4,469 research outputs found
A multi-resolution, non-parametric, Bayesian framework for identification of spatially-varying model parameters
This paper proposes a hierarchical, multi-resolution framework for the
identification of model parameters and their spatially variability from noisy
measurements of the response or output. Such parameters are frequently
encountered in PDE-based models and correspond to quantities such as density or
pressure fields, elasto-plastic moduli and internal variables in solid
mechanics, conductivity fields in heat diffusion problems, permeability fields
in fluid flow through porous media etc. The proposed model has all the
advantages of traditional Bayesian formulations such as the ability to produce
measures of confidence for the inferences made and providing not only
predictive estimates but also quantitative measures of the predictive
uncertainty. In contrast to existing approaches it utilizes a parsimonious,
non-parametric formulation that favors sparse representations and whose
complexity can be determined from the data. The proposed framework in
non-intrusive and makes use of a sequence of forward solvers operating at
various resolutions. As a result, inexpensive, coarse solvers are used to
identify the most salient features of the unknown field(s) which are
subsequently enriched by invoking solvers operating at finer resolutions. This
leads to significant computational savings particularly in problems involving
computationally demanding forward models but also improvements in accuracy. It
is based on a novel, adaptive scheme based on Sequential Monte Carlo sampling
which is embarrassingly parallelizable and circumvents issues with slow mixing
encountered in Markov Chain Monte Carlo schemes
Training deep neural density estimators to identify mechanistic models of neural dynamics
Mechanistic modeling in neuroscience aims to explain observed phenomena in terms of underlying causes. However, determining which model parameters agree with complex and stochastic neural data presents a significant challenge. We address this challenge with a machine learning tool which uses deep neural density estimators-- trained using model simulations-- to carry out Bayesian inference and retrieve the full space of parameters compatible with raw data or selected data features. Our method is scalable in parameters and data features, and can rapidly analyze new data after initial training. We demonstrate the power and flexibility of our approach on receptive fields, ion channels, and Hodgkin-Huxley models. We also characterize the space of circuit configurations giving rise to rhythmic activity in the crustacean stomatogastric ganglion, and use these results to derive hypotheses for underlying compensation mechanisms. Our approach will help close the gap between data-driven and theory-driven models of neural dynamics
A statistical approach to the inverse problem in magnetoencephalography
Magnetoencephalography (MEG) is an imaging technique used to measure the
magnetic field outside the human head produced by the electrical activity
inside the brain. The MEG inverse problem, identifying the location of the
electrical sources from the magnetic signal measurements, is ill-posed, that
is, there are an infinite number of mathematically correct solutions. Common
source localization methods assume the source does not vary with time and do
not provide estimates of the variability of the fitted model. Here, we
reformulate the MEG inverse problem by considering time-varying locations for
the sources and their electrical moments and we model their time evolution
using a state space model. Based on our predictive model, we investigate the
inverse problem by finding the posterior source distribution given the multiple
channels of observations at each time rather than fitting fixed source
parameters. Our new model is more realistic than common models and allows us to
estimate the variation of the strength, orientation and position. We propose
two new Monte Carlo methods based on sequential importance sampling. Unlike the
usual MCMC sampling scheme, our new methods work in this situation without
needing to tune a high-dimensional transition kernel which has a very high
cost. The dimensionality of the unknown parameters is extremely large and the
size of the data is even larger. We use Parallel Virtual Machine (PVM) to speed
up the computation.Comment: Published in at http://dx.doi.org/10.1214/14-AOAS716 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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