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    <id>6123</id>
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    <language>eng</language>
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    <publishedDate>2016-11-30</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Empirical Bayes Methods, Reference Priors, Cross Entropy and the EM Algorithm</title>
    <abstract language="eng">When estimating a probability density within the empirical Bayes framework, the non-parametric maximum likelihood estimate (NPMLE) usually tends to overfit the data. This issue is usually taken care of by regularization - a penalization term is subtracted from the marginal log-likelihood before the maximization step, so that the estimate favors smooth solutions, resulting in the so-called maximum penalized likelihood estimation (MPLE).&#13;
The majority of penalizations currently in use are rather arbitrary brute-force solutions, which lack invariance under transformation of the parameters(reparametrization) and measurements.&#13;
This contradicts the principle that, if the underlying model&#13;
has several equivalent formulations, the methods of inductive inference should lead to consistent results. Motivated by this principle and using an information-theoretic point of view, we suggest an entropy-based penalization term that guarantees this kind of invariance. The resulting density estimate can be seen as a generalization of reference priors. Using the reference prior as a hyperprior, on the other hand, is argued to be a poor choice for regularization. We also present an insightful connection between the NPMLE, the cross entropy&#13;
and the principle of minimum discrimination information suggesting another method of inference that contains the doubly-smoothed maximum likelihood estimation as a special case.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-61230</identifier>
    <author>Ilja Klebanov</author>
    <submitter>Ilja Klebanov</submitter>
    <author>Alexander Sikorski</author>
    <author>Christof Schütte</author>
    <author>Susanna Röblitz</author>
    <series>
      <title>ZIB-Report</title>
      <number>16-56</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>parameter estimation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Bayesian inference</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Bayesian hierarchical modeling</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>hyperparameter</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>hyperprior</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>EM algorithm</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>NPMLE</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>MPLE</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>DS-MLE</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>principle of maximum entropy</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>cross entropy</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>minimum discrimination information</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>reference prior</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Jeffreys prior</value>
    </subject>
    <collection role="ccs" number="F.">Theory of Computation</collection>
    <collection role="msc" number="62-XX">STATISTICS</collection>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="institutes" number="compsys">Computational Systems Biology</collection>
    <collection role="persons" number="susanna.roeblitz">Röblitz, Susanna</collection>
    <collection role="persons" number="schuette">Schütte, Christof</collection>
    <collection role="projects" number="ECMath-CH6">ECMath-CH6</collection>
    <collection role="persons" number="sikorski">Sikorski, Alexander</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/6123/EmpiricalBayesReferencePriorsCrossEntropyEMAlgorithm.pdf</file>
  </doc>
  <doc>
    <id>6130</id>
    <completedYear/>
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    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
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    <edition/>
    <issue/>
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    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2016-12-01</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Empirical Bayes Methods for Prior Estimation in Systems Medicine</title>
    <abstract language="eng">One of the main goals of mathematical modelling in systems medicine related to medical applications is to obtain patient-specific parameterizations and model predictions. In clinical practice, however, the number of available measurements for single patients is usually limited due to time and cost restrictions. This hampers the process of making patient-specific predictions about the outcome of a treatment. On the other hand, data are often available for many patients, in particular if extensive clinical studies have been performed. Therefore, before applying Bayes’ rule separately to the data of each patient (which is typically performed using a non-informative prior), it is meaningful to use empirical Bayes methods in order to construct an informative prior from all available data. We compare the performance of four priors - a non-informative prior and priors chosen by nonparametric maximum likelihood estimation (NPMLE), by maximum penalized lilelihood estimation (MPLE) and by doubly-smoothed maximum likelihood estimation (DS-MLE) - by applying them to a low-dimensional parameter estimation problem in a toy model as well as to a high-dimensional ODE model of the human menstrual cycle, which represents a typical example from systems biology modelling.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-61307</identifier>
