170 research outputs found
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
Bayesian inference of early-universe signals
This thesis focuses on the development and application of Bayesian inference techniques for early-Universe signals and on the advancement of mathematical tools for information retrieval.
A crucial quantity required to gain information from the early Universe is the primordial scalar potential and its statistics. We reconstruct this scalar potential from cosmic microwave background data. Technically, the inference is done by splitting the large inverse problem of such a reconstruction into many, each of them solved by an optimal linear filter.
Once the primordial scalar potential and its correlation structure have been obtained the underlying physics can be directly inferred from it. Small deviations of the scalar potential from Gaussianity, for instance, can be used to study parameters of inflationary models. A method to infer such parameters from non-Gaussianity is presented. To avoid expensive numerical techniques
the method is kept analytical as far as possible. This is achieved by introducing an approximation of the desired posterior probability including a Taylor expansion of a matrix determinant.
The calculation of a determinant is also essential in many other Bayesian approaches, both apart from and within cosmology. In cases where a Taylor approximation fails, its evaluation is usually challenging. The evaluation is in particular difficult, when dealing with big data, where matrices are to huge to be accessible directly, but need to be represented indirectly by a computer routine implementing the action of the matrix. To solve this problem, we develop a method to calculate the determinant of a matrix by using well-known sampling techniques and an integral representation of the log-determinant.
The prerequisite for the presented methods as well as for every data analysis of scientific experiments is a proper calibration of the measurement device. Therefore we advance the theory of self-calibration at the beginning of the
thesis to infer signal and calibration simultaneously from data. This is achieved by successively absorbing more and more portions of calibration uncertainty into the signal inference equations. The result, the Calibration-Uncertainty
Renormalized Estimator, follows from the solution of a coupled differential equation
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
Fast and precise way to calculate the posterior for the local non-Gaussianity parameter from cosmic microwave background observations
We present an approximate calculation of the full Bayesian posterior
probability distribution for the local non-Gaussianity parameter
from observations of cosmic microwave background anisotropies
within the framework of information field theory. The approximation that we
introduce allows us to dispense with numerically expensive sampling techniques.
We use a novel posterior validation method (DIP test) in cosmology to test the
precision of our method. It transfers inaccuracies of the calculated posterior
into deviations from a uniform distribution for a specially constructed test
quantity. For this procedure we study toy cases that use one- and
two-dimensional flat skies, as well as the full spherical sky. We find that we
are able to calculate the posterior precisely under a flat-sky approximation,
albeit not in the spherical case. We argue that this is most likely due to an
insufficient precision of the used numerical implementation of the spherical
harmonic transform, which might affect other non-Gaussianity estimators as
well. Furthermore, we present how a nonlinear reconstruction of the primordial
gravitational potential on the full spherical sky can be obtained in principle.
Using the flat-sky approximation, we find deviations for the posterior of
from a Gaussian shape that become more significant for larger
values of the underlying true . We also perform a comparison to
the well-known estimator of Komatsu et al. [Astrophys. J. 634, 14 (2005)] and
finally derive the posterior for the local non-Gaussianity parameter
as an example of how to extend the introduced formalism to
higher orders of non-Gaussianity
Signal inference with unknown response: Calibration-uncertainty renormalized estimator
The calibration of a measurement device is crucial for every scientific
experiment, where a signal has to be inferred from data. We present CURE, the
calibration uncertainty renormalized estimator, to reconstruct a signal and
simultaneously the instrument's calibration from the same data without knowing
the exact calibration, but its covariance structure. The idea of CURE,
developed in the framework of information field theory, is starting with an
assumed calibration to successively include more and more portions of
calibration uncertainty into the signal inference equations and to absorb the
resulting corrections into renormalized signal (and calibration) solutions.
Thereby, the signal inference and calibration problem turns into solving a
single system of ordinary differential equations and can be identified with
common resummation techniques used in field theories. We verify CURE by
applying it to a simplistic toy example and compare it against existent
self-calibration schemes, Wiener filter solutions, and Markov Chain Monte Carlo
sampling. We conclude that the method is able to keep up in accuracy with the
best self-calibration methods and serves as a non-iterative alternative to it
Quantitative wave function analysis for excited states of transition metal complexes
The character of an electronically excited state is one of the most important
descriptors employed to discuss the photophysics and photochemistry of
transition metal complexes. In transition metal complexes, the interaction
between the metal and the different ligands gives rise to a rich variety of
excited states, including metal-centered, intra-ligand, metal-to-ligand charge
transfer, ligand-to-metal charge transfer, and ligand-to-ligand charge transfer
states. Most often, these excited states are identified by considering the most
important wave function excitation coefficients and inspecting visually the
involved orbitals. This procedure is tedious, subjective, and imprecise.
Instead, automatic and quantitative techniques for excited-state
characterization are desirable. In this contribution we review the concept of
charge transfer numbers---as implemented in the TheoDORE package---and show its
wide applicability to characterize the excited states of transition metal
complexes. Charge transfer numbers are a formal way to analyze an excited state
in terms of electron transitions between groups of atoms based only on the
well-defined transition density matrix. Its advantages are many: it can be
fully automatized for many excited states, is objective and reproducible, and
provides quantitative data useful for the discussion of trends or patterns. We
also introduce a formalism for spin-orbit-mixed states and a method for
statistical analysis of charge transfer numbers. The potential of this
technique is demonstrated for a number of prototypical transition metal
complexes containing Ir, Ru, and Re. Topics discussed include orbital
delocalization between metal and carbonyl ligands, nonradiative decay through
metal-centered states, effect of spin-orbit couplings on state character, and
comparison among results obtained from different electronic structure methods.Comment: 47 pages, 19 figures, including supporting information (7 pages, 1
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