Bayesian signal inference

The way of Bayesian signal inference Marco Selig, Michael R. Bell, Henrik Junklewitz, Niels Oppermann, Martin Reinecke, Maksim Greiner, Carlos Pacha...
Author: Lindsay Bruce
2 downloads 0 Views 7MB Size
The

way of Bayesian signal inference

Marco Selig, Michael R. Bell, Henrik Junklewitz, Niels Oppermann, Martin Reinecke, Maksim Greiner, Carlos Pachajoa, Torsten A. Enßlin Max Planck Institute for Astrophysics

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

1

The

way of Bayesian signal inference

Marco Selig, Michael R. Bell, Henrik Junklewitz, Niels Oppermann, Martin Reinecke, Maksim Greiner, Carlos Pachajoa, Torsten A. Enßlin References: arXiv:1301.4499 and arXiv:1210.6866 NIFTY project homepage: http://www.mpa-garching.mpg.de/ift/nifty/ 2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

2

Outline 1. IFT – Information Field Theory 2. NIFTY – Numerical Information Field Theory 3. Applications 4. Summary

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

3

Theory

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

4

What's the problem? ●





features in the Galactic diffuse γ-ray emission

separation of diffuse and point-like components medical and Galactic tomography

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

6

Theory

NIFTY

Applications

observation

Data

inference

Summary

Signal estimate

• data vector

• signal field

• finite set of numbers

• infinite number of de• grees of freedom

42 23 16

12 4

2013 - 03 - 22

2

0

5

4

6

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

11

Theory

NIFTY

Applications

Summary

Bayes' Theorem ...

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

12

Theory

NIFTY

Applications

Summary

Bayes' Theorem ...

… Information Field Theory 2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

13

Theory

NIFTY

Applications

Summary

Wiener filter ●

data model ◦ linear response ◦ additive Gaussian noise



a priori assumptions ◦ signal



← multidimensional Gaussian prior

information Hamiltonian

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

17

Theory

NIFTY

Applications

Summary

Wiener filter ●

a posteriori solution

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

18

Theory

NIFTY

Applications

Summary

Wiener filter ●

a posteriori solution

observation

signal 2013 - 03 - 22

inference

data

reconstruction

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

19

Theory

NIFTY

Applications

Summary

Discretizing continuous fields

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

21

Theory

NIFTY

Applications

Summary

Discretizing continuous fields

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

22

Theory

NIFTY

Applications

Summary

Discretizing continuous fields

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

24

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

25

Theory

NIFTY

Applications

... ●

Summary

Selig et al. (2013)

is a versatile PYTHON library incorporating CYTHON, C++, and C libraries

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

26

Theory

NIFTY

Applications

Summary

sage R

simplicity

PYTHON

CYTHON

C++ C Fortran performance

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

27

Theory

NIFTY

Applications

Summary

PYTHON is simple In [1]: 1+2 Out[1]: 3 In [2]: func = lambda x: x**2 In [3]: func(3) Out[3]: 9 In [4]: import numpy In [5]: func(numpy.array([1, 2, 3])) Out[5]: array([1, 4, 9]) In [6]:

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

28

Theory

NIFTY

Applications

... ●



Summary

Selig et al. (2013)

is a versatile PYTHON library incorporating CYTHON, C++, and C libraries operates regardless of the underlying spatial grid and its resolution

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

29

Theory

NIFTY

Applications

Summary

Grid independence

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

30

Theory

NIFTY

Applications

Summary

Grid independence

1D

2D 2013 - 03 - 22

128x128

2D

32x32

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

31

Theory

NIFTY

Applications

Summary

Grid independence

1D

2D 2013 - 03 - 22

128x128

2D

32x32

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

32

Theory

NIFTY

Applications

... ●





Summary

Selig et al. (2013)

is a versatile PYTHON library incorporating CYTHON, C++, and C libraries operates regardless of the underlying spatial grid and its resolution abstracts spaces, fields, and operators into an object-orientated framework

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

33

Theory

NIFTY

Applications

                object NIFT Y classes

Summary

Selig et al. (2013)

space   field   operator    • parameters

2013 - 03 - 22

• domain • space

• domain • space

• field values

• target space • instance methods • applying to fields

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

35

Theory

NIFTY

Applications

                object NIFT Y classes

Summary

Selig et al. (2013)

space   field   operator      point_space  unstructured list of points   rg_space     n-dimensional regular grid   lm_space     spherical harmonics   hp_space     Gauss-Legendre grid on the sphere   gl_space     HEALPIX grid on the sphere   nested_space (arbitrary product of grids)

