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

Galactic dust and dynamics

Physics is about building a model of the world. Building a model can have two different interpretations. On the one hand, it can refer to the construction of a model that mimics the behavior of a system, i.e. in the form of a simulation. On the other hand it can denote the process of building something that has properties of the original, i.e. a map. This dissertation contributes to modeling the world in both meanings of the word, and also connects them. We regard a map as a conditional probability, the map has degrees of freedom that constrain the mapped system. Maps of time variable systems have to be updated as the system evolves. Given only the information that a map contains about a system at a previous point in time, and the time evolution of the system, the degrees of freedom of an updated map should be selected such that the least amount of information about the system is lost. Iterating this procedure, one obtains a simulation scheme, as the time evolution of the system is imprinted in the sequence of maps. In this thesis, simulation schemes for a simple fluid dynamic equation are constructed this way from first principles. Of paramount importance is the conditional probability of the system given the map data, as it is the only way to influence the resulting simulation scheme. The second part of this thesis focuses on constructing three dimensional maps of the Galactic dust. In this application one has to specify as well, which statements the map degrees of freedom make about the actual distribution of Galactic dust. We choose to model dust as a correlated field, where the degree of correlation is an additional parameter of the map. To infer the parameters of the map, data about dust in three dimensions is needed. To this end, data from stellar surveys are used, which reflects dust density through the extinction towards millions of sources; sources of which also the distance is known to a limited precision. Three dust maps are presented, one using simulated data through which we verify the validity of our approach, one using data from the most recent and precise stellar survey obtained by the Gaia satellite, and a final map using data from a combination of many larger stellar surveys that are available. Our final result is a map showing the extinction due to Galactic dust up to a distance of about $1000$ light years in three dimensions. The map is of importance for observers, to whom dust extinction comprises a foreground to observations, as well as for astrophysicists interested in the composition and structure of the interstellar medium. Also parameters of simulations of the interstellar medium can be constrained using our derived statistical properties. In conclusion, this thesis demonstrates the importance of models and how they constrain reality, as well as the impact of statistical analyses that are derived from first principles.Physik befasst sich mit der Modellierung der Welt. Ein Modell zu bauen kann zwei Bedeutungen haben: Einerseits kann man damit die Konstruktion eines Modells bezeichnen, das das Verhalten eines Systems imitiert, eine Simulation. Andererseits kann ein Modell etwas bezeichnen, das Aspekte des Originals zeigt, nur nicht so groß ist, z.B. eine Karte. Diese Dissertation beschäftigt sich mit der Modellierung der Welt in beiderlei Bedeutungen, und verbindet diese auch. Wir betrachten eine Karte als bedingte Wahrscheinlichkeit, denn die Karte hat Freiheitsgrade, die Aussagen über das System ermöglichen. Kartografiert man ein zeitveränderliches System, so muss man Karten erneuern wenn das System sich verändert. Kennt man die Zeitevolution des Systems, so kann man Aussagen einer Karte in die Zukunft extrapolieren. Die Freiheitsgrade einer erneuerten Karte sollte man dann so wählen, dass man möglichst wenig Informationen über das System verliert. Folgt man diesem Paradigma wiederholt, so erhält man eine Simulation des Systems, abgebildet durch die Serie an Karten. Auf diese Art und Weise leiten wir Simulationen eines einfachen fluiddynamischen Systems von Grund auf her. Dabei ist die durch die Karte induzierte bedingte Wahrscheinlichkeit entscheidend, da sie die einzige Stellschraube für das resultierende Simulationsschema ist. Der zweite Teil dieser Arbeit behandelt das Erstellen von dreidimensionalen Karten von galaktischem Staub. Auch hierbei spielt die Wahl der bedingten Wahrscheinlichkeit, die von der Karte induziert wird, eine zentrale Rolle. Wir modellieren Staub als ein korreliertes Feld, wobei der Grad der Korrelation ein zusätzlicher Parameter der Karte ist. Um die Parameter der Karte zu inferieren werden Daten über Staub in drei Dimensionen benötigt. Diese beziehen wir aus Sternenkatalogen, die Informationen über die Staubdichte durch Abdunklungswerte von Sternen enthält; Sternen von welchen auch die Positionen zu gewissem Grad bekannt sind. Drei Staubkarten werden hier präsentiert. Die erste Staubkarte verwendete synthetische Daten und dient der Validierung unseres Ansatzes. Der zweiten Staubkarte liegt der neuste und präziseste Katalog von Sternen, durchgeführt von dem Gaia Satelliten, zu Grunde. Die finale Staubkarte benutzt Daten von allen größeren öffentlichen Katalogen von Sternen zusammen. Diese Karte zeigt die Abdunklung durch Staub bis zu einer Distanz von 1000 Lichtjahren in drei Dimensionen. Sie ist sowohl für Beobachter zur Korrektur von Staubabsorption relevant, als auch für Astrophysiker, die sich für die Zusammensetzung des interstellaren Mediums interessieren. Auch Parameter von Simulationen des interstellaren Mediums können durch die hergeleiteten statistischen Eigenschaften eingeschränkt werden. Zusammenfassend demonstriert diese Arbeit die Wichtigkeit von Modellen und deren Aussagen über die Realität, sowie die Bedeutung statistischer Analysen, die von Grund auf hergeleitet werden

