Geometry of
Sensitivity
analysis
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Computational Information Geometry:
Geometry of Model Choice
Paul Marriott
Example
Example
Global
analysis
University of Waterloo
April 6, 2010
Geometry of
Sensitivity
analysis
Big Picture
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Introduction to Computational Information Geometry
Geometry of
Sensitivity
analysis
Big Picture
Paul Marriott
Objectives
SEM
Geometries
The example
• Introduction to Computational Information Geometry
Likelihood in
sparse
simplex
• The overall objective is to construct diagnostic tools to
Least
informative
model
Example
Example
Global
analysis
help understand sensitivity to model choice
Geometry of
Sensitivity
analysis
Big Picture
Paul Marriott
Objectives
SEM
Geometries
The example
• Introduction to Computational Information Geometry
Likelihood in
sparse
simplex
• The overall objective is to construct diagnostic tools to
Least
informative
model
• Targeted at applications where Generalised Linear
Example
Example
Global
analysis
help understand sensitivity to model choice
Models are used
Geometry of
Sensitivity
analysis
Big Picture
Paul Marriott
Objectives
SEM
Geometries
The example
• Introduction to Computational Information Geometry
Likelihood in
sparse
simplex
• The overall objective is to construct diagnostic tools to
Least
informative
model
• Targeted at applications where Generalised Linear
Example
Example
Global
analysis
help understand sensitivity to model choice
Models are used
• Joint work with Karim Anaya-Izquierdo, Frank Critchley
and Paul Vos
Geometry of
Sensitivity
analysis
Big Picture
Paul Marriott
Objectives
SEM
Geometries
The example
• Introduction to Computational Information Geometry
Likelihood in
sparse
simplex
• The overall objective is to construct diagnostic tools to
Least
informative
model
• Targeted at applications where Generalised Linear
Example
Example
Global
analysis
help understand sensitivity to model choice
Models are used
• Joint work with Karim Anaya-Izquierdo, Frank Critchley
and Paul Vos
• Thanks to EPSRC Grant Number EP/E017878/1
Geometry of
Sensitivity
analysis
Problem of Interest
Paul Marriott
Objectives
SEM
Geometries
The data
The example
• Question: what is the
Likelihood in
sparse
simplex
Global
analysis
60
40
effect inference about mean?
• Can geometry of
‘space of all models’ give
a framework for
discussion?
20
Example
• How do modelling assumptions
0
Example
population mean?
Frequency
Least
informative
model
0
5
10
Number of red boxes
15
Geometry of
Sensitivity
analysis
Problem of Interest
Paul Marriott
Objectives
SEM
Geometries
The data
The example
• Question: what is the
Likelihood in
sparse
simplex
Global
analysis
60
40
effect inference about mean?
• Can geometry of
‘space of all models’ give
a framework for
discussion?
20
Example
• How do modelling assumptions
0
Example
population mean?
Frequency
Least
informative
model
0
5
10
Number of red boxes
15
Geometry of
Sensitivity
analysis
Problem of Interest
Paul Marriott
Objectives
SEM
Geometries
The data
The example
• Question: what is the
Likelihood in
sparse
simplex
Global
analysis
60
40
effect inference about mean?
• Can geometry of
‘space of all models’ give
a framework for
discussion?
20
Example
• How do modelling assumptions
0
Example
population mean?
Frequency
Least
informative
model
0
5
10
Number of red boxes
15
Geometry of
Sensitivity
analysis
Paul Marriott
Structured Extended
Multinomials
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Extended multinomials are multinomial but allow cell
probabilities to be zero.
Geometry of
Sensitivity
analysis
Paul Marriott
Structured Extended
Multinomials
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Extended multinomials are multinomial but allow cell
probabilities to be zero.
• Discretizing continuous data gives categorical models
with structure on the cells
Geometry of
Sensitivity
analysis
Paul Marriott
Structured Extended
Multinomials
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Extended multinomials are multinomial but allow cell
probabilities to be zero.
• Discretizing continuous data gives categorical models
with structure on the cells
• Examples of structure:
Geometry of
Sensitivity
analysis
Paul Marriott
Structured Extended
Multinomials
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Extended multinomials are multinomial but allow cell
probabilities to be zero.
• Discretizing continuous data gives categorical models
with structure on the cells
• Examples of structure:
• numerical labels
• ordering
• neighbourhood structures
Geometry of
Sensitivity
analysis
Structured Extended
Multinomials
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Extended multinomials are multinomial but allow cell
probabilities to be zero.
