31,484 research outputs found
Efficient computational strategies for doubly intractable problems with applications to Bayesian social networks
Powerful ideas recently appeared in the literature are adjusted and combined
to design improved samplers for Bayesian exponential random graph models.
Different forms of adaptive Metropolis-Hastings proposals (vertical, horizontal
and rectangular) are tested and combined with the Delayed rejection (DR)
strategy with the aim of reducing the variance of the resulting Markov chain
Monte Carlo estimators for a given computational time. In the examples treated
in this paper the best combination, namely horizontal adaptation with delayed
rejection, leads to a variance reduction that varies between 92% and 144%
relative to the adaptive direction sampling approximate exchange algorithm of
Caimo and Friel (2011). These results correspond to an increased performance
which varies from 10% to 94% if we take simulation time into account. The
highest improvements are obtained when highly correlated posterior
distributions are considered.Comment: 23 pages, 8 figures. Accepted to appear in Statistics and Computin
Measuring edge importance: a quantitative analysis of the stochastic shielding approximation for random processes on graphs
Mathematical models of cellular physiological mechanisms often involve random
walks on graphs representing transitions within networks of functional states.
Schmandt and Gal\'{a}n recently introduced a novel stochastic shielding
approximation as a fast, accurate method for generating approximate sample
paths from a finite state Markov process in which only a subset of states are
observable. For example, in ion channel models, such as the Hodgkin-Huxley or
other conductance based neural models, a nerve cell has a population of ion
channels whose states comprise the nodes of a graph, only some of which allow a
transmembrane current to pass. The stochastic shielding approximation consists
of neglecting fluctuations in the dynamics associated with edges in the graph
not directly affecting the observable states. We consider the problem of
finding the optimal complexity reducing mapping from a stochastic process on a
graph to an approximate process on a smaller sample space, as determined by the
choice of a particular linear measurement functional on the graph. The
partitioning of ion channel states into conducting versus nonconducting states
provides a case in point. In addition to establishing that Schmandt and
Gal\'{a}n's approximation is in fact optimal in a specific sense, we use recent
results from random matrix theory to provide heuristic error estimates for the
accuracy of the stochastic shielding approximation for an ensemble of random
graphs. Moreover, we provide a novel quantitative measure of the contribution
of individual transitions within the reaction graph to the accuracy of the
approximate process.Comment: Added one reference, typos corrected in Equation 6 and Appendix C,
added the assumption that the graph is irreducible to the main theorem
(results unchanged
How to Solve Classification and Regression Problems on High-Dimensional Data with a Supervised Extension of Slow Feature Analysis
Supervised learning from high-dimensional data, e.g., multimedia data, is a challenging task. We propose an extension of slow feature analysis (SFA) for supervised dimensionality reduction called graph-based SFA (GSFA). The algorithm extracts a label-predictive low-dimensional set of features that can be post-processed by typical supervised algorithms to generate the final label or class estimation. GSFA is trained with a so-called training graph, in which the vertices are the samples and the edges represent similarities of the corresponding labels. A new weighted SFA optimization problem is introduced, generalizing the notion of slowness from sequences of samples to such training graphs. We show that GSFA computes an optimal solution to this problem in the considered function space, and propose several types of training graphs. For classification, the most straightforward graph yields features equivalent to those of (nonlinear) Fisher discriminant analysis. Emphasis is on regression, where four different graphs were evaluated experimentally with a subproblem of face detection on photographs. The method proposed is promising particularly when linear models are insufficient, as well as when feature selection is difficult
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