Computational Information Geometry:
Model Sensitivity and Approximate Cuts
Karim Anaya-Izquierdo (The Open University)
joint work with Frank Critchley (Open University),
Paul Marriott (Waterloo) and Paul Vos (East Carolina)
WOGAS 2, April 2010
CIG: Model Sensitivity and Approximate Cuts
Introduction
Overall objective:
provide tools to help understand sensitivity to model choice
CIG: Model Sensitivity and Approximate Cuts
Introduction
Overall objective:
provide tools to help understand sensitivity to model choice
Target:
applications of Generalised Linear Models
CIG: Model Sensitivity and Approximate Cuts
Introduction
Overall objective:
provide tools to help understand sensitivity to model choice
Target:
applications of Generalised Linear Models
Delivered via ...
Computational Information Geometry
CIG: Model Sensitivity and Approximate Cuts
Introduction
Overall objective:
provide tools to help understand sensitivity to model choice
Target:
applications of Generalised Linear Models
Delivered via ...
Computational Information Geometry
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Key ingredients
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Key ingredients
θ quantity/parameter of interest
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Key ingredients
θ quantity/parameter of interest
x data
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Key ingredients
θ quantity/parameter of interest
x data
f (x; θ ) base/working model
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Key ingredients
θ quantity/parameter of interest
x data
f (x; θ ) base/working model
⇒
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Key ingredients
θ quantity/parameter of interest
x data
f (x; θ ) base/working model
⇒ inference about θ
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Key ingredients
θ quantity/parameter of interest
x data
f (x; θ ) base/working model
⇒ inference about θ
What is the effect on inference about θ when considering
models apart from f (x; θ )?
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Key ingredients
θ quantity/parameter of interest
x data
f (x; θ ) base/working model
⇒ inference about θ
What is the effect on inference about θ when considering
models apart from f (x; θ )?
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Perturb base model f (x; θ )
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Perturb base model f (x; θ )
‘Pick a direction’ η
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Perturb base model f (x; θ )
‘Pick a direction’ η
Perturb
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Perturb base model f (x; θ )
‘Pick a direction’ η
Perturb
f (x; θ ) → g (x; θ, η )
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Perturb base model f (x; θ )
‘Pick a direction’ η
Perturb
f (x; θ ) → g (x; θ, η )
so that
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Perturb base model f (x; θ )
‘Pick a direction’ η
Perturb
f (x; θ ) → g (x; θ, η )
so that
1
θ does not lose its meaning
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Perturb base model f (x; θ )
‘Pick a direction’ η
Perturb
f (x; θ ) → g (x; θ, η )
so that
1
2
θ does not lose its meaning
g (x; θ, η0 ) = f (x; θ ) for some η0
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Perturb base model f (x; θ )
‘Pick a direction’ η
Perturb
f (x; θ ) → g (x; θ, η )
so that
1
2
θ does not lose its meaning
g (x; θ, η0 ) = f (x; θ ) for some η0
Examples of ‘directions’ η
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Perturb base model f (x; θ )
‘Pick a direction’ η
Perturb
f (x; θ ) → g (x; θ, η )
so that
1
2
θ does not lose its meaning
g (x; θ, η0 ) = f (x; θ ) for some η0
Examples of ‘directions’ η
introduce extra variability (mixture model)
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Perturb base model f (x; θ )
‘Pick a direction’ η
Perturb
f (x; θ ) → g (x; θ, η )
so that
1
2
θ does not lose its meaning
g (x; θ, η0 ) = f (x; θ ) for some η0
Examples of ‘directions’ η
introduce extra variability (mixture model)
introduce skewness
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Perturb base model f (x; θ )
‘Pick a direction’ η
Perturb
f (x; θ ) → g (x; θ, η )
so that
1
2
θ does not lose its meaning
g (x; θ, η0 ) = f (x; θ ) for some η0
Examples of ‘directions’ η
introduce extra variability (mixture model)
introduce skewness
introduce heavier tails
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Perturb base model f (x; θ )
‘Pick a direction’ η
Perturb
f (x; θ ) → g (x; θ, η )
so that
1
2
θ does not lose its meaning
g (x; θ, η0 ) = f (x; θ ) for some η0
Examples of ‘directions’ η
introduce
introduce
introduce
introduce
extra variability (mixture model)
skewness
heavier tails
dependence
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Perturb base model f (x; θ )
‘Pick a direction’ η
Perturb
f (x; θ ) → g (x; θ, η )
so that
1
2
θ does not lose its meaning
g (x; θ, η0 ) = f (x; θ ) for some η0
Examples of ‘directions’ η
introduce
introduce
introduce
introduce
extra variability (mixture model)
skewness
heavier tails
dependence
What is the effect on inference about θ when perturbing along
the direction η?
