10,303 research outputs found
Consistent estimation of the spectrum of trace class data augmentation algorithms
Markov chain Monte Carlo is widely used in a variety of scientific
applications to generate approximate samples from intractable distributions. A
thorough understanding of the convergence and mixing properties of these Markov
chains can be obtained by studying the spectrum of the associated Markov
operator. While several methods to bound/estimate the second largest eigenvalue
are available in the literature, very few general techniques for consistent
estimation of the entire spectrum have been proposed. Existing methods for this
purpose require the Markov transition density to be available in closed form,
which is often not true in practice, especially in modern statistical
applications. In this paper, we propose a novel method to consistently estimate
the entire spectrum of a general class of Markov chains arising from a popular
and widely used statistical approach known as Data Augmentation. The transition
densities of these Markov chains can often only be expressed as intractable
integrals. We illustrate the applicability of our method using real and
simulated data.Comment: 43 pages (including Appendix), 3 figures; final versio
Efficient Bayesian estimation of Markov model transition matrices with given stationary distribution
Direct simulation of biomolecular dynamics in thermal equilibrium is
challenging due to the metastable nature of conformation dynamics and the
computational cost of molecular dynamics. Biased or enhanced sampling methods
may improve the convergence of expectation values of equilibrium probabilities
and expectation values of stationary quantities significantly. Unfortunately
the convergence of dynamic observables such as correlation functions or
timescales of conformational transitions relies on direct equilibrium
simulations. Markov state models are well suited to describe both, stationary
properties and properties of slow dynamical processes of a molecular system, in
terms of a transition matrix for a jump process on a suitable discretiza- tion
of continuous conformation space. Here, we introduce statistical estimation
methods that allow a priori knowledge of equilibrium probabilities to be
incorporated into the estimation of dynamical observables. Both, maximum
likelihood methods and an improved Monte Carlo sampling method for reversible
transition ma- trices with fixed stationary distribution are given. The
sampling approach is applied to a toy example as well as to simulations of the
MR121-GSGS-W peptide, and is demonstrated to converge much more rapidly than a
previous approach in [F. Noe, J. Chem. Phys. 128, 244103 (2008)]Comment: 15 pages, 8 figure
Parameter estimation in stochastic systems: some recent results and applications
Some recent work on the characterization of almost sure limit sets for maximum likelihood estimates for stochastic systems is reviewed. Applications to allied topics such as input selection for identification, model selection, self-tuning etc. are briefly discussed
Theory and inference for a Markov switching GARCH model
We develop a Markov-switching GARCH model (MS-GARCH) wherein the conditional mean and variance switch in time from one GARCH process to another. The switching is governed by a hidden Markov chain. We provide sufficient conditions for geometric ergodicity and existene of moments of the process. Because of path dependence, maximum likelihood estimation is not feasible. By enlarging the parameter space to include the state variables, Bayesian estimation using a Gibbs sampling algorithm is feasible. We illustrate the model on SP500 daily returns.GARCH, Markov-switching, Bayesian inference
"Computing Densities: A Conditional Monte Carlo Estimator"
We propose a generalized conditional Monte Carlo technique for computing densities in economic models. Global consistency and functional asymptotic normality are established under ergodicity assumptions on the simulated process. The asymptotic normality result allows us to characterize the asymptotic distribution of the error in density space, and implies faster convergence than nonparametric kernel density estimators. We show that our results nest several other well-known density estimators, and illustrate potential applications.
Computing Densities: A Conditional Monte Carlo Estimator
We propose a generalized conditional Monte Carlo technique for computing densities in economic models. Global consistency and functional asymptotic normality are established under ergodicity assumptions on the simulated process. The asymptotic normality result allows us to characterize the asymptotic distribution of the error in density space, and implies faster convergence than nonparametric kernel density estimators. We show that our results nest several other well-known density estimators, and illustrate potential applications.
Generalized Look-Ahead Methods for Computing Stationary Densities
The look-ahead estimator is used to compute densities associated with Markov processes via simulation. We study a framework that extends the look-ahead estimator to a much broader range of applications. We provide a general asymptotic theory for the estimator, where both L1 consistency and L2 asymptotic normality are established.
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