• Discretizing continuous data gives categorical models
with structure on the cells
• Examples of structure:
• numerical labels
• ordering
• neighbourhood structures
• Structured Extended Multinomials (SEM) include this
structure
Geometry of
Sensitivity
analysis
Structured Extended
Multinomials
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Extended multinomials are multinomial but allow cell
probabilities to be zero.
• Discretizing continuous data gives categorical models
with structure on the cells
• Examples of structure:
• numerical labels
• ordering
• neighbourhood structures
• Structured Extended Multinomials (SEM) include this
structure
• SEM proxy for universal space of all distributions
• Can be finite or infinite dimensional
Geometry of
Sensitivity
analysis
−1 -simplical structure
Paul Marriott
Objectives
SEM
Geometries
The example
1.0
0.8
be on boundary
• Union of exponential
Example
0.2
• Different support sets
Global
analysis
0.0
Example
• Mean (−1) parameters can
0.6
Least
informative
model
mean parametrisation
0.4
Likelihood in
sparse
simplex
0.0
0.2
0.4
0.6
0.8
1.0
families each with
corresponding natural (+1)
parameters
Geometry of
Sensitivity
analysis
−1 -simplical structure
Paul Marriott
Objectives
SEM
Geometries
The example
1.0
0.8
be on boundary
• Union of exponential
Example
0.2
• Different support sets
Global
analysis
0.0
Example
• Mean (−1) parameters can
0.6
Least
informative
model
meanSupport
parametrisation
sets
●
0.4
Likelihood in
sparse
simplex
●
0.0
●
0.2
0.4
0.6
0.8
1.0
families each with
corresponding natural (+1)
parameters
Geometry of
Sensitivity
analysis
−1 -simplical structure
Paul Marriott
Objectives
SEM
Geometries
The example
1.0
0.8
be on boundary
• Union of exponential
Example
0.2
• Different support sets
Global
analysis
0.0
Example
• Mean (−1) parameters can
0.6
Least
informative
model
meanSupport
parametrisation
sets
●
0.4
Likelihood in
sparse
simplex
●
0.0
●
0.2
0.4
0.6
0.8
1.0
families each with
corresponding natural (+1)
parameters
Geometry of
Sensitivity
analysis
+1 simplex structure
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• How can we define +1 structure on the extended
multinomial?
Geometry of
Sensitivity
analysis
+1 simplex structure
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
• How can we define +1 structure on the extended
Least
informative
model
• Problem: the support and the moment structure
Example
Example
Global
analysis
multinomial?
changes across SEM
Geometry of
Sensitivity
analysis
+1 simplex structure
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
• How can we define +1 structure on the extended
Least
informative
model
• Problem: the support and the moment structure
Example
Example
Global
analysis
multinomial?
changes across SEM
• Need to glue together different exponential families
Geometry of
Sensitivity
analysis
+1 simplex structure
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
• How can we define +1 structure on the extended
Least
informative
model
• Problem: the support and the moment structure
Example
Example
Global
analysis
multinomial?
changes across SEM
• Need to glue together different exponential families
• Use the dual structure of information geometry
Geometry of
Sensitivity
analysis
Dual Parameterisations
Paul Marriott
(a) !1!geodesics in !1!simplex
(b) !1!geodesics in +1!simplex
Likelihood in
sparse
simplex
Least
informative
model
10
5
0
!5
The example
!10
SEM
Geometries
0.0 0.2 0.4 0.6 0.8 1.0
Objectives
0.0
0.2
0.4
0.6
0.8
1.0
!10
!5
0
5
10
Example
(c) +1!geodesics in !1!simplex
!5
0
5
10
(d) +1!geodesics in +1!simplex
!10
Global
analysis
0.0 0.2 0.4 0.6 0.8 1.0
Example
0.0
0.2
0.4
0.6
0.8
1.0
!10
!5
0
5
10
Geometry of
Sensitivity
analysis
Paul Marriott
Computational Information
Geometry
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Want to be able to numerically compute in high
dimensional SEM
Geometry of
Sensitivity
analysis
Paul Marriott
Computational Information
Geometry
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Want to be able to numerically compute in high
dimensional SEM
• Need to get the topology and geometry right
Geometry of
Sensitivity
analysis
Paul Marriott
Computational Information
Geometry
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Want to be able to numerically compute in high
dimensional SEM
• Need to get the topology and geometry right
• Information Geometry sits naturally on these simplicial
structures not manifolds
Geometry of
Sensitivity
analysis
Paul Marriott
Computational Information
Geometry
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Want to be able to numerically compute in high
dimensional SEM
• Need to get the topology and geometry right