CIG: Model Sensitivity and Approximate Cuts
Model sensitivity
Perturb base model f (x; θ )
‘Pick a direction’ η
Perturb
f (x; θ ) → g (x; θ, η )
so that
1
2
θ does not lose its meaning
g (x; θ, η0 ) = f (x; θ ) for some η0
Examples of ‘directions’ η
introduce
introduce
introduce
introduce
extra variability (mixture model)
skewness
heavier tails
dependence
What is the effect on inference about θ when perturbing along
the direction η?
CIG: Model Sensitivity and Approximate Cuts
Cuts
Model g (x; θ, η )
CIG: Model Sensitivity and Approximate Cuts
Cuts
Model g (x; θ, η )
θ is of interest
CIG: Model Sensitivity and Approximate Cuts
Cuts
Model g (x; θ, η )
θ is of interest
η is nuisance
CIG: Model Sensitivity and Approximate Cuts
Cuts
Model g (x; θ, η )
θ is of interest
η is nuisance
A cut is a statistic which allow inference about θ
independently of the values of η
CIG: Model Sensitivity and Approximate Cuts
Cuts
Model g (x; θ, η )
θ is of interest
η is nuisance
A cut is a statistic which allow inference about θ
independently of the values of η
Key idea is factorisation of the likelihood
CIG: Model Sensitivity and Approximate Cuts
Cuts
Model g (x; θ, η )
θ is of interest
η is nuisance
A cut is a statistic which allow inference about θ
independently of the values of η
Key idea is factorisation of the likelihood
if (T (x ), S (x )) ⇔ x,
CIG: Model Sensitivity and Approximate Cuts
Cuts
Model g (x; θ, η )
θ is of interest
η is nuisance
A cut is a statistic which allow inference about θ
independently of the values of η
Key idea is factorisation of the likelihood
if (T (x ), S (x )) ⇔ x, (θ, η ) ∈ Θ × Λ
CIG: Model Sensitivity and Approximate Cuts
Cuts
Model g (x; θ, η )
θ is of interest
η is nuisance
A cut is a statistic which allow inference about θ
independently of the values of η
Key idea is factorisation of the likelihood
if (T (x ), S (x )) ⇔ x, (θ, η ) ∈ Θ × Λ
L(θ, η; x ) =
CIG: Model Sensitivity and Approximate Cuts
Cuts
Model g (x; θ, η )
θ is of interest
η is nuisance
A cut is a statistic which allow inference about θ
independently of the values of η
Key idea is factorisation of the likelihood
if (T (x ), S (x )) ⇔ x, (θ, η ) ∈ Θ × Λ
L(θ, η; x ) = Li (θ; T (x )) ×
CIG: Model Sensitivity and Approximate Cuts
Cuts
Model g (x; θ, η )
θ is of interest
η is nuisance
A cut is a statistic which allow inference about θ
independently of the values of η
Key idea is factorisation of the likelihood
if (T (x ), S (x )) ⇔ x, (θ, η ) ∈ Θ × Λ
L(θ, η; x ) = Li (θ; T (x )) × Ln (η; S (x ))
∀ θ, η, x
CIG: Model Sensitivity and Approximate Cuts
Cuts
Model g (x; θ, η )
θ is of interest
η is nuisance
A cut is a statistic which allow inference about θ
independently of the values of η
Key idea is factorisation of the likelihood
if (T (x ), S (x )) ⇔ x, (θ, η ) ∈ Θ × Λ
L(θ, η; x ) = Li (θ; T (x )) × Ln (η; S (x ))
∀ θ, η, x
then T is a cut
CIG: Model Sensitivity and Approximate Cuts
Cuts
Model g (x; θ, η )
θ is of interest
η is nuisance
A cut is a statistic which allow inference about θ
independently of the values of η
Key idea is factorisation of the likelihood
if (T (x ), S (x )) ⇔ x, (θ, η ) ∈ Θ × Λ
L(θ, η; x ) = Li (θ; T (x )) × Ln (η; S (x ))
∀ θ, η, x
then T is a cut
Very tight definition valid in very special cases:
CIG: Model Sensitivity and Approximate Cuts
Cuts
Model g (x; θ, η )
θ is of interest
η is nuisance
A cut is a statistic which allow inference about θ
independently of the values of η
Key idea is factorisation of the likelihood
if (T (x ), S (x )) ⇔ x, (θ, η ) ∈ Θ × Λ
L(θ, η; x ) = Li (θ; T (x )) × Ln (η; S (x ))
∀ θ, η, x
then T is a cut
Very tight definition valid in very special cases:
Exponential Families with so-called mixed parametrisation
CIG: Model Sensitivity and Approximate Cuts
Cuts for model sensitivity
Inferences on θ are made using T (x ), for example
CIG: Model Sensitivity and Approximate Cuts
Cuts for model sensitivity
Inferences on θ are made using T (x ), for example
an interval estimate for θ
CIG: Model Sensitivity and Approximate Cuts
Cuts for model sensitivity
Inferences on θ are made using T (x ), for example
an interval estimate for θ
the rejection region of a hypothesis test
CIG: Model Sensitivity and Approximate Cuts
Cuts for model sensitivity
Inferences on θ are made using T (x ), for example
an interval estimate for θ
the rejection region of a hypothesis test
Ideal case:
CIG: Model Sensitivity and Approximate Cuts
Cuts for model sensitivity
Inferences on θ are made using T (x ), for example
an interval estimate for θ
the rejection region of a hypothesis test
Ideal case:
there exists an enlarged model g (x; θ, η ) with cut
(T (x ), S (x ))
CIG: Model Sensitivity and Approximate Cuts