    <identifier type="arxiv">1612.01403</identifier>
    <author>Ilja Klebanov</author>
    <submitter>Ilja Klebanov</submitter>
    <author>Alexander Sikorski</author>
    <author>Christof Schütte</author>
    <author>Susanna Röblitz</author>
    <series>
      <title>ZIB-Report</title>
      <number>16-57</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Parameter estimation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Bayesian inference</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Bayesian hierarchical modelling</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>NPMLE</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>MPLE</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>DS-MLE</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>EM algorithm</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Jeffreys prior</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>reference prior</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>hyperparameter</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>hyperprior</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>principle of maximum entropy</value>
    </subject>
    <collection role="msc" number="62-XX">STATISTICS</collection>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="institutes" number="compsys">Computational Systems Biology</collection>
    <collection role="persons" number="susanna.roeblitz">Röblitz, Susanna</collection>
    <collection role="persons" number="schuette">Schütte, Christof</collection>
    <collection role="projects" number="ECMath-CH6">ECMath-CH6</collection>
    <collection role="persons" number="sikorski">Sikorski, Alexander</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/6130/EmpiricalBayesSystemsMedicine.pdf</file>
  </doc>
  <doc>
    <id>5596</id>
    <completedYear/>
    <publishedYear>2015</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume>12</volume>
    <type>book</type>
    <publisherName>Springer</publisherName>
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    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">A guide to numerical modelling in systems biology</title>
    <identifier type="isbn">978-3-319-20058-3</identifier>
    <identifier type="doi">10.1007/978-3-319-20059-0</identifier>
    <enrichment key="Series">Texts in computational science and engineering</enrichment>
    <author>Peter Deuflhard</author>
    <submitter>Regine Kossick</submitter>
    <author>Susanna Röblitz</author>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="institutes" number="compsys">Computational Systems Biology</collection>
    <collection role="persons" number="deuflhard">Deuflhard, Peter</collection>
    <collection role="persons" number="susanna.roeblitz">Röblitz, Susanna</collection>
    <collection role="projects" number="BovSys">BovSys</collection>
    <collection role="projects" number="ECMath-CH5">ECMath-CH5</collection>
    <collection role="projects" number="ECMath-CH6">ECMath-CH6</collection>
  </doc>
  <doc>
    <id>5014</id>
    <completedYear/>
    <publishedYear>2014</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>29</pageFirst>
    <pageLast>44</pageLast>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume>1</volume>
    <type>incollection</type>
    <publisherName>European Mathematical Society</publisherName>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Advanced mathematical modeling in systems biology</title>
    <parentTitle language="eng">MATHEON-Mathematics for Key Technologies</parentTitle>
    <enrichment key="Series">Series in Industrial and Applied Mathematics</enrichment>
    <enrichment key="PeerReviewed">no</enrichment>
    <author>Alexander Bockmayr</author>
    <editor>Peter Deuflhard</editor>
    <submitter>Erlinda Körnig</submitter>
    <author>Heike Siebert</author>
    <editor>Martin Grötschel</editor>
    <author>Susanna Röblitz</author>
    <editor>Dietmar Hömberg</editor>
    <author>Christof Schütte</author>
    <editor>Jürg Kramer</editor>
    <author>Peter Deuflhard</author>
    <editor>Volker Mehrmann</editor>
    <editor>Konrad Polthier</editor>
    <editor>Frank Schmidt</editor>
    <editor>Christof Schütte</editor>
    <editor>Martin Skutela</editor>
    <editor>Jürgen Sprekels</editor>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="institutes" number="compsys">Computational Systems Biology</collection>
    <collection role="persons" number="susanna.roeblitz">Röblitz, Susanna</collection>
    <collection role="persons" number="schuette">Schütte, Christof</collection>
    <collection role="projects" number="ECMath-CH6">ECMath-CH6</collection>
    <collection role="projects" number="SFB1114-C3">SFB1114-C3</collection>
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  <doc>
    <id>5747</id>
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    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
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    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2016-02-16</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Prior estimation and Bayesian inference from large cohort data sets</title>
    <abstract language="eng">One of the main goals of mathematical modelling in systems biology related to medical applications is to obtain patient-specific parameterisations and model predictions.&#13;
