A Markovian Model for Joint Observations, Bell's Inequality and Hidden States

While the standard approach to quantum systems studies length preserving linear transformations of wave functions, the Markov picture focuses on trace preserving operators on the space of Hermitian (self-adjoint) matrices. The Markov approach extends the standard one and provides a refined analysis of measurements and quantum Markov chains. In particular, Bell's inequality becomes structurally clear. It turns out that hidden state models are natural in the Markov context. In particular, a violation of Bell's inequality is seen to be compatible with the existence of hidden states. The Markov model moreover clarifies the role of the "negative probabilities" in Feynman's analysis of the EPR paradox.Comment: 14 page

Sensitive Long-Indel-Aware Alignment of Sequencing Reads

The tremdendous advances in high-throughput sequencing technologies have made population-scale sequencing as performed in the 1000 Genomes project and the Genome of the Netherlands project possible. Next-generation sequencing has allowed genom-wide discovery of variations beyond single-nucleotide polymorphisms (SNPs), in particular of structural variations (SVs) like deletions, insertions, duplications, translocations, inversions, and even more complex rearrangements. Here, we design a read aligner with special emphasis on the following properties: (1) high sensitivity, i.e. find all (reasonable) alignments; (2) ability to find (long) indels; (3) statistically sound alignment scores; and (4) runtime fast enough to be applied to whole genome data. We compare performance to BWA, bowtie2, stampy and find that our methods is especially advantageous on reads containing larger indels

On Hidden States in Quantum Random Walks

It was recently pointed out that identifiability of quantum random walks and hidden Markov processes underlie the same principles. This analogy immediately raises questions on the existence of hidden states also in quantum random walks and their relationship with earlier debates on hidden states in quantum mechanics. The overarching insight was that not only hidden Markov processes, but also quantum random walks are finitary processes. Since finitary processes enjoy nice asymptotic properties, this also encourages to further investigate the asymptotic properties of quantum random walks. Here, answers to all these questions are given. Quantum random walks, hidden Markov processes and finitary processes are put into a unifying model context. In this context, quantum random walks are seen to not only enjoy nice ergodic properties in general, but also intuitive quantum-style asymptotic properties. It is also pointed out how hidden states arising from our framework relate to hidden states in earlier, prominent treatments on topics such as the EPR paradoxon or Bell's inequalities.Comment: 26 page

On analytic properties of entropy rate

Entropy rate of discrete random sources are a real valued functional on the space of probability measures associated with the random sources. If one equips this space with a topology one can ask for the analytic properties of the entropy rates. A natural choice is the topology, which is induced by the norm of total variation. A central result is that entropy rate is Lipschitz continuous relative to this topology. The consequences are manifold. First, corollaries are obtained that refer to prevalent objects of probability theory. Second, the result is extended to entropy rate of dynamical systems. Third, it is shown how to exploit the proof schemes to give a direct and elementary proof for the existence of entropy rate of asymptotically mean stationary random sources

Diskretwertige stochastische Vektorräume . Grundlagen, Ergodentheorie und Darstellungen

Inhalt dieser Arbeit sind eine Ausarbeitung eines neuen auf Vektorräumen basierten Formalismus zur Untersuchung stochastischer Prozesse sowie, darauf aufbauend, Dimensionsbegriffe, die zur Definition neuer oder wenig analysierter Prozessklassen führen. Diese Prozessklassen enthalten die weithin beliebte Klasse der Hidden-Markov-Prozesse, die intuitiv leicht zugänglich sind, sich rein mathematischen Untersuchungen gegenber jedoch unzugänglich erweisen. Wir zeigen in dieser Arbeit, dass mit Hilfe des neuen Formalismus deutlich einfachere Beweise für bereits bestehende Erkenntnisse erhalten werden können und fügen eine Reihe neuer Ergebnisse hinzu. Wir zeigen darüberhinaus Wege zum konkreten Einsatz der der neuen Klassen entstammenden Prozesse für Lernalgorithmen auf

Generic identification of binary-valued hidden Markov processes

The generic identification problem is to decide whether a stochastic process $(X_t)$ is a hidden Markov process and if yes to infer its parameters for all but a subset of parametrizations that form a lower-dimensional subvariety in parameter space. Partial answers so far available depend on extra assumptions on the processes, which are usually centered around stationarity. Here we present a general solution for binary-valued hidden Markov processes. Our approach is rooted in algebraic statistics hence it is geometric in nature. We find that the algebraic varieties associated with the probability distributions of binary-valued hidden Markov processes are zero sets of determinantal equations which draws a connection to well-studied objects from algebra. As a consequence, our solution allows for algorithmic implementation based on elementary (linear) algebraic routines.Comment: 28 page

Pair HMM based gap statistics for re-evaluation of indels in alignments with affine gap penalties: Extended Version

Although computationally aligning sequence is a crucial step in the vast majority of comparative genomics studies our understanding of alignment biases still needs to be improved. To infer true structural or homologous regions computational alignments need further evaluation. It has been shown that the accuracy of aligned positions can drop substantially in particular around gaps. Here we focus on re-evaluation of score-based alignments with affine gap penalty costs. We exploit their relationships with pair hidden Markov models and develop efficient algorithms by which to identify gaps which are significant in terms of length and multiplicity. We evaluate our statistics with respect to the well-established structural alignments from SABmark and find that indel reliability substantially increases with their significance in particular in worst-case twilight zone alignments. This points out that our statistics can reliably complement other methods which mostly focus on the reliability of match positions.Comment: 17 pages, 7 figure

Next Generation Cluster Editing

This work aims at improving the quality of structural variant prediction from the mapped reads of a sequenced genome. We suggest a new model based on cluster editing in weighted graphs and introduce a new heuristic algorithm that allows to solve this problem quickly and with a good approximation on the huge graphs that arise from biological datasets

Characterization of ergodic hidden Markov sources

An algebraic criterim for the ergodicity of discrete random sources is presented. For finite-dimensional sources, which contain hidden Markov sources as a subclass, the criterium can be effectively computed. This result is obtained on the background of a novel, elementary theory of discrete random sources, which is based on linear spaces spanned by word functions, and linear operators on these spaces. An outline of basic elements of this theory is provided