Cuts for model sensitivity
Inferences on θ are made using T (x ), for example
an interval estimate for θ
the rejection region of a hypothesis test
Ideal case:
there exists an enlarged model g (x; θ, η ) with cut
(T (x ), S (x ))
⇒ perturbation η0 → η does not affect inferences about θ
CIG: Model Sensitivity and Approximate Cuts
Cuts for model sensitivity
Inferences on θ are made using T (x ), for example
an interval estimate for θ
the rejection region of a hypothesis test
Ideal case:
there exists an enlarged model g (x; θ, η ) with cut
(T (x ), S (x ))
⇒ perturbation η0 → η does not affect inferences about θ
In practice relax conditions to define an approximate cut:
CIG: Model Sensitivity and Approximate Cuts
Cuts for model sensitivity
Inferences on θ are made using T (x ), for example
an interval estimate for θ
the rejection region of a hypothesis test
Ideal case:
there exists an enlarged model g (x; θ, η ) with cut
(T (x ), S (x ))
⇒ perturbation η0 → η does not affect inferences about θ
In practice relax conditions to define an approximate cut:
Inferences about θ based on T (x ) ‘do not change much’ when
perturbing in the direction η
CIG: Model Sensitivity and Approximate Cuts
Cuts for model sensitivity
Inferences on θ are made using T (x ), for example
an interval estimate for θ
the rejection region of a hypothesis test
Ideal case:
there exists an enlarged model g (x; θ, η ) with cut
(T (x ), S (x ))
⇒ perturbation η0 → η does not affect inferences about θ
In practice relax conditions to define an approximate cut:
Inferences about θ based on T (x ) ‘do not change much’ when
perturbing in the direction η
data x is fixed
CIG: Model Sensitivity and Approximate Cuts
Cuts for model sensitivity
Inferences on θ are made using T (x ), for example
an interval estimate for θ
the rejection region of a hypothesis test
Ideal case:
there exists an enlarged model g (x; θ, η ) with cut
(T (x ), S (x ))
⇒ perturbation η0 → η does not affect inferences about θ
In practice relax conditions to define an approximate cut:
Inferences about θ based on T (x ) ‘do not change much’ when
perturbing in the direction η
data x is fixed
For (θ, η ) values where data x supports the use of g (x; θ, η )
CIG: Model Sensitivity and Approximate Cuts
Cuts for model sensitivity
Inferences on θ are made using T (x ), for example
an interval estimate for θ
the rejection region of a hypothesis test
Ideal case:
there exists an enlarged model g (x; θ, η ) with cut
(T (x ), S (x ))
⇒ perturbation η0 → η does not affect inferences about θ
In practice relax conditions to define an approximate cut:
Inferences about θ based on T (x ) ‘do not change much’ when
perturbing in the direction η
data x is fixed
For (θ, η ) values where data x supports the use of g (x; θ, η )
By complement we can find directions η with large effect on
inference:
CIG: Model Sensitivity and Approximate Cuts
Cuts for model sensitivity
Inferences on θ are made using T (x ), for example
an interval estimate for θ
the rejection region of a hypothesis test
Ideal case:
there exists an enlarged model g (x; θ, η ) with cut
(T (x ), S (x ))
⇒ perturbation η0 → η does not affect inferences about θ
In practice relax conditions to define an approximate cut:
Inferences about θ based on T (x ) ‘do not change much’ when
perturbing in the direction η
data x is fixed
For (θ, η ) values where data x supports the use of g (x; θ, η )
By complement we can find directions η with large effect on
inference:
Confidence interval length substantially increased
CIG: Model Sensitivity and Approximate Cuts
Cuts for model sensitivity
Inferences on θ are made using T (x ), for example
an interval estimate for θ
the rejection region of a hypothesis test
Ideal case:
there exists an enlarged model g (x; θ, η ) with cut
(T (x ), S (x ))
⇒ perturbation η0 → η does not affect inferences about θ
In practice relax conditions to define an approximate cut:
Inferences about θ based on T (x ) ‘do not change much’ when
perturbing in the direction η
data x is fixed
For (θ, η ) values where data x supports the use of g (x; θ, η )
By complement we can find directions η with large effect on
inference:
Confidence interval length substantially increased
Do not reject instead of rejecting
CIG: Model Sensitivity and Approximate Cuts
Our approach
Use Structured Extended Multinomial models (SEM’s)
Extended multinomials are multinomials but allow cell
probabilities to be zero
Discretising continuous data gives multinomials with structure
on the cells
SEM proxy for universal space of all distributions
CIG: Model Sensitivity and Approximate Cuts
Example
Question: what is the population mean θ?
