1) The document derives an extended formalism for the Fisher matrix that accounts for correlations between observables and errors in both the dependent and independent variables. 2) It shows that accounting for these additional sources of error results in a modified covariance matrix, R, that replaces the standard covariance matrix, C, in the Fisher matrix formulation. 3) The extended formalism generalizes the standard treatment of errors in the Fisher matrix and allows for a more comprehensive prediction of constraints on cosmological parameters.

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