• Information Geometry sits naturally on these simplicial
structures not manifolds
• Almost all of the information geometry on SEM is
numerically easy
Geometry of
Sensitivity
analysis
Paul Marriott
Computational Information
Geometry
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Want to be able to numerically compute in high
dimensional SEM
• Need to get the topology and geometry right
• Information Geometry sits naturally on these simplicial
structures not manifolds
• Almost all of the information geometry on SEM is
numerically easy
• Hard part: the mixed parameterisation
Geometry of
Sensitivity
analysis
Paul Marriott
Computational Information
Geometry
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Want to be able to numerically compute in high
dimensional SEM
• Need to get the topology and geometry right
• Information Geometry sits naturally on these simplicial
structures not manifolds
• Almost all of the information geometry on SEM is
numerically easy
• Hard part: the mixed parameterisation
• This is our computational framework
Geometry of
Sensitivity
analysis
The example
Paul Marriott
Objectives
SEM
Geometries
The data
The example
Example
40
20
Example
Frequency
Least
informative
model
60
Likelihood in
sparse
simplex
0
Global
analysis
0
5
10
Number of red boxes
15
Geometry of
Sensitivity
analysis
The example
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Data is 200 integers
Geometry of
Sensitivity
analysis
The example
Paul Marriott
Objectives
SEM
Geometries
The example
• Data is 200 integers
Likelihood in
sparse
simplex
• Told each is total number of red squares out of 25 red
Least
informative
model
Example
Example
Global
analysis
or blue
Geometry of
Sensitivity
analysis
The example
Paul Marriott
Objectives
SEM
Geometries
The example
• Data is 200 integers
Likelihood in
sparse
simplex
• Told each is total number of red squares out of 25 red
Least
informative
model
Example
Example
Global
analysis
or blue
• Told each comes from unrelated experiments
Geometry of
Sensitivity
analysis
The example
Paul Marriott
Objectives
SEM
Geometries
The example
• Data is 200 integers
Likelihood in
sparse
simplex
• Told each is total number of red squares out of 25 red
or blue
Least
informative
model
• Told each comes from unrelated experiments
Example
• Statistician A: Binomial model as working problem
Example
Global
analysis
formulation
Geometry of
Sensitivity
analysis
The example
Paul Marriott
Objectives
SEM
Geometries
The example
• Data is 200 integers
Likelihood in
sparse
simplex
• Told each is total number of red squares out of 25 red
or blue
Least
informative
model
• Told each comes from unrelated experiments
Example
• Statistician A: Binomial model as working problem
Example
Global
analysis
formulation
• Passes Goodness of Fit tests but for ‘outlier’
Geometry of
Sensitivity
analysis
The example
Paul Marriott
Objectives
SEM
Geometries
The example
• Data is 200 integers
Likelihood in
sparse
simplex
• Told each is total number of red squares out of 25 red
or blue
Least
informative
model
• Told each comes from unrelated experiments
Example
• Statistician A: Binomial model as working problem
Example
Global
analysis
formulation
• Passes Goodness of Fit tests but for ‘outlier’
• Statistician B: Looks at ‘raw’ data and talks to scientists
Geometry of
Sensitivity
analysis
The Example
Least
informative
model
6
5
0
1
2
3
4
5
4
0
Likelihood in
sparse
simplex
3
The example
2
SEM
Geometries
1
Objectives
6
Paul Marriott
0
1
2
3
4
5
6
0
1
2
3
4
5
6
0
1
2
3
4
5
6
0
1
2
3
4
5
6
Example
6
5
4
0
1
2
3
4
3
2
1
0
Global
analysis
5
6
Example
Geometry of
Sensitivity
analysis
The example
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Statistician B: uses theory which has equilibrium
distribution from a spatial Markov model
Geometry of
Sensitivity
analysis
The example
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Statistician B: uses theory which has equilibrium
distribution from a spatial Markov model
• Statistician C: is non parametric
Geometry of
Sensitivity
analysis
The example
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Statistician B: uses theory which has equilibrium
distribution from a spatial Markov model
• Statistician C: is non parametric
• Statistician D: uses robust methods
Geometry of
Sensitivity
analysis
The example
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Statistician B: uses theory which has equilibrium
distribution from a spatial Markov model
• Statistician C: is non parametric
• Statistician D: uses robust methods
• Main Question: can the universality and geometry the
SEM give a framework in which A, B, C and D can
communicate their differing views about the mean
number of red boxes?
