This is the presentation for FMExtended in the Reloaded Conference

1. Fishing for Errors
Extending the Treatment of Errors in the
Fisher Matrix Formalism
!
ded Moumita Aich
a Mohammed El-Mufti
elo Eli Kasai
Brian Nord
R Marina Seikel
Sahba Yahya
Alan Heavens
Bruce Bassett
12 April 2012

2. Fisher Power!
The Fisher Matrix forecasts astronomical constraints
for model parameters: it can be used to predict 68% confidence that parameters
confidence contours cosmological parameter! lie within the blue (dashed)
contour.
Ingredients required:
a parametrized, physical model for an
θA
observable [ y = f(x;θ) ]
a set of errors in the observable [ σy ]
Example: Baryon Acoustic Oscillations [e.g., Big BOSS] 99% confidence
Model and parameters: dA(z; H0, Ωk) and
H(z; H0, Ωk, Ωm)
a set of errors in the observable variables: σdA, σH
θB

3. Fisher Power!
The Fisher Matrix forecasts astronomical constraints
for model parameters: it can be used to predict 68% confidence that parameters
confidence contours cosmological parameter! lie within the blue (dashed)
contour.
Ingredients required:
a parametrized, physical model for an
θA
observable [ y = f(x;θ) ]
a set of errors in the observable [ σy ]
Example: Baryon Acoustic Oscillations [e.g., Big BOSS] 99% confidence
Model and parameters: dA(z; H0, Ωk) and
H(z; H0, Ωk, Ωm)
a set of errors in the observable variables: σdA, σH
θB
General Formalism:
⌧ 2
@ lnL [definition]
FAB = y = f (x; ✓) model
{
@✓A @✓B
✓ ◆ ✓ ◆ covariance of
@f (x) 1 @f (x) 1 1 @C 1 @C 2
FAB = C + Tr C C y1 0 data variable
@✓A @✓B 2 @✓A @✓B 2
0 yN errors

4. Fisher Power!
The Fisher Matrix forecasts astronomical constraints
for model parameters: it can be used to predict 68% confidence that parameters
confidence contours cosmological parameter! lie within the blue (dashed)
contour.
Ingredients required:
a parametrized, physical model for an
θA
observable [ y = f(x;θ) ]
a set of errors in the observable [ σy ]
Example: Baryon Acoustic Oscillations [e.g., Big BOSS] 99% confidence
Model and parameters: dA(z; H0, Ωk) and
H(z; H0, Ωk, Ωm)
a set of errors in the observable variables: σdA, σH
θB
General Formalism:
⌧ 2
@ lnL [definition]
FAB = y = f (x; ✓) model
{
@✓A @✓B
✓ ◆ ✓ ◆ covariance of
@f (x) 1 @f (x) 1 1 @C 1 @C 2
FAB = C + Tr C C y1 0 data variable
@✓A @✓B 2 @✓A @✓B 2
0 yN errors
This work focuses on the covariance matrix, C

5. Primary Goals and Questions
• Will the errors in the independent variable [e.g.,
redshift] impact predicted constraints on model
parameters?
• What is the impact of the dependent and
independent variables being correlated?
• Can we account for multi-peaked distributions in the
independent variable? --e.g., double-peaked
distributions for photometric redshifts. [Next time]

6. Primary Goals and Questions
✓ 2
◆
0
y1
0 2 • Will the errors in the independent variable [e.g.,
yN redshift] impact predicted constraints on model
parameters?
✓ ◆
2
xx 0 • What is the impact of the dependent and
0 2
yy independent variables being correlated?
• Can we account for multi-peaked distributions in the
✓ ◆ independent variable? --e.g., double-peaked
2 2
xx xy distributions for photometric redshifts. [Next time]
2 2
yx yy

7. Outline
• The motivation: Enhance FM predictions with more
comprehensive error accounting?
• General approach: Derive FM from scratch.
• Introducing Covariances in Observables.

8. Re-Derive FM From first principles
Setup [Observables]
Measured Observables: {Xi } , {Yi } ; i = 1, . . . N
True values of Observables: {xi } , {yi } ; i = 1, . . . N
Setup [Model]
Errors: X and Y are gaussian-distributed about true values, x and y, respectively.
Mean Model: x and y are related by y = f (x)

9. Re-Derive FM From first principles
Setup [Observables]
Measured Observables: {Xi } , {Yi } ; i = 1, . . . N
True values of Observables: {xi } , {yi } ; i = 1, . . . N
Setup [Model]
Errors: X and Y are gaussian-distributed about true values, x and y, respectively.
Mean Model: x and y are related by y = f (x)
Calculate Likelihood
L(✓) = p(X, Y |✓) / p(✓|X, Y ) (Likelihood of a parameter via Bayes’ Thm.)
Z
L= p(X, Y |x, y, ✓)p(y|x, ✓)p(x|✓) dN x dN y (Unpack all the conditional probabilities)
Let y always be the function of x: Assume uniform distribution for x:
p(y|x, ✓) = (y f (x)) xi ⇠ U = p(xi |✓)