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

36

Theory

NIFTY

Applications

                object NIFT Y classes

Summary

Selig et al. (2013)

space   field   operator      point_space  unstructured list of points   rg_space     n-dimensional regular grid   lm_space     spherical harmonics   hp_space     Gauss-Legendre grid on the sphere   gl_space     HEALPIX grid on the sphere   nested_space (arbitrary product of grids)

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

37

Theory

NIFTY

Applications

                object NIFT Y classes

Summary

Selig et al. (2013)

space   field   operator      point_space  unstructured list of points   rg_space     n-dimensional regular grid   lm_space     spherical harmonics   hp_space     Gauss-Legendre grid on the sphere   gl_space     HEALPIX grid on the sphere   nested_space (arbitrary product of grids)

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

38

Theory

NIFTY

Applications

                object NIFT Y classes

Summary

Selig et al. (2013)

space   field   operator      point_space  unstructured list of points   rg_space     n-dimensional regular grid   lm_space     spherical harmonics   hp_space     Gauss-Legendre grid on the sphere   gl_space     HEALPIX grid on the sphere   nested_space (arbitrary product of grids)

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

39

Theory

NIFTY

Applications

                object NIFT Y classes

Summary

Selig et al. (2013)

space   field   operator   probing   point_space     diagonal_operator   rg_space          power_operator   lm_space        projection_operator   hp_space        vecvec_operator   gl_space        response_operator   nested_space

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

41

Theory

NIFTY

Applications

... ●







Summary

Selig et al. (2013)

is a versatile PYTHON library incorporating CYTHON, C++, and C libraries operates regardless of the underlying spatial grid and its resolution abstracts spaces, fields, and operators into an object-orientated framework allows the user the abstract formulation and programming of signal inference algorithms

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

42

Theory

NIFTY

Applications

Wiener filtering

Summary

Selig et al. (2013)

from nifty import * from scipy.sparse.linalg import LinearOperator as lo from scipy.sparse.linalg import cg class propagator(operator):                                           # define propagator class     _matvec = (lambda self, x: self.inverse_times(x).val.flatten())     def _multiply(self, x):         # some numerical invertion technique; here, conjugate gradient         A = lo(shape=tuple(self.dim()), matvec=self._matvec)         b = x.val.flatten()         x_, info = cg(A, b, M=None)         return x_     def _inverse_multiply(self, x):         S, N, R = self.para         return S.inverse_times(x) + R.adjoint_times(N.inverse_times(R.times(x))) # some signal space; e.g., a one­dimensional regular grid s_space = rg_space(512, zerocenter=False, dist=0.002)                 # define signal space # or      rg_space([256, 256]) # or      hp_space(128) k_space = s_space.get_codomain()                                      # get conjugate space kindex, rho = k_space.get_power_index(irreducible=True) # some power spectrum power = [42 / (kk + 1) ** 3 for kk in kindex] S = power_operator(k_space, spec=power)                               # define signal covariance s = S.get_random_field(domain=s_space)                                # generate signal R = response_operator(s_space, sigma=0.0, mask=1.0, assign=None)      # define response d_space = R.target                                                    # get data space # some noise variance; e.g., 1 N = diagonal_operator(d_space, diag=1, bare=True)                     # define noise covariance n = N.get_random_field(domain=d_space)                                # generate noise d = R(s) + n                                                          # compute data j = R.adjoint_times(N.inverse_times(d))                               # define source D = propagator(s_space, sym=True, imp=True, para=[S,N,R])             # define propagator m = D(j)                                                              # reconstruct map s.plot(title="signal")                                                # plot signal d.cast_domain(s_space) d.plot(title="data", vmin=s.val.min(), vmax=s.val.max())              # plot data m.plot(title="reconstructed map", vmin=s.val.min(), vmax=s.val.max()) # plot map

