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
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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
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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
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Theory
2013 - 03 - 22
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
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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
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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
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Theory
NIFTY
Applications
Summary
Bayes' Theorem ...
2013 - 03 - 22
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
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Theory
NIFTY
Applications
Summary
Bayes' Theorem ...
… Information Field Theory 2013 - 03 - 22
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
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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
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Theory
NIFTY
Applications
Summary
Wiener filter ●
a posteriori solution
2013 - 03 - 22
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
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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
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Theory
NIFTY
Applications
Summary
Discretizing continuous fields
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The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
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Theory
NIFTY
Applications
Summary
Discretizing continuous fields
2013 - 03 - 22
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
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Theory
NIFTY
Applications
Summary
Discretizing continuous fields
2013 - 03 - 22
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
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2013 - 03 - 22
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
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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
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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
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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
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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
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Theory
NIFTY
Applications
Summary
Grid independence
2013 - 03 - 22
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Applications
2013 - 03 - 22
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
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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
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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
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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
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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
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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
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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
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Theory
NIFTY
Applications
Summary
Computer tomography
2013 - 03 - 22
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
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Theory
NIFTY
Applications
Summary
Computer tomography
2013 - 03 - 22
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
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Theory
NIFTY
Applications
Summary
Computer tomography
2013 - 03 - 22
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
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Theory
NIFTY
Applications
Summary
Computer tomography
2013 - 03 - 22
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
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Theory
NIFTY
Applications
Summary
Computer tomography
2013 - 03 - 22
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
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Theory
NIFTY
Applications
Summary
Computer tomography
2013 - 03 - 22
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
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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
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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
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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
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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
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Theory
2013 - 03 - 22
NIFTY
Applications
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
Summary
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Theory
2013 - 03 - 22
NIFTY
Applications
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
Summary
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Theory
2013 - 03 - 22
NIFTY
Applications
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
Summary
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Theory
2013 - 03 - 22
NIFTY
Applications
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
Summary
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Summary
2013 - 03 - 22
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
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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
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2013 - 03 - 22
The NIFT NIFTY Y way of Bayesian signal inference by Marco Selig
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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