0.6
Histogram of toy.data
0.5
Data
0.4
Model N (θ, 1)
0.3
Model N (θ, σ2 )
0.2
Model N (θ, 1) remove outlier (Base)
0.1
Model Logistic(µ(θ ), σ )
0.0
Model log-Normal(µ0 (θ ), σ2 )
−2
0
2
4
6
8
Data
CIG: Model Sensitivity and Approximate Cuts
Example
Question: what is the population mean θ?
0.6
Histogram of toy.data
0.5
Data
0.4
Model N (θ, 1)
0.3
Model N (θ, σ2 )
0.2
Model N (θ, 1) remove outlier (Base)
0.1
Model Logistic(µ(θ ), σ )
0.0
Model log-Normal(µ0 (θ ), σ2 )
−2
0
2
4
6
8
Data
CIG: Model Sensitivity and Approximate Cuts
Example
Question: what is the population mean θ?
0.6
Histogram of toy.data
0.5
Data
0.4
Model N (θ, 1)
0.3
Model N (θ, σ2 )
0.2
Model N (θ, 1) remove outlier (Base)
0.1
Model Logistic(µ(θ ), σ )
0.0
Model log-Normal(µ0 (θ ), σ2 )
−2
0
2
4
6
8
Data
CIG: Model Sensitivity and Approximate Cuts
Example
Question: what is the population mean θ?
0.6
Histogram of toy.data
0.5
Data
0.4
Model N (θ, 1)
0.3
Model N (θ, σ2 )
0.2
Model N (θ, 1) remove outlier (Base)
0.1
Model Logistic(µ(θ ), σ )
Model log-Normal(µ0 (θ ), σ2 )
0.0
Outlier
−2
0
2
4
6
8
Data
CIG: Model Sensitivity and Approximate Cuts
Example
Question: what is the population mean θ?
0.6
Histogram of toy.data
0.5
Data
0.4
Model N (θ, 1)
0.3
Model N (θ, σ2 )
0.2
Model N (θ, 1) remove outlier (Base)
0.1
Model Logistic(µ(θ ), σ )
Model log-Normal(µ0 (θ ), σ2 )
0.0
Outlier
−2
0
2
4
6
8
Data
CIG: Model Sensitivity and Approximate Cuts
Example
Question: what is the population mean θ?
0.6
Histogram of toy.data
0.5
Data
0.4
Model N (θ, 1)
0.3
Model N (θ, σ2 )
0.2
Model N (θ, 1) remove outlier (Base)
0.1
Model Logistic(µ(θ ), σ )
Model log-Normal(µ0 (θ ), σ2 )
0.0
Outlier
−2
0
2
4
6
8
Data
CIG: Model Sensitivity and Approximate Cuts
Insensitive direction: CUT
0.4
1
0
2
3
4
2
6
−3.0
−4
−2.5
−2
5
−1.0
7
1.6
8
1.
−1.5
8
log−likelihood
6
2.2
2
0.
−2.0
4
1.4
0.8
−0.5
1.2
1
η
Log−likelihood
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Insensitive direction: CUT
0.4
1
0
2
3
4
2
6
−3.0
−4
−2.5
−2
5
−1.0
7
1.6
8
1.
−1.5
8
log−likelihood
6
2.2
2
0.
−2.0
4
1.4
0.8
−0.5
1.2
1
η
Log−likelihood
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Insensitive direction: CUT
0.4
1
0
2
3
4
2
6
−3.0
−4
−2.5
−2
5
−1.0
7
1.6
8
1.
−1.5
8
log−likelihood
6
2.2
2
0.
−2.0
4
1.4
0.8
−0.5
1.2
1
η
Log−likelihood
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Insensitive direction: CUT
0.4
1
0
2
3
4
2
6
−3.0
−4
−2.5
−2
5
−1.0
7
1.6
8
1.