Geometry of
Sensitivity
analysis
Shape of likelihood in SEM
Paul Marriott
Objectives
SEM
Geometries
• Inference on population mean
The example
• Working in high dimensional
Histogram
15
10
Frequency
parameters, µ = Nπ
Example
5
• Likelihood in mean
• Likelihood in natural
Global
analysis
0
Least
informative
model
multinomial with many
zero counts
20
Likelihood in
sparse
simplex
Example
parameters, η: no MLE
0
1
2
3
Data
4
5
• Likelihood in natural
parameters η: regular case
Geometry of
Sensitivity
analysis
Shape of likelihood in SEM
Paul Marriott
Objectives
SEM
Geometries
• Inference on population mean
The example
• Working in high dimensional
Histogram
3 20
15
• Likelihood in mean
51
10 2
Example
Global
analysis
0
Example
Frequency
Least
informative
model
multinomial with many
zero counts
4
Likelihood in
sparse
simplex
parameters, µ = Nπ
• Likelihood in natural
parameters, η: no MLE
00
11
22
33
Data
4
5
• Likelihood in natural
parameters η: regular case
Geometry of
Sensitivity
analysis
Shape of likelihood in SEM
Paul Marriott
Objectives
SEM
Geometries
• Inference on population mean
The example
• Working in high dimensional
Likelihood: Histogram
mean parametrisation
20
30.8
51
0.2
Frequency
pi21
Example
Global
analysis
0.0
0
Example
multinomial with many
zero counts
10 2 0.6
15
0.4
Least
informative
model
1.0
4
Likelihood in
sparse
simplex
• Likelihood in mean
parameters, µ = Nπ
• Likelihood in natural
parameters, η: no MLE
0
0
0.0
0.2
11
0.4
22
0.6
33
Data
π1
0.8
4
5
1.0
• Likelihood in natural
parameters η: regular case
Geometry of
Sensitivity
analysis
Shape of likelihood in SEM
Paul Marriott
Objectives
SEM
Geometries
• Inference on population mean
The example
Histogram
Likelihood:
Likelihood:natural
mean parametrisation
parametrisation
Example
Global
analysis
20
30.8
0
Frequency
pi2
η21
5 −10
10 2 −50.6
15
0.2
1
0.4
Example
• Working in high dimensional
multinomial with many
zero counts
• Likelihood in mean
parameters, µ = Nπ
• Likelihood in natural
−15
0.0
0
Least
informative
model
1.0
4
Likelihood in
sparse
simplex
parameters, η: no MLE
0
0
0.0
−3
−2
0.2
11
−1 0.4
22
0
Data
η
π1
0.6
33 1
0.8
42
5
1.0
3
• Likelihood in natural
parameters η: regular case
Geometry of
Sensitivity
analysis
Shape of likelihood in SEM
Paul Marriott
Objectives
SEM
Geometries
• Inference on population mean
The example
Histogram
Likelihood:
Likelihood:natural
mean parametrisation
parametrisation
Example
Global
analysis
Frequency
pi2
η21
5 −10
10 0
151 30.8
20
−2
0.2
1 −10.4
2 −50.6
0
2
Example
• Working in high dimensional
multinomial with many
zero counts
• Likelihood in mean
parameters, µ = Nπ
• Likelihood in natural
−15
0.0
−3
0
Least
informative
model
1.0
4
3
Likelihood in
sparse
simplex
parameters, η: no MLE
0
0
0.0
−3
−6
−2
0.2
11−4
−1 0.4
22 −2 0
Data
η
π1
0.6
33 0
1
0.8
42 2
5
1.0
3
• Likelihood in natural
parameters η: regular case
Geometry of
Sensitivity
analysis
Region of interest
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• The universality of the SEM is too rich to be the desired
framework for communication
Geometry of
Sensitivity
analysis
Region of interest
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
• The universality of the SEM is too rich to be the desired
Least
informative
model
• Only want to look at models which are data-supported
Example
Example
Global
analysis
framework for communication
Geometry of
Sensitivity
analysis
Region of interest
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
• The universality of the SEM is too rich to be the desired
Least
informative
model
• Only want to look at models which are data-supported
Example
Example
Global
analysis
framework for communication
• There are many types of goodness-of-fit tests on
simplex
Geometry of
Sensitivity
analysis
Region of interest
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
• The universality of the SEM is too rich to be the desired
Least
informative
model
• Only want to look at models which are data-supported
Example
Example
Global
analysis
framework for communication
• There are many types of goodness-of-fit tests on
simplex
• Such tests are necessary but not sufficient · · ·
Geometry of
Sensitivity
analysis
Thought experiment
Paul Marriott
Objectives
−1−geometry
+1−geometry
Least
informative
model
η2
−5
−10
π2
●
●
0.0
Likelihood in
sparse