In clinical practice, however, the number of available measurements for single patients is usually limited due to time and cost restrictions. This hampers the process of making patient-specific predictions about the outcome of a treatment. On the other hand, data are often available for many patients, in particular if extensive clinical studies have been performed. Using these population data, we propose an iterative algorithm for contructing an informative prior distribution, which then serves as the basis for computing patient-specific posteriors and obtaining individual predictions. We demonsrate the performance of our method by applying it to a low-dimensional parameter estimation problem in a toy model as well as to a high-dimensional ODE model of the human menstrual cycle, which represents a typical example from systems biology modelling.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-57475</identifier>
    <author>Ilja Klebanov</author>
    <submitter>Ilja Klebanov</submitter>
    <author>Alexander Sikorski</author>
    <author>Christof Schütte</author>
    <author>Susanna Röblitz</author>
    <series>
      <title>ZIB-Report</title>
      <number>16-09</number>
    </series>
    <collection role="msc" number="62-XX">STATISTICS</collection>
    <collection role="msc" number="65-XX">NUMERICAL ANALYSIS</collection>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="institutes" number="compsys">Computational Systems Biology</collection>
    <collection role="persons" number="susanna.roeblitz">Röblitz, Susanna</collection>
    <collection role="persons" number="schuette">Schütte, Christof</collection>
    <collection role="projects" number="ECMath-CH6">ECMath-CH6</collection>
    <collection role="persons" number="sikorski">Sikorski, Alexander</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/5747/ZR-16-09_revised_version.pdf</file>
  </doc>
  <doc>
    <id>5631</id>
    <completedYear/>
    <publishedYear>2015</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>1</pageFirst>
    <pageLast>12</pageLast>
    <pageNumber/>
    <edition/>
    <issue>67</issue>
    <volume>9</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Solution of the chemical master equation by radial basis functions approximation with interface tracking</title>
    <abstract language="eng">Background.&#13;
The chemical master equation is the fundamental equation of stochastic chemical kinetics. This differential-difference equation describes temporal evolution of the probability density function for states of a chemical system. A state of the system, usually encoded as a vector, represents the number of entities or copy numbers of interacting species, which are changing according to a list of possible reactions. It is often the case, especially when the state vector is high-dimensional, that the number of possible states the system may occupy is too large to be handled computationally. One way to get around this problem is to consider only those states that are associated with probabilities that are greater than a certain threshold level. &#13;
&#13;
Results.&#13;
We introduce an algorithm that significantly reduces computational resources and is especially powerful when dealing with multi-modal distributions. The algorithm is built according to two key principles. Firstly, when performing time integration, the algorithm keeps track of the subset of states with significant probabilities (essential support). Secondly, the probability distribution that solves the equation is parametrised with a small number of coefficients using collocation on Gaussian radial basis functions. The system of basis functions is chosen in such a way that the solution is approximated only on the essential support instead of the whole state space.&#13;
&#13;
Discussion.&#13;
In order to demonstrate the effectiveness of the method, we consider four application examples: a) the self-regulating gene model, b) the 2-dimensional bistable toggle switch, c) a generalisation of the bistable switch to a 3-dimensional tristable problem, and d) a 3-dimensional cell differentiation model that, depending on parameter values, may operate in bistable or tristable modes. In all multidimensional examples the manifold containing the system states with significant probabilities undergoes drastic transformations over time. This fact makes the examples especially challenging for numerical methods.&#13;
&#13;
Conclusions.&#13;
The proposed method is a new numerical approach permitting to approximately solve a wide range of problems that have been hard to tackle until now. A full representation of multi-dimensional distributions is recovered. The method is especially attractive when dealing with models that yield solutions of a complex structure, for instance, featuring multi-stability.&#13;
&#13;
Electronic version: http://www.biomedcentral.com/1752-0509/9/67</abstract>
    <parentTitle language="eng">BMC Systems Biology</parentTitle>
    <identifier type="doi">10.1186/s12918-015-0210-y</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="FulltextUrl">http://www.biomedcentral.com/1752-0509/9/67</enrichment>
    <author>Ivan Kryven</author>
    <submitter>Erlinda Koernig</submitter>
    <author>Susanna Röblitz</author>
    <author>Christof Schütte</author>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="institutes" number="compsys">Computational Systems Biology</collection>
    <collection role="persons" number="susanna.roeblitz">Röblitz, Susanna</collection>
    <collection role="persons" number="schuette">Schütte, Christof</collection>
    <collection role="projects" number="ECMath-CH6">ECMath-CH6</collection>
    <collection role="projects" number="MODAL-MedLab">MODAL-MedLab</collection>
    <collection role="projects" number="SFB1114-C3">SFB1114-C3</collection>
    <collection role="projects" number="MODAL-Gesamt">MODAL-Gesamt</collection>
  </doc>
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