−1.5
8
log−likelihood
6
2.2
2
0.
−2.0
4
1.4
0.8
−0.5
1.2
1
η
Log−likelihood
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Insensitive direction: CUT
0.4
1
0
2
3
4
2
6
−3.0
−4
−2.5
−2
5
−1.0
7
1.6
8
1.
−1.5
8
log−likelihood
6
2.2
2
0.
−2.0
4
1.4
0.8
−0.5
1.2
1
η
Log−likelihood
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Insensitive direction: CUT
0.4
1
0
2
3
4
2
6
−3.0
−4
−2.5
−2
5
−1.0
7
1.6
8
1.
−1.5
8
log−likelihood
6
2.2
2
0.
−2.0
4
1.4
0.8
−0.5
1.2
1
η
Log−likelihood
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Insensitive direction: CUT
0.4
1
0
2
3
4
2
6
−3.0
−4
−2.5
−2
5
−1.0
7
1.6
8
1.
−1.5
8
log−likelihood
6
2.2
2
0.
−2.0
4
1.4
0.8
−0.5
1.2
1
η
Log−likelihood
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Insensitive direction: CUT
0.4
1
0
2
3
4
2
6
−3.0
−4
−2.5
−2
5
−1.0
7
1.6
8
1.
−1.5
8
log−likelihood
6
2.2
2
0.
−2.0
4
1.4
0.8
−0.5
1.2
1
η
Log−likelihood
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Insensitive direction: CUT
0.4
1
0
2
3
4
2
6
−3.0
−4
−2.5
−2
5
−1.0
7
1.6
8
1.
−1.5
8
log−likelihood
6
2.2
2
0.
−2.0
4
1.4
0.8
−0.5
1.2
1
η
Log−likelihood
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Insensitive direction: CUT
0.4
1
0
2
3
4
2
6
−3.0
−4
−2.5
−2
5
−1.0
7
1.6
8
1.
−1.5
8
log−likelihood
6
2.2
2
0.
−2.0
4
1.4
0.8
−0.5
1.2
1
η
Log−likelihood
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Insensitive direction: CUT
0.4
1
0
2
3
4
2
6
−3.0
−4
−2.5
−2
5
−1.0
7
1.6
8
1.
−1.5
8
log−likelihood
6
2.2
2
0.
−2.0
4
1.4
0.8
−0.5
1.2
1
η
Log−likelihood
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Insensitive direction: CUT
Log−likelihood
0.0
Natural Parameters
−3.0
−4
−2.5
−2
5
−1.0
2
2
7
log−likelihood
0
η2
0.4
8
6
4 3
1
2.2
2
8
1.
6
−1.5
1.6
0.8
0.
−2.0
4
1.4
−0.5
1.2
1
−2
−1
0
1
2
1.0
1.2
1.4
η1
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Insensitive direction: CUT
Log−likelihood
0.0
Natural Parameters
−3.0
−4
−2.5
−2
5
−1.0
2
2
7
log−likelihood
0
η2
0.4
8
6
4 3
1
2.2
2
8
1.
6
−1.5
1.6
0.8
0.
−2.0
4
1.4
−0.5
1.2
1
−2
−1
0
1
2
1.0
1.2
1.4
η1
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Insensitive direction: CUT
Log−likelihood
0.0
Natural Parameters
−3.0
−4
−2.5
−2
5
−1.0
2
2
7
log−likelihood
0
η2
0.4
8
6
4 3
1
2.2
2
8
1.
6
−1.5
1.6
0.8
0.
−2.0
4
1.4
−0.5
1.2
1
−2
−1
0
1
2
1.0
1.2
1.4
η1
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Insensitive direction: CUT
Log−likelihood
0.0
Natural Parameters
−3.0
−4
−2.5
−2
5
−1.0
2
2
7
log−likelihood
0
η2
0.4
8
6
4 3
1
2.2
2
8
1.
6
−1.5
1.6
0.8
0.
−2.0
4
1.4
−0.5
1.2
1
−2
−1
0
1
2
1.0
1.2
1.4
η1
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Insensitive direction: CUT
Log−likelihood
0.0
Natural Parameters
−3.0
−4
−2.5
−2
5
−1.0
2
2
7
log−likelihood
0
η2
0.4
8
6
4 3
1
2.2
2
8
1.
6
−1.5
1.6
0.8
0.
−2.0
4
1.4
−0.5
1.2
1
−2
−1
0
1
2
1.0
1.2
1.4
η1
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Insensitive direction: CUT
Log−likelihood
0.0
Natural Parameters
−3.0
−4
−2.5
−2
5
−1.0
2
2
7
log−likelihood
0
η2
0.4
8
6
4 3
1
2.2
2
8
1.