simplex
0.4
The example
0
5
0.8
10
SEM
Geometries
0.0
0.2
0.4
0.6
0.8
1.0
−10
−5
Example
Log−Likelihood
−1
−2
−3
−4
Log−likelihood
0
Example
Global
analysis
0
η1
π1
1.5
2.0
2.5
mean
3.0
3.5
5
10
Geometry of
Sensitivity
analysis
Thought experiment
Paul Marriott
Objectives
−1−geometry
+1−geometry
Least
informative
model
η2
−5
−10
π2
●
●
0.0
Likelihood in
sparse
simplex
0.4
The example
0
5
0.8
10
SEM
Geometries
0.0
0.2
0.4
0.6
0.8
1.0
−10
−5
Example
Log−Likelihood
−1
−2
−3
−4
Log−likelihood
0
Example
Global
analysis
0
η1
π1
1.5
2.0
2.5
mean
3.0
3.5
5
10
Geometry of
Sensitivity
analysis
Thought experiment
Paul Marriott
Objectives
−1−geometry
+1−geometry
Least
informative
model
η2
−5
−10
π2
●
●
0.0
Likelihood in
sparse
simplex
0.4
The example
0
5
0.8
10
SEM
Geometries
0.0
0.2
0.4
0.6
0.8
1.0
−10
−5
Example
Log−Likelihood
−1
−2
−3
−4
Log−likelihood
0
Example
Global
analysis
0
η1
π1
1.5
2.0
2.5
mean
3.0
3.5
5
10
Geometry of
Sensitivity
analysis
Thought experiment
Paul Marriott
Objectives
−1−geometry
+1−geometry
Least
informative
model
η2
−5
−10
π2
●
●
0.0
Likelihood in
sparse
simplex
0.4
The example
0
5
0.8
10
SEM
Geometries
0.0
0.2
0.4
0.6
0.8
1.0
−10
−5
Example
Log−Likelihood
−1
−2
−3
−4
Log−likelihood
0
Example
Global
analysis
0
η1
π1
1.5
2.0
2.5
mean
3.0
3.5
5
10
Geometry of
Sensitivity
analysis
Thought experiment
Paul Marriott
Objectives
−1−geometry
+1−geometry
Least
informative
model
η2
−5
−10
π2
●
●
0.0
Likelihood in
sparse
simplex
0.4
The example
0
5
0.8
10
SEM
Geometries
0.0
0.2
0.4
0.6
0.8
1.0
−10
−5
Example
Log−Likelihood
−1
−2
−3
−4
Log−likelihood
0
Example
Global
analysis
0
η1
π1
1.5
2.0
2.5
mean
3.0
3.5
5
10
Geometry of
Sensitivity
analysis
Thought experiment
Paul Marriott
Objectives
−1−geometry
+1−geometry
Least
informative
model
η2
−5
−10
π2
●
●
0.0
Likelihood in
sparse
simplex
0.4
The example
0
5
0.8
10
SEM
Geometries
0.0
0.2
0.4
0.6
0.8
1.0
−10
−5
Example
Log−Likelihood
−1
−2
−3
−4
Log−likelihood
0
Example
Global
analysis
0
η1
π1
1.5
2.0
2.5
mean
3.0
3.5
5
10
Geometry of
Sensitivity
analysis
Thought experiment
Paul Marriott
Objectives
−1−geometry
+1−geometry
Least
informative
model
η2
−5
−10
π2
●
●
0.0
Likelihood in
sparse
simplex
0.4
The example
0
5
0.8
10
SEM
Geometries
0.0
0.2
0.4
0.6
0.8
1.0
−10
−5
Example
Log−Likelihood
−1
−2
−3
−4
Log−likelihood
0
Example
Global
analysis
0
η1
π1
1.5
2.0
2.5
mean
3.0
3.5
5
10
Geometry of
Sensitivity
analysis
Thought experiment
Paul Marriott
Objectives
−1−geometry
+1−geometry
Least
informative
model
η2
−5
−10
π2
●
●
0.0
Likelihood in
sparse
simplex
0.4
The example
0
5
0.8
10
SEM
Geometries
0.0
0.2
0.4
0.6
0.8
1.0
−10
−5
Example
Log−Likelihood
−1
−2
−3
−4
Log−likelihood
0
Example
Global
analysis
0
η1
π1
1.5
2.0
2.5
mean
3.0
3.5
5
10
Geometry of
Sensitivity
analysis
Thought experiment
Paul Marriott
Objectives
−1−geometry
+1−geometry
Least
informative
model
η2
−5
−10
π2
●
●
0.0
Likelihood in
sparse
simplex
0.4
The example
0
5
0.8
10
SEM
Geometries
0.0
0.2
0.4
0.6
0.8
1.0
−10
−5
Example
Log−Likelihood
−1
−2
−3
−4
Log−likelihood
0
Example
Global
analysis
0
η1
π1
1.5
2.0
2.5
mean
3.0
3.5
5
10
Geometry of
Sensitivity
analysis
Thought experiment
Paul Marriott
Objectives
−1−geometry
+1−geometry
Least
informative
model
η2
−5
−10
π2
●
●
0.0
Likelihood in
sparse
simplex
0.4
The example
0
5
0.8
10
SEM
Geometries
0.0
0.2
0.4
0.6
0.8
1.0
−10
−5
Example
Log−Likelihood
−1
−2
−3
−4
Log−likelihood
0
Example
Global
analysis
0
η1
π1
1.5
2.0
2.5
mean
3.0
3.5
5
10
Geometry of
Sensitivity
analysis
Least informative model
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Since rotation is through data generation process
(DGP) can’t use goodness-of-fit tests to distinguish
between models
Geometry of
Sensitivity
analysis
Least informative model
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Since rotation is through data generation process
(DGP) can’t use goodness-of-fit tests to distinguish
between models
• Rotation changes the mode and the shape of the
likelihood
Geometry of
Sensitivity
analysis
Least informative model
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Since rotation is through data generation process
(DGP) can’t use goodness-of-fit tests to distinguish
between models
• Rotation changes the mode and the shape of the
likelihood
• Smallest Expected Fisher information at DGP is when