Geometric variational inference

Efficiently accessing the information contained in non-linear and high dimensional probability distributions remains a core challenge in modern statistics. Traditionally, estimators that go beyond point estimates are either categorized as Variational Inference (VI) or Markov-Chain Monte-Carlo (MCMC) techniques. While MCMC methods that utilize the geometric properties of continuous probability distributions to increase their efficiency have been proposed, VI methods rarely use the geometry. This work aims to fill this gap and proposes geometric Variational Inference (geoVI), a method based on Riemannian geometry and the Fisher information metric. It is used to construct a coordinate transformation that relates the Riemannian manifold associated with the metric to Euclidean space. The distribution, expressed in the coordinate system induced by the transformation, takes a particularly simple form that allows for an accurate variational approximation by a normal distribution. Furthermore, the algorithmic structure allows for an efficient implementation of geoVI which is demonstrated on multiple examples, ranging from low-dimensional illustrative ones to non-linear, hierarchical Bayesian inverse problems in thousands of dimensions.Comment: 42 pages, 18 figures, accepted by Entrop

Towards Bayesian Data Compression

In order to handle large data sets omnipresent in modern science, efficient compression algorithms are necessary. Here, a Bayesian data compression (BDC) algorithm that adapts to the specific measurement situation is derived in the context of signal reconstruction. BDC compresses a data set under conservation of its posterior structure with minimal information loss given the prior knowledge on the signal, the quantity of interest. Its basic form is valid for Gaussian priors and likelihoods. For constant noise standard deviation, basic BDC becomes equivalent to a Bayesian analog of principal component analysis. Using Metric Gaussian Variational Inference, BDC generalizes to non-linear settings. In its current form, BDC requires the storage of effective instrument response functions for the compressed data and corresponding noise encoding the posterior covariance structure. Their memory demand counteract the compression gain. In order to improve this, sparsity of the compressed responses can be obtained by separating the data into patches and compressing them separately. The applicability of BDC is demonstrated by applying it to synthetic data and radio astronomical data. Still the algorithm needs further improvement as the computation time of the compression and subsequent inference exceeds the time of the inference with the original data.Comment: 39 pages, 15 figures, 1 table, for code, see https://gitlab.mpcdf.mpg.de/jharthki/bd

Bayesian decomposition of the Galactic multi-frequency sky using probabilistic autoencoders