6
−1.5
1.6
0.8
0.
−2.0
4
1.4
−0.5
1.2
1
−2
−1
0
1
2
1.0
1.2
1.4
η1
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction
Log−likelihood
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
0
η
1
2
4
5
7
−2
−3.0
−2.5
−1
8
−1.0
1.4
−1.5
1.2
log−likelihood
1 0
.8
−2.0
0.4
.6
1
0.2
0
−0.5
2
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction
Log−likelihood
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
0
η
1
2
4
5
7
−2
−3.0
−2.5
−1
8
−1.0
1.4
−1.5
1.2
log−likelihood
1 0
.8
−2.0
0.4
.6
1
0.2
0
−0.5
2
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction
Log−likelihood
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
0
η
1
2
4
5
7
−2
−3.0
−2.5
−1
8
−1.0
1.4
−1.5
1.2
log−likelihood
1 0
.8
−2.0
0.4
.6
1
0.2
0
−0.5
2
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction
Log−likelihood
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
0
η
1
2
4
5
7
−2
−3.0
−2.5
−1
8
−1.0
1.4
−1.5
1.2
log−likelihood
1 0
.8
−2.0
0.4
.6
1
0.2
0
−0.5
2
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction
Log−likelihood
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
0
η
1
2
4
5
7
−2
−3.0
−2.5
−1
8
−1.0
1.4
−1.5
1.2
log−likelihood
1 0
.8
−2.0
0.4
.6
1
0.2
0
−0.5
2
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction
Log−likelihood
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
0
η
1
2
4
5
7
−2
−3.0
−2.5
−1
8
−1.0
1.4
−1.5
1.2
log−likelihood
1 0
.8
−2.0
0.4
.6
1
0.2
0
−0.5
2
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction
Log−likelihood
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
0
η
1
2
4
5
7
−2
−3.0
−2.5
−1
8
−1.0
1.4
−1.5
1.2
log−likelihood
1 0
.8
−2.0
0.4
.6
1
0.2
0
−0.5
2
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction
Log−likelihood
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
0
η
1
2
4
5
7
−2
−3.0
−2.5
−1
8
−1.0
1.4
−1.5
1.2
log−likelihood
1 0
.8
−2.0
0.4
.6
1
0.2
0
−0.5
2
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction
Log−likelihood
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
0
η
1
2
4
5
7
−2
−3.0
−2.5
−1
8
−1.0
1.4
−1.5
1.2
log−likelihood
1 0
.8
−2.0
0.4
.6
1
0.2
0
−0.5
2
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction
Log−likelihood
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
0
η
1
2
4
5
7
−2
−3.0
−2.5
−1
8
−1.0
1.4
−1.5
1.2
log−likelihood
1 0
.8
−2.0
0.4
.6
1
0.2
0
−0.5
2
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction
Log−likelihood
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
0
η
1
2
4
5
7
−2
−3.0
−2.5
−1
8
−1.0
1.4
−1.5
1.2
log−likelihood
1 0
.8
−2.0
0.4
.6
1
0.2
0
−0.5
2
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction
Log−likelihood
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
0
η
1
2
4
5
7
−2
−3.0
−2.5
−1
8
−1.0
1.4
−1.5
1.2
log−likelihood
1 0
.8
−2.0
0.4
.6
1
0.2
0
−0.5
2
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction
Log−likelihood
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
0
η
1
2
4
5
7
−2
−3.0
−2.5
−1
8
−1.0
1.4
−1.5
1.2
log−likelihood
1 0
.8
−2.0
0.4
.6
1
0.2
0
−0.5
2
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction
Log−likelihood
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
0
η
1
2
4
5
7
−2
−3.0
−2.5
−1
8
−1.0
1.4
−1.5
1.2
log−likelihood
1 0
.8
−2.0
0.4
.6
1
0.2
0
−0.5
2
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction
Log−likelihood
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
0
η
1
2
4
5
7
−2
−3.0
−2.5
−1
8
−1.0
1.4
−1.5
1.2
log−likelihood
1 0
.8
−2.0
0.4
.6
1
0.2
0
−0.5
2
0.0
Natural Parameters
−2
−1
0
1
2
1.0
1.2
1.4
λ
1.6
1.8
2.0
θ
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction: Interpretation
PMF
1.2
1.4
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
●
0
η
1
4
5
7
−2
0.00
−1
8
0.04
2
0.08
1 0
.8
Probability
0.4
.6
1
0.2
0
0.12
2
Natural Parameters
−2
−1
0
1
2
−2
0
λ
2
4
6
x
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction: Interpretation
PMF
1.2
1.4
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