exponential model is orthogonal to level sets of mean
Geometry of
Sensitivity
analysis
Least informative model
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Since rotation is through data generation process
(DGP) can’t use goodness-of-fit tests to distinguish
between models
• Rotation changes the mode and the shape of the
likelihood
• Smallest Expected Fisher information at DGP is when
exponential model is orthogonal to level sets of mean
• Models with this orthogonality property we call least
informative models
Geometry of
Sensitivity
analysis
Least informative model
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Since rotation is through data generation process
(DGP) can’t use goodness-of-fit tests to distinguish
between models
• Rotation changes the mode and the shape of the
likelihood
• Smallest Expected Fisher information at DGP is when
exponential model is orthogonal to level sets of mean
• Models with this orthogonality property we call least
informative models
• Information in inference comes from two sources: (i)
data and (ii) modelling assumptions. To be
conservative minimise (ii) relative to (i)
Geometry of
Sensitivity
analysis
Effect of translation
Paul Marriott
Objectives
−1−geometry
+1−geometry
Least
informative
model
η2
−10
−5
0.4
0.0
Likelihood in
sparse
simplex
π2
The example
0
5
0.8
10
SEM
Geometries
0.0
0.2
0.4
0.6
0.8
1.0
−10
−5
Example
Log−Likelihood
−1
−2
−3
−4
Log−likelihood
0
Example
Global
analysis
0
η1
π1
1.5
2.0
2.5
mean
3.0
3.5
5
10
Geometry of
Sensitivity
analysis
Effect of translation
Paul Marriott
Objectives
−1−geometry
+1−geometry
Least
informative
model
η2
−10
−5
0.4
0.0
Likelihood in
sparse
simplex
π2
The example
0
5
0.8
10
SEM
Geometries
0.0
0.2
0.4
0.6
0.8
1.0
−10
−5
Example
Log−Likelihood
−1
−2
−3
−4
Log−likelihood
0
Example
Global
analysis
0
η1
π1
1.5
2.0
2.5
mean
3.0
3.5
5
10
Geometry of
Sensitivity
analysis
Effect of translation
Paul Marriott
Objectives
−1−geometry
+1−geometry
Least
informative
model
η2
−10
−5
0.4
0.0
Likelihood in
sparse
simplex
π2
The example
0
5
0.8
10
SEM
Geometries
0.0
0.2
0.4
0.6
0.8
1.0
−10
−5
Example
Log−Likelihood
−1
−2
−3
−4
Log−likelihood
0
Example
Global
analysis
0
η1
π1
1.5
2.0
2.5
mean
3.0
3.5
5
10
Geometry of
Sensitivity
analysis
Effect of translation
Paul Marriott
Objectives
−1−geometry
+1−geometry
Least
informative
model
η2
−10
−5
0.4
0.0
Likelihood in
sparse
simplex
π2
The example
0
5
0.8
10
SEM
Geometries
0.0
0.2
0.4
0.6
0.8
1.0
−10
−5
Example
Log−Likelihood
−1
−2
−3
−4
Log−likelihood
0
Example
Global
analysis
0
η1
π1
1.5
2.0
2.5
mean
3.0
3.5
5
10
Geometry of
Sensitivity
analysis
Effect of translation
Paul Marriott
Objectives
−1−geometry
+1−geometry
Least
informative
model
η2
−10
−5
0.4
0.0
Likelihood in
sparse
simplex
π2
The example
0
5
0.8
10
SEM
Geometries
0.0
0.2
0.4
0.6
0.8
1.0
−10
−5
Example
Log−Likelihood
−1
−2
−3
−4
Log−likelihood
0
Example
Global
analysis
0
η1
π1
1.5
2.0
2.5
mean
3.0
3.5
5
10
Geometry of
Sensitivity
analysis
Effect of translation
Paul Marriott
Objectives
−1−geometry
+1−geometry
Least
informative
model
η2
−10
−5
0.4
0.0
Likelihood in
sparse
simplex
π2
The example
0
5
0.8
10
SEM
Geometries
0.0
0.2
0.4
0.6
0.8
1.0
−10
−5
Example
Log−Likelihood
−1
−2
−3
−4
Log−likelihood
0
Example
Global
analysis
0
η1
π1
1.5
2.0
2.5
mean
3.0
3.5
5
10
Geometry of
Sensitivity
analysis
Effect of translation
Paul Marriott
Objectives
−1−geometry
+1−geometry
Least
informative
model
η2
−10
−5
0.4
0.0
Likelihood in
sparse
simplex
π2
The example
0
5
0.8
10
SEM
Geometries
0.0
0.2
0.4
0.6
0.8
1.0
−10
−5
Example
Log−Likelihood
−1
−2
−3
−4
Log−likelihood
0
Example
Global
analysis
0
η1
π1
1.5
2.0
2.5
mean
3.0
3.5
5
10
Geometry of
Sensitivity
analysis
Effect of translation
Paul Marriott
Objectives
−1−geometry
+1−geometry
Least
informative
model
η2
−10
−5
0.4
0.0
Likelihood in
sparse
simplex
π2
The example
0
5
0.8
10
SEM
Geometries
0.0
0.2
0.4
0.6
0.8
1.0
−10
−5
Example
Log−Likelihood
−1
−2
−3
−4
Log−likelihood
0
Example
Global
analysis
0
η1
π1
1.5
2.0
2.5
mean
3.0
3.5
5
10
Geometry of
Sensitivity
analysis
Sensitive perturbations
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• There exists large perturbations of models which have
no effect on inference
Geometry of
Sensitivity
analysis
Sensitive perturbations
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• There exists large perturbations of models which have