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

44

Theory

NIFTY

Applications

Wiener filtering

Summary

Selig et al. (2013)

from nifty import * from scipy.sparse.linalg import LinearOperator as lo from scipy.sparse.linalg import cg class propagator(operator):                                           # define propagator class     _matvec = (lambda self, x: self.inverse_times(x).val.flatten())     def _multiply(self, x):         # some numerical invertion technique; here, conjugate gradient         A = lo(shape=tuple(self.dim()), matvec=self._matvec)         b = x.val.flatten()         x_, info = cg(A, b, M=None)         return x_     def _inverse_multiply(self, x):         S, N, R = self.para         return S.inverse_times(x) + R.adjoint_times(N.inverse_times(R.times(x))) # some signal space; e.g., a one­dimensional regular grid s_space = rg_space(512, zerocenter=False, dist=0.002)                 # define signal space # or      rg_space([256, 256]) # or      hp_space(128) k_space = s_space.get_codomain()                                      # get conjugate space kindex, rho = k_space.get_power_index(irreducible=True) # some power spectrum power = [42 / (kk + 1) ** 3 for kk in kindex] S = power_operator(k_space, spec=power)                               # define signal covariance s = S.get_random_field(domain=s_space)                                # generate signal R = response_operator(s_space, sigma=0.0, mask=1.0, assign=None)      # define response d_space = R.target                                                    # get data space # some noise variance; e.g., 1 N = diagonal_operator(d_space, diag=1, bare=True)                     # define noise covariance n = N.get_random_field(domain=d_space)                                # generate noise d = R(s) + n                                                          # compute data j = R.adjoint_times(N.inverse_times(d))                               # define source D = propagator(s_space, sym=True, imp=True, para=[S,N,R])             # define propagator m = D(j)                                                              # reconstruct map s.plot(title="signal")                                                # plot signal d.cast_domain(s_space) d.plot(title="data", vmin=s.val.min(), vmax=s.val.max())              # plot data m.plot(title="reconstructed map", vmin=s.val.min(), vmax=s.val.max()) # plot map

d = R(s) + n j = R.adjoint_times(N.inverse_times(d)) m = D(j) 

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

46

Theory

NIFTY

Applications

Wiener filtering

Summary

Selig et al. (2013)

from nifty import * from scipy.sparse.linalg import LinearOperator as lo from scipy.sparse.linalg import cg class propagator(operator):                                           # define propagator class     _matvec = (lambda self, x: self.inverse_times(x).val.flatten())     def _multiply(self, x):         # some numerical invertion technique; here, conjugate gradient         A = lo(shape=tuple(self.dim()), matvec=self._matvec)         b = x.val.flatten()         x_, info = cg(A, b, M=None)         return x_     def _inverse_multiply(self, x):         S, N, R = self.para         return S.inverse_times(x) + R.adjoint_times(N.inverse_times(R.times(x))) # some signal space; e.g., a one­dimensional regular grid s_space = rg_space(512, zerocenter=False, dist=0.002)                 # define signal space # or      rg_space([256, 256]) # or      hp_space(128) k_space = s_space.get_codomain()                                      # get conjugate space kindex, rho = k_space.get_power_index(irreducible=True) # some power spectrum power = [42 / (kk + 1) ** 3 for kk in kindex] S = power_operator(k_space, spec=power)                               # define signal covariance s = S.get_random_field(domain=s_space)                                # generate signal R = response_operator(s_space, sigma=0.0, mask=1.0, assign=None)      # define response d_space = R.target                                                    # get data space # some noise variance; e.g., 1 N = diagonal_operator(d_space, diag=1, bare=True)                     # define noise covariance n = N.get_random_field(domain=d_space)                                # generate noise d = R(s) + n                                                          # compute data j = R.adjoint_times(N.inverse_times(d))                               # define source D = propagator(s_space, sym=True, imp=True, para=[S,N,R])             # define propagator m = D(j)                                                              # reconstruct map s.plot(title="signal")                                                # plot signal d.cast_domain(s_space) d.plot(title="data", vmin=s.val.min(), vmax=s.val.max())              # plot data m.plot(title="reconstructed map", vmin=s.val.min(), vmax=s.val.max()) # plot map

s_space = rg_space(512, ...)

d = R(s) + n j = R.adjoint_times(N.inverse_times(d)) m = D(j) 