●
0
η
1
4
5
7
−2
0.00
−1
8
0.04
2
0.08
1 0
.8
Probability
0.4
.6
1
0.2
0
0.12
2
Natural Parameters
−2
−1
0
1
2
−2
0
λ
2
4
6
x
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction: Interpretation
PMF
1.2
1.4
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
●
0
η
1
4
5
7
−2
0.00
−1
8
0.04
2
0.08
1 0
.8
Probability
0.4
.6
1
0.2
0
0.12
2
Natural Parameters
−2
−1
0
1
2
−2
0
λ
2
4
6
x
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction: Interpretation
PMF
1.2
1.4
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
●
0
η
1
4
5
7
−2
0.00
−1
8
0.04
2
0.08
1 0
.8
Probability
0.4
.6
1
0.2
0
0.12
2
Natural Parameters
−2
−1
0
1
2
−2
0
λ
2
4
6
x
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction: Interpretation
PMF
1.2
1.4
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
●
0
η
1
4
5
7
−2
0.00
−1
8
0.04
2
0.08
1 0
.8
Probability
0.4
.6
1
0.2
0
0.12
2
Natural Parameters
−2
−1
0
1
2
−2
0
λ
2
4
6
x
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction: Interpretation
PMF
1.2
1.4
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
●
0
η
1
4
5
7
−2
0.00
−1
8
0.04
2
0.08
1 0
.8
Probability
0.4
.6
1
0.2
0
0.12
2
Natural Parameters
−2
−1
0
1
2
−2
0
λ
2
4
6
x
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction: Interpretation
PMF
1.2
1.4
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
1
0
η
●
4
5
7
−2
0.00
−1
8
0.04
2
0.08
1 0
.8
Probability
0.4
.6
1
0.2
0
0.12
2
Natural Parameters
−2
−1
0
1
2
−2
0
λ
2
4
6
x
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction: Interpretation
PMF
1.2
1.4
1.6
3.6
1.8
6
3.8
2.2
2.6
2
3.4
3.2
2.8
2.4
3
1
0
η
●3
4
5
7
−2
0.00
−1
8
0.04
2
0.08
1 0
.8
Probability
0.4
.6
1
0.2
0
0.12
2
Natural Parameters
−2
−1
0
1
2
−2
0
λ
2
4
6
x
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction: Interpretation
PMF
1.2
1.4
1.6
3.6
1.8
●
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
0
η
1
4
5
7
−2
0.00
−1
8
0.04
2
0.08
1 0
.8
Probability
0.4
.6
1
0.2
0
0.12
2
Natural Parameters
−2
−1
0
1
2
−2
0
λ
2
4
6
x
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction: Interpretation
PMF
1.2
1.4
1.6
3.6
1.8
●
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
0
η
1
4
5
7
−2
0.00
−1
8
0.04
2
0.08
1 0
.8
Probability
0.4
.6
1
0.2
0
0.12
2
Natural Parameters
−2
−1
0
1
2
−2
0
λ
2
4
6
x
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction: Interpretation
PMF
1.2
1.4
1.6
3.6
●
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
0
η
1
4
5
7
−2
0.00
−1
8
0.04
2
0.08
1 0
.8
Probability
0.4
.6
1
0.2
0
0.12
2
Natural Parameters
−2
−1
0
1
2
−2
0
λ
2
4
6
x
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction: Interpretation
PMF
1.2
1.4
1.6
●3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
0
η
1
4
5
7
−2
0.00
−1
8
0.04
2
0.08
1 0
.8
Probability
0.4
.6
1
0.2
0
0.12
2
Natural Parameters
−2
−1
0
1
2
−2
0
λ
2
4
6
x
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Highly sensitive direction: Interpretation
PMF
1.2
1.4
1.6
●3.6
1.8
6
3.8
2.2
2.6
3
2
3.4
3.2
2.8
2.4
3
0
η
1
4
5
7
−2
0.00
−1
8
0.04
2
0.08
1 0
.8
Probability
0.4
.6
1
0.2
0
0.12
2
Natural Parameters
−2
−1
0
1
2
−2
0
λ
2
4
6
x
Blue = mean; Black = Likelihood; Red = Base Model
data \ outlier
CIG: Model Sensitivity and Approximate Cuts
Sensitive direction: Mixed Parameters (Duality)
2
Mixed Parameters
2
Natural Parameters
1.2
1.4
2
1
4
1.2 1.4
1.6
3.6
1.8
6
3.8
2.2
2.6
3
2
1.8
3.4
2.2
2.4
2.6
2.8 3
3.4
3.2
2.8
2.4
8
3
5
3
4
7
1
3.6
0
0
η
1
3.4
3.2
3
3.8
4
2
2
5
6
7
−2
−1
8
−1
4
−2
η
1.6
1
1 0
.8
3
3.2
0.4
.6
1
0.2
0
−2
−1
0
λ
1
2
0.5
1.0
1.5
2.0
2.5
3.0
θ
CIG: Model Sensitivity and Approximate Cuts
Insensitive direction: Mixed Parameters (Duality) CUT
Natural Parameters
1.2
1.6
1.8
1.6
432
1.8
8
7
4
0
2
3
2
6
6
7
−4
−4
−2
8
−2
5
1
η
0.4
1
5
2.2
4321
3
4
0
2
2
0.6
1.2
2.2
2
2.4
1.4
0.8
4
4
1
η
Mixed Parameters
−2
−1
0
λ
1
2
0.5
1.0
1.5
2.0
2.5
3.0
θ
CIG: Model Sensitivity and Approximate Cuts
Perturbation Space
What is mean?