no effect on inference
• Limit of these translation exists-use the correct topology
Geometry of
Sensitivity
analysis
Sensitive perturbations
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• There exists large perturbations of models which have
no effect on inference
• Limit of these translation exists-use the correct topology
• Limit is Profile Likelihood
Geometry of
Sensitivity
analysis
Sensitive perturbations
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• There exists large perturbations of models which have
no effect on inference
• Limit of these translation exists-use the correct topology
• Limit is Profile Likelihood
• Shows link between least informative parametric
inference and non-parametric inference
Geometry of
Sensitivity
analysis
Sensitive perturbations
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• There exists large perturbations of models which have
no effect on inference
• Limit of these translation exists-use the correct topology
• Limit is Profile Likelihood
• Shows link between least informative parametric
inference and non-parametric inference
• So SEM captures both Statistician A and C views
Geometry of
Sensitivity
analysis
Sensitive perturbations
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
• There exists large perturbations of models which have
no effect on inference
• Limit of these translation exists-use the correct topology
• Limit is Profile Likelihood
• Shows link between least informative parametric
inference and non-parametric inference
Example
• So SEM captures both Statistician A and C views
Global
analysis
• The number of perturbations which matter for inference
about the mean can be surprising small
Geometry of
Sensitivity
analysis
Sensitive perturbations
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
• There exists large perturbations of models which have
no effect on inference
• Limit of these translation exists-use the correct topology
• Limit is Profile Likelihood
• Shows link between least informative parametric
inference and non-parametric inference
Example
• So SEM captures both Statistician A and C views
Global
analysis
• The number of perturbations which matter for inference
about the mean can be surprising small
• We can compute these directions: see Karim’s talk on
approximate cuts
Geometry of
Sensitivity
analysis
Sensitive perturbations
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
• There exists large perturbations of models which have
no effect on inference
• Limit of these translation exists-use the correct topology
• Limit is Profile Likelihood
• Shows link between least informative parametric
inference and non-parametric inference
Example
• So SEM captures both Statistician A and C views
Global
analysis
• The number of perturbations which matter for inference
about the mean can be surprising small
• We can compute these directions: see Karim’s talk on
approximate cuts
Geometry of
Sensitivity
analysis
Information from Model
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Parametric Models: (A) Binomial (B) local Markov
model
Geometry of
Sensitivity
analysis
Information from Model
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
• Parametric Models: (A) Binomial (B) local Markov
Least
informative
model
• Both data consistent
Example
Example
Global
analysis
model
Geometry of
Sensitivity
analysis
Information from Model
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
• Parametric Models: (A) Binomial (B) local Markov
Least
informative
model
• Both data consistent
Example
• Binomial is a least informative model
Example
Global
analysis
model
Geometry of
Sensitivity
analysis
Information from Model
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
• Parametric Models: (A) Binomial (B) local Markov
Least
informative
model
• Both data consistent
Example
• Binomial is a least informative model
Example
• Local Markov model is not · · ·
Global
analysis
model
Geometry of
Sensitivity
analysis
Information from Model
Paul Marriott
Objectives
SEM
Geometries
Moment structure of models
The example
Example
Global
analysis
6.5
Variance
5.5
5.0
Example
4.5
Least
informative
model
Binomial
Markov
LIM
6.0
Likelihood in
sparse
simplex
5.5
6.0
6.5
7.0
Mean
7.5
8.0
8.5
Geometry of
Sensitivity
analysis
