Stochastic simulation is a widely used method for estimating quantities in models of chemical reaction networks where uncertainty plays a crucial role. However, reducing the statistical uncertainty of the corresponding estimators requires the generation of a large number of simulation runs, which is computationally expensive. To reduce the number of necessary runs, we propose a variance reduction technique based on control variates. We exploit constraints on the statistical moments of the stochastic process to reduce the estimators' variances. We develop an algorithm that selects appropriate control variates in an on-line fashion and demonstrate the efficiency of our approach on several case studies.
If the state-space is finite and small enough one can deal with the underlying Markov chain directly. But there are also cases where the transient distribution has an infinitely large support and one can still deal with explicit state probabilities. To this end, one can fix a finite state-space, that should contain most of the probability @cite_6 . Refinements of the method work dynamically and adjust the state-space according to the transient distributions @cite_9 @cite_7 @cite_23 .
@cite_9 @cite_23 @cite_7 @cite_6
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Consider a Markov chain on an infinite tree T = (V,E) rooted at . In such a chain, once the initial root state ( ) is chosen, each vertex iteratively chooses its state from the one of its parent by an application of a Markov transition rule (and all such applications are independent). Let µj denote the resulting measure for ( ) = j. µj is defined on configurations = ( (x))x2V 2 A V , where A is some finite set. Let µ n denote the restriction of µ to the sigma-algebra generated by the variables (x) where x is at distance exactly n from . Letting n = maxi,j2A dTV (µ n ,µ n ), where dTV denotes total variation distance, we say that the reconstruction problem is solvable if liminfn!1 n > 0. Reconstruction solvability roughly means that the n’th level of the tree contains a non-vanishing amount of information on the root of the tree, as n ! 1. In this paper we study the problem of robust reconstruction. Let be a non degenerate distribution on A and > 0. Let be chosen according to µ n and 0 be obtained from by letting for each node independently (v) = 0 (v) with probability 1 and 0 (v) to be an independent sample from otherwise. We denote by µ n [, ] the resulting measure on 0 . The measure µ n [, ] is a perturbation of the measure µ n . Letting n(, ) = maxi,j2A dTV (µ n [, ],µ n [, ]), we say that the reconstruction problem is -robust-solvable if liminfn!1 n(, ) > 0 for all 0 1, where 2(M) is the second largest eigenvalue of M (in absolute value), then for all non-degenerate , the reconstruction problem is -robust-solvable. We prove a converse and show that the reconstruction problem is not -robust-solvable if B| 2(M)| 2 < 1. This proves a conjecture by the second author and Peres. We also consider other models of noise and general trees. We introduce a highly efficient method for solving continuous partially-observable Markov decision processes (POMDPs) in which beliefs can be modeled using Gaussian distributions over the state space. Our method enables fast solutions to sequential decision making under uncertainty for a variety of problems involving noisy or incomplete observations and stochastic actions. We present an efficient approach to compute locally-valid approximations to the value function over continuous spaces in time polynomial (O[n4]) in the dimension n of the state space. To directly tackle the intractability of solving general POMDPs, we leverage the assumption that beliefs are Gaussian distributions over the state space, approximate the belief update using an extended Kalman filter (EKF), and represent the value function by a function that is quadratic in the mean and linear in the variance of the belief. Our approach iterates towards a linear control policy over the state space that is locally-optimal with respect to a user defined cost function, and is approximately valid in the vicinity of a nominal trajectory through belief space. We demonstrate the scalability and potential of our approach on problems inspired by robot navigation under uncertainty for state spaces of up to 128 dimensions. We consider a discrete time hidden Markov model where the signal is a stationary Markov chain. When conditioned on the observations, the signal is a Markov chain in a random environment under the conditional measure. It is shown that this conditional signal is weakly ergodic when the signal is ergodic and the observations are nondegenerate. This permits a delicate exchange of the intersection and supremum of σ-fields, which is key for the stability of the nonlinear filter and partially resolves a long-standing gap in the proof of a result of Kunita [J. Multivariate Anal. 1 (1971) 365―393]. A similar result is obtained also in the continuous time setting. The proofs are based on an ergodic theorem for Markov chains in random environments in a general state space. We consider finite state space stationary hidden Markov models (HMMs) in the situation where the number of hidden states is unknown. We provide a frequentist asymptotic evaluation of Bayesian analysis methods. Our main result gives posterior concentration rates for the marginal densities, that is for the density of a fixed number of consecutive observations. Using conditions on the prior, we are then able to define a consistent Bayesian estimator of the number of hidden states. It is known that the likelihood ratio test statistic for overfitted HMMs has a nonstandard behaviour and is unbounded. Our conditions on the prior may be seen as a way to penalize parameters to avoid this phenomenon. Inference of parameters is a much more difficult task than inference of marginal densities, we still provide a precise description of the situation when the observations are i.i.d. and we allow for 2 possible hidden states.