0.1
0.2
0.3
0.4
Data \ outlier
0.0
Outlier
−2
0
2
4
6
8
Data
CIG: Model Sensitivity and Approximate Cuts
Perturbation Space
What is mean?
0.1
0.2
0.3
0.4
Data \ outlier
0.0
Outlier
−2
0
2
4
6
8
Data
CIG: Model Sensitivity and Approximate Cuts
Perturbation Space
What is mean?
0.1
0.2
0.3
0.4
Data \ outlier and
base model N (µ, 1):
0.0
Outlier
−2
0
2
4
6
8
Data
CIG: Model Sensitivity and Approximate Cuts
Perturbation Space
What is mean?
0.1
0.2
0.3
0.4
Data \ outlier and
base model N (µ, 1): only one
direction is highly sensitive
0.0
Outlier
−2
0
2
4
6
8
Data
CIG: Model Sensitivity and Approximate Cuts
Perturbation Space
What is mean?
0.3
0.4
Data \ outlier and
base model N (µ, 1): only one
direction is highly sensitive
0.1
0.2
This corresponds to changing
variance giving model
N (µ, σ2 )
0.0
Outlier
−2
0
2
4
6
8
Data
CIG: Model Sensitivity and Approximate Cuts
Perturbation Space
What is mean?
0.3
0.4
Data \ outlier and
base model N (µ, 1): only one
direction is highly sensitive
0.1
0.2
This corresponds to changing
variance giving model
N (µ, σ2 )
0.0
Outlier
−2
0
2
4
6
8
If include outlier one
more parameter needed
Data
CIG: Model Sensitivity and Approximate Cuts
Other types of perturbations
We can perturb by:
CIG: Model Sensitivity and Approximate Cuts
Other types of perturbations
We can perturb by:
Translations · · · Least informative directions
CIG: Model Sensitivity and Approximate Cuts
Other types of perturbations
We can perturb by:
Translations · · · Least informative directions
Rotations another story · · · bias, variance trade-off · · ·
CIG: Model Sensitivity and Approximate Cuts
Other types of perturbations
We can perturb by:
Translations · · · Least informative directions
Rotations another story · · · bias, variance trade-off · · ·
Adding nuisance parameters · · ·
CIG: Model Sensitivity and Approximate Cuts
Other types of perturbations
We can perturb by:
Translations · · · Least informative directions
Rotations another story · · · bias, variance trade-off · · ·
Adding nuisance parameters · · ·
Down-weight/delete outliers · · ·
CIG: Model Sensitivity and Approximate Cuts
Other types of perturbations
We can perturb by:
Translations · · · Least informative directions
Rotations another story · · · bias, variance trade-off · · ·
Adding nuisance parameters · · ·
Down-weight/delete outliers · · ·
CIG: Model Sensitivity and Approximate Cuts
Computation
Discretise and work in SEM
CIG: Model Sensitivity and Approximate Cuts
Computation
Discretise and work in SEM
Extension g is an exponential family
CIG: Model Sensitivity and Approximate Cuts
Computation
Discretise and work in SEM
Extension g is an exponential family
Construct mixed parametrisation
CIG: Model Sensitivity and Approximate Cuts
Computation
Discretise and work in SEM
Extension g is an exponential family
Construct mixed parametrisation
Fisher Information (variance) is block diagonal
CIG: Model Sensitivity and Approximate Cuts
Computation
Discretise and work in SEM
Extension g is an exponential family
Construct mixed parametrisation
Fisher Information (variance) is block diagonal
Choose directions where θ block changes most with η
CIG: Model Sensitivity and Approximate Cuts
Summary
Overall objective:
provide tools to help understand sensitivity to model choice
Target:
applications of Generalised Linear Models
Delivered via ...
Computational Information Geometry (hidden from the user)
Work supported by EPSRC grant EP/E017878/1
CIG: Model Sensitivity and Approximate Cuts