All-sky observations of the Milky Way show both Galactic and non-Galactic diffuse emission, for example from interstellar matter or the cosmic microwave background (CMB). The different emitters are partly superimposed in the measurements, partly they obscure each other, and sometimes they dominate within a certain spectral range. The decomposition of the underlying radiative components from spectral data is a signal reconstruction problem and often associated with detailed physical modeling and substantial computational effort. We aim to build an effective and self-instructing algorithm detecting the essential spectral information contained Galactic all-sky data covering spectral bands from $\gamma$-ray to radio waves. Utilizing principles from information theory, we develop a state-of-the-art variational autoencoder specialized on the adaption to Gaussian noise statistics. We first derive a generic generative process that leads from a low-dimensional set of emission features to the observed high-dimensional data. We formulate a posterior distribution of these features using Bayesian methods and approximate this posterior with variational inference. The algorithm efficiently encodes the information of 35 Galactic emission data sets in ten latent feature maps. These contain the essential information required to reconstruct the initial data with high fidelity and are ranked by the algorithm according to their significance for data regeneration. The three most significant feature maps encode astrophysical components: (1) The dense interstellar medium (ISM), (2) the hot and dilute regions of the ISM and (3) the CMB. The machine-assisted and data-driven dimensionality reduction of spectral data is able to uncover the physical features encoding the input data. Our algorithm is able to extract the dense and dilute Galactic regions, as well as the CMB, from the sky brightness values only.Comment: 25 pages, 8 figures, 3 tables. Submitted to Astronomy & Astrophysic

M87* in space, time, and frequency

Observing the dynamics of compact astrophysical objects provides insights into their inner workings, thereby probing physics under extreme conditions. The immediate vicinity of an active supermassive black hole with its event horizon, photon ring, accretion disk, and relativistic jets is a perfect pace to study general relativity, magneto-hydrodynamics, and high energy plasma physics. The recent observations of the black hole shadow of M87* with Very Long Baseline Interferometry (VLBI) by the Event Horizon Telescope (EHT) open the possibility to investigate its dynamical processes on time scales of days. In this regime, radio astronomical imaging algorithms are brought to their limits. Compared to regular radio interferometers, VLBI networks typically have fewer antennas and low signal to noise ratios (SNRs). If the source is variable during the observational period, one cannot co-add data on the sky brightness distribution from different time frames to increase the SNR. Here, we present an imaging algorithm that copes with the data scarcity and the source's temporal evolution, while simultaneously providing uncertainty quantification on all results. Our algorithm views the imaging task as a Bayesian inference problem of a time-varying brightness, exploits the correlation structure between time frames, and reconstructs an entire, $\mathbf{2+1+1}$ dimensional time-variable and spectrally resolved image at once. The degree of correlation in the spatial and the temporal domains is inferred from the data and no form of correlation is excluded a priori. We apply this method to the EHT observation of M87* and validate our approach on synthetic data. The time- and frequency-resolved reconstruction of M87* confirms variable structures on the emission ring on a time scale of days. The reconstruction indicates extended and time-variable emission structures outside the ring itself.Comment: 43 pages, 15 figures, 6 table

Spatially Resolved Ultraviolet Spectroscopy of the Great Dimming of Betelgeuse

The bright supergiant, Betelgeuse (Alpha Orionis, HD 39801) experienced a visual dimming during 2019 December and the first quarter of 2020 reaching an historic minimum 2020 February 7$-$13. During 2019 September-November, prior to the optical dimming event, the photosphere was expanding. At the same time, spatially resolved ultraviolet spectra using the Hubble Space Telescope/Space Telescope Imaging Spectrograph revealed a substantial increase in the ultraviolet spectrum and Mg II line emission from the chromosphere over the southern hemisphere of the star. Moreover, the temperature and electron density inferred from the spectrum and C II diagnostics also increased in this hemisphere. These changes happened prior to the Great Dimming Event. Variations in the Mg II k-line profiles suggest material moved outwards in response to the passage of a pulse or acoustic shock from 2019 September through 2019 November. It appears that this extraordinary outflow of material from the star, likely initiated by convective photospheric elements, was enhanced by the coincidence with the outward motions in this phase of the $\sim$400 day pulsation cycle. These ultraviolet observations appear to provide the connecting link between the known large convective cells in the photosphere and the mass ejection event that cooled to form the dust cloud in the southern hemisphere imaged in 2019 December, and led to the exceptional optical dimming of Betelgeuse in 2020 February.Comment: 11 pages, 8 figures, Astrophysical Journal, accepte