TY - GEN A1 - Klebanov, Ilja A1 - Sikorski, Alexander A1 - Schütte, Christof A1 - Röblitz, Susanna T1 - Empirical Bayes Methods, Reference Priors, Cross Entropy and the EM Algorithm N2 - 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). The majority of penalizations currently in use are rather arbitrary brute-force solutions, which lack invariance under transformation of the parameters(reparametrization) and measurements. This contradicts the principle that, if the underlying model 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 and the principle of minimum discrimination information suggesting another method of inference that contains the doubly-smoothed maximum likelihood estimation as a special case. T3 - ZIB-Report - 16-56 KW - parameter estimation KW - Bayesian inference KW - Bayesian hierarchical modeling KW - hyperparameter KW - hyperprior KW - EM algorithm KW - NPMLE KW - MPLE KW - DS-MLE KW - principle of maximum entropy KW - cross entropy KW - minimum discrimination information KW - reference prior KW - Jeffreys prior Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-61230 SN - 1438-0064 ER - TY - GEN A1 - Klebanov, Ilja A1 - Sikorski, Alexander A1 - Schütte, Christof A1 - Röblitz, Susanna T1 - Empirical Bayes Methods for Prior Estimation in Systems Medicine N2 - 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. T3 - ZIB-Report - 16-57 KW - Parameter estimation KW - Bayesian inference KW - Bayesian hierarchical modelling KW - NPMLE KW - MPLE KW - DS-MLE KW - EM algorithm KW - Jeffreys prior KW - reference prior KW - hyperparameter KW - hyperprior KW - principle of maximum entropy Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-61307 SN - 1438-0064 ER - TY - BOOK A1 - Deuflhard, Peter A1 - Röblitz, Susanna T1 - A guide to numerical modelling in systems biology Y1 - 2015 SN - 978-3-319-20058-3 U6 - https://doi.org/10.1007/978-3-319-20059-0 VL - 12 PB - Springer ER - TY - GEN A1 - Bockmayr, Alexander A1 - Siebert, Heike A1 - Röblitz, Susanna A1 - Schütte, Christof A1 - Deuflhard, Peter ED - Deuflhard, Peter ED - Grötschel, Martin ED - Hömberg, Dietmar ED - Kramer, Jürg ED - Mehrmann, Volker ED - Polthier, Konrad ED - Schmidt, Frank ED - Schütte, Christof ED - Skutela, Martin ED - Sprekels, Jürgen T1 - Advanced mathematical modeling in systems biology T2 - MATHEON-Mathematics for Key Technologies Y1 - 2014 VL - 1 SP - 29 EP - 44 PB - European Mathematical Society ER - TY - GEN A1 - Klebanov, Ilja A1 - Sikorski, Alexander A1 - Schütte, Christof A1 - Röblitz, Susanna T1 - Prior estimation and Bayesian inference from large cohort data sets N2 - One of the main goals of mathematical modelling in systems biology related to medical applications is to obtain patient-specific parameterisations 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. 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. T3 - ZIB-Report - 16-09 Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-57475 SN - 1438-0064 ER - TY - JOUR A1 - Kryven, Ivan A1 - Röblitz, Susanna A1 - Schütte, Christof T1 - Solution of the chemical master equation by radial basis functions approximation with interface tracking JF - BMC Systems Biology N2 - Background. 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. Results. 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. Discussion. 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. Conclusions. 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. Electronic version: http://www.biomedcentral.com/1752-0509/9/67 Y1 - 2015 U6 - https://doi.org/10.1186/s12918-015-0210-y VL - 9 IS - 67 SP - 1 EP - 12 ER -