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.