10. Re-Derive FM From first principles
Setup [Observables]
Measured Observables: {Xi } , {Yi } ; i = 1, . . . N
True values of Observables: {xi } , {yi } ; i = 1, . . . N
Setup [Model]
Errors: X and Y are gaussian-distributed about true values, x and y, respectively.
Mean Model: x and y are related by y = f (x)
Calculate Likelihood
L(✓) = p(X, Y |✓) / p(✓|X, Y ) (Likelihood of a parameter via Bayes’ Thm.)
Z
L= p(X, Y |x, y, ✓)p(y|x, ✓)p(x|✓) dN x dN y (Unpack all the conditional probabilities)
Let y always be the function of x: Assume uniform distribution for x:
p(y|x, ✓) = (y f (x)) xi ⇠ U = p(xi |✓)

11. Re-Derive FM From first principles
Setup [Observables]
Measured Observables: {Xi } , {Yi } ; i = 1, . . . N
True values of Observables: {xi } , {yi } ; i = 1, . . . N
Setup [Model]
Errors: X and Y are gaussian-distributed about true values, x and y, respectively.
Mean Model: x and y are related by y = f (x)
Calculate Likelihood
L(✓) = p(X, Y |✓) / p(✓|X, Y ) (Likelihood of a parameter via Bayes’ Thm.)
Z
L= p(X, Y |x, y, ✓)p(y|x, ✓)p(x|✓) dN x dN y (Unpack all the conditional probabilities)
Let y always be the function of x: Assume uniform distribution for x:
p(y|x, ✓) = (y f (x)) xi ⇠ U = p(xi |✓)
Z
N N
) L= p(X, Y |x, f , ✓) d x d y

12. Calculate Likelihood (II.)
Z
The distributions in X and Y are Normal,
L = p(X, Y |x, f , ✓) dN x dN y but not generally analytically soluble.
Therefore, we Taylor-expand the model...
assume that it is linear across the width of f (xi ) ! f ⇤ (xi ) = f (Xi ) + (xi Xi )f 0 (Xi )
gaussian distribution of x, retaining only
linear terms.:

13. Calculate Likelihood (II.)
Z
The distributions in X and Y are Normal,
L = p(X, Y |x, f , ✓) dN x dN y but not generally analytically soluble.
Therefore, we Taylor-expand the model...
assume that it is linear across the width of f (xi ) ! f ⇤ (xi ) = f (Xi ) + (xi Xi )f 0 (Xi )
gaussian distribution of x, retaining only
linear terms.:
{zi , Zi } = {xi , Xi } for i  N
Let Z be a 2N-dimensional vector, containing
both measured and true observables! {zi , Zi } = {f ⇤ , Yi } for i > N
Z 
This provides the canonical form of the 1 1
multi-variate normal distribution: )L/ p exp (Z z)T C 1
(Z z)
detC 2
✓ ◆
With {z,Z} as vectors, the covariance matrix CXX CXY
can be written in block form: CXYT CY Y

14. Calculate Likelihood (II.)
Z
The distributions in X and Y are Normal,
L = p(X, Y |x, f , ✓) dN x dN y but not generally analytically soluble.
Therefore, we Taylor-expand the model...
assume that it is linear across the width of f (xi ) ! f ⇤ (xi ) = f (Xi ) + (xi Xi )f 0 (Xi )
gaussian distribution of x, retaining only
linear terms.:
{zi , Zi } = {xi , Xi } for i  N
Let Z be a 2N-dimensional vector, containing
both measured and true observables! {zi , Zi } = {f ⇤ , Yi } for i > N
Z 
This provides the canonical form of the 1 1
multi-variate normal distribution: )L/ p exp (Z z)T C 1
(Z z)
detC 2
✓ ◆
With {z,Z} as vectors, the covariance matrix CXX CXY
can be written in block form: CXYT CY Y
Notice that this form natively contains covariance
among X’s among Y’s and between X’s and Y’s

15. Calculate Likelihood (III.)
Z 
1 1
L/ exp (Z z)T C 1
(Z z) Evaluating the exponent and simplifying,
detC 2

1 1 eT 1e
)L/ p exp Y R Y
detR 2
(where R is a function of Cij and f*) R = CY Y + CXYT T + T CXY + T CXX T
✓ ◆
df (x)
T = diag
dx x=X

16. Calculate Likelihood (III.)
Z 
1 1
L/ exp (Z z)T C 1
(Z z) Evaluating the exponent and simplifying,
detC 2

1 1 eT 1e
)L/ p exp Y R Y
detR 2
(where R is a function of Cij and f*) R = CY Y + CXYT T + T CXY + T CXX T
✓ ◆
df (x)
T = diag
dx x=X
Result : Where the x data are irrelevant,
i.e., when derivatives of f are zero,
i.e., when CXX (or σx) = 0,
We recover the original form R→ C = CYY

17. Main
Result :

18. Main
Result :
With the help of Tegmark, Taylor and Heavens (1997), R then takes the place of the
covariance, C, in the canonical formulation:
T
✓ ◆
@f 1 @f 1 1 @R 1 @R
FAB = R + Tr R R
@A @B 2 @A @B

19. Main
Result :
With the help of Tegmark, Taylor and Heavens (1997), R then takes the place of the
covariance, C, in the canonical formulation:
T
✓ ◆
@f 1 @f 1 1 @R 1 @R
FAB = R + Tr R R
@A @B 2 @A @B
Even if C does not dependent on the parameters, R does depend on
them [via f]. The trace term is in general non-zero.

20. Summary: An Extended Formalism
• Development of general process for evaluating arbitrary
model functions to 1st order in the FM formalism.
• Incorporation of correlated errors among observables.
Next Steps
• Application: Double-peaked Error distributions
• Check: Compare to MCMC
• Cosmological Application
• Incorporate into Fisher4Cast