With
Andrew Stuart and
Martin Hairer I am
exploring how infinite dimensional Langevin sampling can be used to sample
from conditioned distributions of solutions of stochastic differential
equations. Our method is based on constructing a partial stochastic
differential equation which has the conditioned target density as its
stationary distribution.
Example: the following picture shows a path from the solution of a
stochastic differential equation on the time interval [0,100], conditioned
on having the value -1 at time 0 and having the value +1 at time 100.
The drift has -1 and +1 as stable equilibrium points and 0 as an unstable
equilibrium point.
This method yields a new algorithm for the (non-linear) Kalman
filter/smoother. One can view the pair of signal and observation as the
solution of a two-dimensional stochastic differential equation. Our method
can then be used to study the distribution of the signal conditioned on a
given observation.
A.M. Stuart, J. Voss and P. Wiberg:
Conditional Path Sampling of SDEs and the Langevin MCMC Method.
Communications in Mathematical Sciences, vol. 2, no. 4,
pp. 685–697, 2004.
link, preprint, more…
M. Hairer, A.M. Stuart, J. Voss and P. Wiberg:
Analysis of SPDEs arising in Path Sampling, Part I: The Gaussian Case.
Communications in Mathematical Sciences, vol. 3, no. 4,
pp. 587–603, 2005.
link, preprint, more…
M. Hairer, A.M. Stuart and J. Voss:
Analysis of SPDEs Arising in Path Sampling, Part II: The Nonlinear Case.
Annals of Applied Probability, vol. 17, no. 5,
pp. 1657–1706, 2007.
online, preprint, more…
A. Apte, M. Hairer, A.M. Stuart and J. Voss:
Sampling The Posterior: An Approach to Non-Gaussian Data Assimilation.
Physica D: Nonlinear Phenomena, vol. 230, no. 1–2,
pp. 50–64, 2007.
online, preprint, more…
A. Apte, C.K.R.T. Jones, A.M. Stuart and J. Voss:
Data Assimilation: Mathematical and Statistical Perspectives.
International Journal for Numerical Methods in Fluids, vol. 56,
no. 8, pp. 1033–1046, 2008.
online, preprint, more…
A. Beskos, G.O. Roberts, A.M. Stuart and J. Voss:
MCMC Methods for Diffusion Bridges.
Stochastics and Dynamics, vol. 8, no. 3, pp. 319–350,
2008.
online, preprint, more…
M. Hairer, A.M. Stuart and J. Voss:
Sampling Conditioned Diffusions.
Pages 159–186 in Trends in Stochastic Analysis,
Cambridge University Press,
vol. 353 of London Mathematical Society Lecture Note Series, 2009.
link, preprint, more…
M. Hairer, A.M. Stuart and J. Voss:
Signal Processing Problems on Function Space: Bayesian Formulation, Stochastic PDEs and Effective MCMC Methods.
To appear in The Oxford Handbook of Nonlinear Filtering (editors Dan Crisan and Boris Rozovsky),
2009.
preprint, more…
M. Hairer, A.M. Stuart and J. Voss:
Sampling Conditioned Hypoelliptic Diffusions.
Submitted, 2009.
preprint, more…
Diffusion Processes
Together with my brother
Andreas Voss
I work on a parameter estimation problem for diffusion processes which
arises in psychology when modelling speeded binary decision processes.
A. Voss, K. Rothermund and J. Voss:
Interpreting the Parameters of the Diffusion Model: An Empirical Validation.
Memory & Cognition, vol. 32, no. 7, pp. 1206–1220,
2004.
link, preprint, more…
A. Voss and J. Voss:
Fast-Dm: a Free Programm for Efficient Diffusion Model Analysis.
Behavior Research Methods, vol. 39, no. 4, pp. 767–775,
2007.
link, preprint, more…
A. Voss and J. Voss:
A Fast Numerical Algorithm for the Estimation of Diffusion-Model Parameters.
Journal of Mathematical Psychology, vol. 52, pp. 1–9, 2008.
online, preprint, more…
A. Voss, J. Voss and K.C. Klauer:
Separating Response-Execution Bias from Decision Bias: Arguments for an Additional Parameter in Ratcliff's Diffusion Model.
To appear in the British Journal of Mathematical and Statistical Psychology,
2009.
preprint, more…
We published our fast-dm programm for
parameter estimation in the diffusion model.
My Diplomarbeit (approximately equivalent to an MSc thesis)
deals with a topic related to diffusion processes: There I consider the
question how fast one can distinguish between two different given
diffusions, when observing a single path over long intervals of time.
J. Voß:
Über die Asymptotik des Bayesrisikos bei Diffusionsprozessen.
Diplomarbeit, Universität Kaiserslautern, 1997.
link, preprint, more…
Large Deviations
In my PhD thesis I prove a large deviation result about the behaviour
of diffusions under a strong drift.
J. Voss:
Some Large Deviation Results for Diffusion Processes.
PhD thesis, University of Kaiserslautern, Germany, 2004.
link, preprint, more…
J. Voss:
Large Deviations for One Dimensional Diffusions with a Strong Drift.
Electronic Journal of Probability, vol. 13, no. 53,
pp. 1479–1526, 2008.
link, preprint, more…
J. Voss:
Upper and Lower Bounds in Exponential Tauberian Theorems.
Tbilisi Mathematical Journal, vol. 2, pp. 41–50, 2009.
preprint, more…
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