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

48

Theory

NIFTY

Applications

Summary

from nifty import * from scipy.sparse.linalg import LinearOperator as lo from scipy.sparse.linalg import cg class propagator(operator):                                           # define propagator class     _matvec = (lambda self, x: self.inverse_times(x).val.flatten())     def _multiply(self, x):         # some numerical invertion technique; here, conjugate gradient         A = lo(shape=tuple(self.dim()), matvec=self._matvec)         b = x.val.flatten()         x_, info = cg(A, b, M=None)         return x_     def _inverse_multiply(self, x):         S, N, R = self.para         return S.inverse_times(x) + R.adjoint_times(N.inverse_times(R.times(x))) # some signal space; e.g., a one­dimensional regular grid s_space = rg_space(512, zerocenter=False, dist=0.002)                 # define signal space # or      rg_space([256, 256]) # or      hp_space(128) k_space = s_space.get_codomain()                                      # get conjugate space kindex, rho = k_space.get_power_index(irreducible=True) # some power spectrum power = [42 / (kk + 1) ** 3 for kk in kindex] S = power_operator(k_space, spec=power)                               # define signal covariance s = S.get_random_field(domain=s_space)                                # generate signal R = response_operator(s_space, sigma=0.0, mask=1.0, assign=None)      # define response d_space = R.target                                                    # get data space # some noise variance; e.g., 1 N = diagonal_operator(d_space, diag=1, bare=True)                     # define noise covariance n = N.get_random_field(domain=d_space)                                # generate noise d = R(s) + n                                                          # compute data j = R.adjoint_times(N.inverse_times(d))                               # define source D = propagator(s_space, sym=True, imp=True, para=[S,N,R])             # define propagator m = D(j)                                                              # reconstruct map s.plot(title="signal")                                                # plot signal d.cast_domain(s_space) d.plot(title="data", vmin=s.val.min(), vmax=s.val.max())              # plot data m.plot(title="reconstructed map", vmin=s.val.min(), vmax=s.val.max()) # plot map

s_space = rg_space([256, 256])

d = R(s) + n j = R.adjoint_times(N.inverse_times(d)) m = D(j) 

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

49

Theory

NIFTY

Applications

Wiener filtering

Summary

Selig et al. (2013)

from nifty import * from scipy.sparse.linalg import LinearOperator as lo from scipy.sparse.linalg import cg class propagator(operator):                                           # define propagator class     _matvec = (lambda self, x: self.inverse_times(x).val.flatten())     def _multiply(self, x):         # some numerical invertion technique; here, conjugate gradient         A = lo(shape=tuple(self.dim()), matvec=self._matvec)         b = x.val.flatten()         x_, info = cg(A, b, M=None)         return x_     def _inverse_multiply(self, x):         S, N, R = self.para         return S.inverse_times(x) + R.adjoint_times(N.inverse_times(R.times(x))) # some signal space; e.g., a one­dimensional regular grid s_space = rg_space(512, zerocenter=False, dist=0.002)                 # define signal space # or      rg_space([256, 256]) # or      hp_space(128) k_space = s_space.get_codomain()                                      # get conjugate space kindex, rho = k_space.get_power_index(irreducible=True) # some power spectrum power = [42 / (kk + 1) ** 3 for kk in kindex] S = power_operator(k_space, spec=power)                               # define signal covariance s = S.get_random_field(domain=s_space)                                # generate signal R = response_operator(s_space, sigma=0.0, mask=1.0, assign=None)      # define response d_space = R.target                                                    # get data space # some noise variance; e.g., 1 N = diagonal_operator(d_space, diag=1, bare=True)                     # define noise covariance n = N.get_random_field(domain=d_space)                                # generate noise d = R(s) + n                                                          # compute data j = R.adjoint_times(N.inverse_times(d))                               # define source D = propagator(s_space, sym=True, imp=True, para=[S,N,R])             # define propagator m = D(j)                                                              # reconstruct map s.plot(title="signal")                                                # plot signal d.cast_domain(s_space) d.plot(title="data", vmin=s.val.min(), vmax=s.val.max())              # plot data m.plot(title="reconstructed map", vmin=s.val.min(), vmax=s.val.max()) # plot map

s_space = hp_space(128)

d = R(s) + n j = R.adjoint_times(N.inverse_times(d)) m = D(j) 

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

50

Theory

NIFTY

Applications

Wiener filtering and more

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

Summary

Selig et al. (2013)

51

Theory

NIFTY

Applications

Wiener filtering and more

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

Summary

Selig et al. (2013)

52

NIFTY

Applications

Summary

source: USC-SIPI

Theory

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

53

Theory

NIFTY

Applications

... ●









Summary

Selig et al. (2013)

is a versatile PYTHON library incorporating CYTHON, C++, and C libraries operates regardless of the underlying spatial grid and its resolution abstracts spaces, fields, and operators into an object-orientated framework allows the user the abstract formulation and programming of signal inference algorithms provides an extensive online documentation

