10,513 research outputs found
Phase Transitions in Nonlinear Filtering
It has been established under very general conditions that the ergodic
properties of Markov processes are inherited by their conditional distributions
given partial information. While the existing theory provides a rather complete
picture of classical filtering models, many infinite-dimensional problems are
outside its scope. Far from being a technical issue, the infinite-dimensional
setting gives rise to surprising phenomena and new questions in filtering
theory. The aim of this paper is to discuss some elementary examples,
conjectures, and general theory that arise in this setting, and to highlight
connections with problems in statistical mechanics and ergodic theory. In
particular, we exhibit a simple example of a uniformly ergodic model in which
ergodicity of the filter undergoes a phase transition, and we develop some
qualitative understanding as to when such phenomena can and cannot occur. We
also discuss closely related problems in the setting of conditional Markov
random fields.Comment: 51 page
Algebraic Geometry of Matrix Product States
We quantify the representational power of matrix product states (MPS) for
entangled qubit systems by giving polynomial expressions in a pure quantum
state's amplitudes which hold if and only if the state is a translation
invariant matrix product state or a limit of such states. For systems with few
qubits, we give these equations explicitly, considering both periodic and open
boundary conditions. Using the classical theory of trace varieties and trace
algebras, we explain the relationship between MPS and hidden Markov models and
exploit this relationship to derive useful parameterizations of MPS. We make
four conjectures on the identifiability of MPS parameters
Subspace estimation and prediction methods for hidden Markov models
Hidden Markov models (HMMs) are probabilistic functions of finite Markov
chains, or, put in other words, state space models with finite state space. In
this paper, we examine subspace estimation methods for HMMs whose output lies a
finite set as well. In particular, we study the geometric structure arising
from the nonminimality of the linear state space representation of HMMs, and
consistency of a subspace algorithm arising from a certain factorization of the
singular value decomposition of the estimated linear prediction matrix. For
this algorithm, we show that the estimates of the transition and emission
probability matrices are consistent up to a similarity transformation, and that
the -step linear predictor computed from the estimated system matrices is
consistent, i.e., converges to the true optimal linear -step predictor.Comment: Published in at http://dx.doi.org/10.1214/09-AOS711 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Deterministic Mean-field Ensemble Kalman Filtering
The proof of convergence of the standard ensemble Kalman filter (EnKF) from
Legland etal. (2011) is extended to non-Gaussian state space models. A
density-based deterministic approximation of the mean-field limit EnKF
(DMFEnKF) is proposed, consisting of a PDE solver and a quadrature rule. Given
a certain minimal order of convergence between the two, this extends
to the deterministic filter approximation, which is therefore asymptotically
superior to standard EnKF when the dimension . The fidelity of
approximation of the true distribution is also established using an extension
of total variation metric to random measures. This is limited by a Gaussian
bias term arising from non-linearity/non-Gaussianity of the model, which exists
for both DMFEnKF and standard EnKF. Numerical results support and extend the
theory
Data-driven satisficing measure and ranking
We propose an computational framework for real-time risk assessment and
prioritizing for random outcomes without prior information on probability
distributions. The basic model is built based on satisficing measure (SM) which
yields a single index for risk comparison. Since SM is a dual representation
for a family of risk measures, we consider problems constrained by general
convex risk measures and specifically by Conditional value-at-risk. Starting
from offline optimization, we apply sample average approximation technique and
argue the convergence rate and validation of optimal solutions. In online
stochastic optimization case, we develop primal-dual stochastic approximation
algorithms respectively for general risk constrained problems, and derive their
regret bounds. For both offline and online cases, we illustrate the
relationship between risk ranking accuracy with sample size (or iterations).Comment: 26 Pages, 6 Figure
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