼ N(0, 1).

1 / 1

SLIDE 20

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

Basic setup

The inferential output for an assertion A about θ is the belief/plausibility pair (by(A), py(A)) where by(A) = PS(Θy(S) ⊆ A), and py(A) = 1 − by(Ac).

1 / 1

SLIDE 21

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

Basic setup

The inferential output for an assertion A about θ is the belief/plausibility pair (by(A), py(A)) where by(A) = PS(Θy(S) ⊆ A), and py(A) = 1 − by(Ac). Example: for the normal example the plausibility contour function may be written py({ϑ}) = 2(1 − Φ(|y − ϑ|))

1 / 1

SLIDE 22

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

Presentation Outline

1 Commercials 2 Statistical inference based on belief/plausibility 3 Validity 4 Main problem

1 / 1

SLIDE 23

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

Validity property

We insist the inferential model output is valid. We rarely place high belief on false assertions. equivalently We rarely place low plausibility on true assertions.

1 / 1

SLIDE 24

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

Validity property

Precisely, sup

θ∈A

PY |θ(pY (A) ≤ α) ≤ α for every true A and every α ∈ (0, 1). By rare we mean calibrated to a uniform distribution.

1 / 1

SLIDE 25

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

Valid inferential models

Our previous construction provides a valid inferential model whenever S is valid, details omitted. Not difficult to find a valid S.

1 / 1

SLIDE 26

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

Presentation Outline

1 Commercials 2 Statistical inference based on belief/plausibility 3 Validity 4 Main problem

1 / 1

SLIDE 27

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

Combination rules preserving validity

We can construct a valid IM given one data point (piece of evidence), and produce belief/plausibility for any assertion.

1 / 1

SLIDE 28

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

Combination rules preserving validity

We can construct a valid IM given one data point (piece of evidence), and produce belief/plausibility for any assertion. If we get a second data point we can repeat and obtain another IM, but now we should combine in some fashion.

1 / 1

SLIDE 29

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

Combination rules preserving validity

We can construct a valid IM given one data point (piece of evidence), and produce belief/plausibility for any assertion. If we get a second data point we can repeat and obtain another IM, but now we should combine in some fashion. How to do this?

1 / 1

SLIDE 30

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

Combination rules preserving validity

We can construct a valid IM given one data point (piece of evidence), and produce belief/plausibility for any assertion. If we get a second data point we can repeat and obtain another IM, but now we should combine in some fashion. How to do this?

1. Combine output? (the belief/plausibility functions)

1 / 1

SLIDE 31

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

Combination rules preserving validity

We can construct a valid IM given one data point (piece of evidence), and produce belief/plausibility for any assertion. If we get a second data point we can repeat and obtain another IM, but now we should combine in some fashion. How to do this?

1. Combine output? (the belief/plausibility functions)
2. Combine input? (the data)

1 / 1

SLIDE 32

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

Possible strategies

1. Combining plausibility (contour) functions - via Dempster’s

Rule, (a version of) Dubois and Prade’s Rule, perhaps others?

2. Combining data - using ideas from statistics like sufficiency,

and a related PDE technique.

1 / 1

SLIDE 33

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

See me at my poster!

1 / 1