Information from Model
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• The information from the model is captured using the
mixed parametrisation in the universal SEM
Geometry of
Sensitivity
analysis
Information from Model
Paul Marriott
Objectives
SEM
Geometries
• The information from the model is captured using the
mixed parametrisation in the universal SEM
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Look at angle between parametric model and level set
of mean in SEM
Geometry of
Sensitivity
analysis
Information from Model
Paul Marriott
Objectives
SEM
Geometries
• The information from the model is captured using the
mixed parametrisation in the universal SEM
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Look at angle between parametric model and level set
of mean in SEM
• The smaller the angle the larger the model information
about parameter of interest
Geometry of
Sensitivity
analysis
Information from Model
Paul Marriott
Objectives
SEM
Geometries
• The information from the model is captured using the
mixed parametrisation in the universal SEM
The example
Likelihood in
sparse
simplex
• Look at angle between parametric model and level set
of mean in SEM
Least
informative
model
• The smaller the angle the larger the model information
Example
• If the models assumptions are correct increase
Example
Global
analysis
about parameter of interest
information
Geometry of
Sensitivity
analysis
Information from Model
Paul Marriott
Objectives
SEM
Geometries
• The information from the model is captured using the
mixed parametrisation in the universal SEM
The example
Likelihood in
sparse
simplex
• Look at angle between parametric model and level set
of mean in SEM
Least
informative
model
• The smaller the angle the larger the model information
Example
• If the models assumptions are correct increase
Example
Global
analysis
about parameter of interest
information
• Errors in model assumptions generate bias
Geometry of
Sensitivity
analysis
Information from Model
Paul Marriott
Objectives
SEM
Geometries
• The information from the model is captured using the
mixed parametrisation in the universal SEM
The example
Likelihood in
sparse
simplex
• Look at angle between parametric model and level set
of mean in SEM
Least
informative
model
• The smaller the angle the larger the model information
Example
• If the models assumptions are correct increase
Example
Global
analysis
about parameter of interest
information
• Errors in model assumptions generate bias
• The set of sensitive directions defines a framework in
which the four Statisticians can communicate, see
Karim’s talk
Geometry of
Sensitivity
analysis
Local to Global
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• The basic geometries of SEM are affine and convex,
rather than differential geometric
Geometry of
Sensitivity
analysis
Local to Global
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
• The basic geometries of SEM are affine and convex,
Least
informative
model
• Topology allows limits on boundaries to be taken
Example
Example
Global
analysis
rather than differential geometric
Geometry of
Sensitivity
analysis
Local to Global
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
• The basic geometries of SEM are affine and convex,
Least
informative
model
• Topology allows limits on boundaries to be taken
Example
Example
Global
analysis
rather than differential geometric
• Affine geometry allows downweight/delete outlier
c.f.Statistician D
Geometry of
Sensitivity
analysis
Local to Global
Paul Marriott
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
• The basic geometries of SEM are affine and convex,
Least
informative
model
• Topology allows limits on boundaries to be taken
Example
Example
Global
analysis
rather than differential geometric
• Affine geometry allows downweight/delete outlier
c.f.Statistician D
• Affine geometry allows local and global analysis
Geometry of
Sensitivity
analysis
Paul Marriott
Computation Information
Geometry
Objectives
SEM
Geometries
The example
Likelihood in
sparse
simplex
Least
informative
model
Example
Example
Global
analysis
• Computational Information Geometry
• The overall objective is to construct diagnostic tools to
help understand sensitivity to model choice
• Targeted at applications where Generalised Linear
Models are used