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

55

Theory

NIFTY

Applications

Summary

NIFTY project homepage: http://www.mpa-garching.mpg.de/ift/nifty/

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

56

Applications

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

57

Theory

NIFTY

Applications

Summary

The Galactic free electron density ●

Greiner et al. (in prep.)

data model

◦ dispersion measures from different lines of sight ◦ additive Gaussian noise

1 of 67 pulsars

Sol 3

2013 - 03 - 22

lar l e t rs i nt e iu m d e m

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

58

Theory

NIFTY

Applications

Summary

The Galactic free electron density ●

data model

Greiner et al. (in prep.)

◦ dispersion measures from different lines of sight ◦ additive Gaussian noise



a priori assumptions ◦ electron density ← log-normal prior ◦ unknown correlations

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

59

Theory

NIFTY

Applications

Summary

The Galactic free electron density Greiner et al. (in prep.)

[ 10-3 cm-3 ]

2013 - 03 - 22

[ 10-3 cm-3 ]

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

60

Theory

NIFTY

Applications

Summary

Computer tomography ●

data model ◦ absorption along the line of sight ◦ Poissonian noise

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

61

Theory

NIFTY

Applications

Summary

Computer tomography ●

data model ◦ absorption along the line of sight ◦ Poissonian noise

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

62

Theory

NIFTY

Applications

Summary

Computer tomography ●

data model ◦ absorption along the line of sight ◦ Poissonian noise



a priori assumptions ◦ matter density ← log-normal prior ◦ known correlations ← medical databases

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

63

Theory

NIFTY

Applications

Summary

Computer tomography

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

64

Theory

NIFTY

Applications

Summary

Computer tomography

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

65

Theory

NIFTY

Applications

Summary

Computer tomography

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

66

Theory

NIFTY

Applications

Summary

Computer tomography

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

67

Theory

NIFTY

Applications

Summary

Computer tomography

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

68

Theory

NIFTY

Applications

Summary

Computer tomography

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

69

Theory

NIFTY

Applications

Summary

The Fermi γ-ray sky Selig et al. (in prep.)

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

70

Theory

NIFTY

Applications

Summary

The Fermi γ-ray sky ●

data model

Selig et al. (in prep.)

◦ uneven survey coverage ◦ Poissonian noise



a priori assumptions ◦ diffuse flux ← log-normal prior ◦ unknown correlations (but spectral smoothness)

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

72

Theory

NIFTY

Applications

Summary

The Fermi γ-ray sky Selig et al. (in prep.)

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

73

Theory

NIFTY

Applications

Summary

Component separation ●

data model

Selig et al. (in prep.)

◦ superposition of flux components ◦ complex instrument response function ◦ Poissonian noise



a priori assumptions ◦ diffuse flux ← log-normal prior ◦ known correlations ◦ point source flux ← inverse-Gamma priors

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

75

Theory

2013 - 03 - 22

NIFTY

Applications

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

Summary

76

Theory

2013 - 03 - 22

NIFTY

Applications

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

Summary

77

Theory

2013 - 03 - 22

NIFTY

Applications

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

Summary

78

Theory

2013 - 03 - 22

NIFTY

Applications

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

Summary

79

Theory

2013 - 03 - 22

NIFTY

Applications

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

Summary

80

Theory

2013 - 03 - 22

NIFTY

Applications

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

Summary

81

Summary

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

82

Theory

NIFTY

Applications

Summary

Summary ●

effective IFT framework ◦ inference on continuous signal fields ◦ treatment of unknown correlations



useful NIFTY library ◦ ◦ ◦ ◦

2013 - 03 - 22

versatile toolbox for signal inference algorithms grid and resolution independence applicability to real-life problems extensive documentation (including tutorials)

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

83

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

85

Theory

NIFTY

Applications

Summary

Spectral smoothness prior Oppermann et al. (2012) ●

unknown signal correlations



inverse-Gamma prior

2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

86

Theory

NIFTY

Applications

Summary

Spectral smoothness prior Oppermann et al. (2012) ●

unknown signal correlations



inverse-Gamma prior and spectral smoothness prior

σ2 = 1000 σ2 = 10 2013 - 03 - 22

The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig

87

Suggest Documents