132 research outputs found
Noisy independent component analysis of auto-correlated components
We present a new method for the separation of superimposed, independent,
auto-correlated components from noisy multi-channel measurement. The presented
method simultaneously reconstructs and separates the components, taking all
channels into account and thereby increases the effective signal-to-noise ratio
considerably, allowing separations even in the high noise regime.
Characteristics of the measurement instruments can be included, allowing for
application in complex measurement situations. Independent posterior samples
can be provided, permitting error estimates on all desired quantities. Using
the concept of information field theory, the algorithm is not restricted to any
dimensionality of the underlying space or discretization scheme thereof
Stochastic determination of matrix determinants
Matrix determinants play an important role in data analysis, in particular
when Gaussian processes are involved. Due to currently exploding data volumes,
linear operations - matrices - acting on the data are often not accessible
directly but are only represented indirectly in form of a computer routine.
Such a routine implements the transformation a data vector undergoes under
matrix multiplication. While efficient probing routines to estimate a matrix's
diagonal or trace, based solely on such computationally affordable
matrix-vector multiplications, are well known and frequently used in signal
inference, there is no stochastic estimate for its determinant. We introduce a
probing method for the logarithm of a determinant of a linear operator. Our
method rests upon a reformulation of the log-determinant by an integral
representation and the transformation of the involved terms into stochastic
expressions. This stochastic determinant determination enables large-size
applications in Bayesian inference, in particular evidence calculations, model
comparison, and posterior determination.Comment: 8 pages, 5 figure
Reply to "Comment on `Inference with minimal Gibbs free energy in information field theory'" by Iatsenko, Stefanovska and McClintock
We endorse the comment on our recent paper [En{\ss}lin and Weig, Phys. Rev. E
82, 051112 (2010)] by Iatsenko, Stefanovska and McClintock [Phys. Rev. E 85
033101 (2012)] and we try to clarify the origin of the apparent controversy on
two issues. The aim of the minimal Gibbs free energy approach to provide a
signal estimate is not affected by their Comment. However, if one wants to
extend the method to also infer the a posteriori signal uncertainty any
tempering of the posterior has to be undone at the end of the calculations, as
they correctly point out. Furthermore, a distinction is made here between
maximum entropy, the maximum entropy principle, and the so-called maximum
entropy method in imaging, hopefully clarifying further the second issue of
their Comment paper.Comment: 1 page, no figures, Reply to Comment pape
The Galactic Faraday depth sky revisited
The Galactic Faraday depth sky is a tracer for both the Galactic magnetic
field and the thermal electron distribution. It has been previously
reconstructed from polarimetric measurements of extra-galactic point sources.
Here, we improve on these works by using an updated inference algorithm as well
as by taking into account the free-free emission measure map from the Planck
survey. In the future, the data situation will improve drastically with the
next generation Faraday rotation measurements from SKA and its pathfinders.
Anticipating this, the aim of this paper is to update the map reconstruction
method with the latest development in imaging based on information field
theory. We demonstrate the validity of the new algorithm by applying it to the
Oppermann et al. (2012) data compilation and compare our results to the
previous map.\\ Despite using exactly the previous data set, a number of novel
findings are made: A non-parametric reconstruction of an overall amplitude
field resembles the free-free emission measure map of the Galaxy. Folding this
free-free map into the analysis allows for more detailed predictions. The joint
inference enables us to identify regions with deviations from the assumed
correlations between the free-free and Faraday data, thereby pointing us to
Galactic structures with distinguishably different physics. We e.g. find
evidence for an alignment of the magnetic field within the line of sights along
both directions of the Orion arm.Comment: 16 pages, 15 figure
Optimal Belief Approximation
In Bayesian statistics probability distributions express beliefs. However,
for many problems the beliefs cannot be computed analytically and
approximations of beliefs are needed. We seek a loss function that quantifies
how "embarrassing" it is to communicate a given approximation. We reproduce and
discuss an old proof showing that there is only one ranking under the
requirements that (1) the best ranked approximation is the non-approximated
belief and (2) that the ranking judges approximations only by their predictions
for actual outcomes. The loss function that is obtained in the derivation is
equal to the Kullback-Leibler divergence when normalized. This loss function is
frequently used in the literature. However, there seems to be confusion about
the correct order in which its functional arguments, the approximated and
non-approximated beliefs, should be used. The correct order ensures that the
recipient of a communication is only deprived of the minimal amount of
information. We hope that the elementary derivation settles the apparent
confusion. For example when approximating beliefs with Gaussian distributions
the optimal approximation is given by moment matching. This is in contrast to
many suggested computational schemes.Comment: made improvements on the proof and the languag
Diagnostics for insufficiencies of posterior calculations in Bayesian signal inference
We present an error-diagnostic validation method for posterior distributions
in Bayesian signal inference, an advancement of a previous work. It transfers
deviations from the correct posterior into characteristic deviations from a
uniform distribution of a quantity constructed for this purpose. We show that
this method is able to reveal and discriminate several kinds of numerical and
approximation errors, as well as their impact on the posterior distribution.
For this we present four typical analytical examples of posteriors with
incorrect variance, skewness, position of the maximum, or normalization. We
show further how this test can be applied to multidimensional signals
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