Standard State Space Models of Unawareness

The impossibility theorem of Dekel, Lipman and Rustichini has been thought to demonstrate that standard state-space models cannot be used to represent unawareness. We first show that Dekel, Lipman and Rustichini do not establish this claim. We then distinguish three notions of awareness, and argue that although one of them may not be adequately modeled using standard state spaces, there is no reason to think that standard state spaces cannot provide models of the other two notions. In fact, standard space models of these forms of awareness are attractively simple. They allow us to prove completeness and decidability results with ease, to carry over standard techniques from decision theory, and to add propositional quantifiers straightforwardly

Approximate Methods for State-Space Models

State-space models provide an important body of techniques for analyzing time-series, but their use requires estimating unobserved states. The optimal estimate of the state is its conditional expectation given the observation histories, and computing this expectation is hard when there are nonlinearities. Existing filtering methods, including sequential Monte Carlo, tend to be either inaccurate or slow. In this paper, we study a nonlinear filter for nonlinear/non-Gaussian state-space models, which uses Laplace's method, an asymptotic series expansion, to approximate the state's conditional mean and variance, together with a Gaussian conditional distribution. This {\em Laplace-Gaussian filter} (LGF) gives fast, recursive, deterministic state estimates, with an error which is set by the stochastic characteristics of the model and is, we show, stable over time. We illustrate the estimation ability of the LGF by applying it to the problem of neural decoding and compare it to sequential Monte Carlo both in simulations and with real data. We find that the LGF can deliver superior results in a small fraction of the computing time.Comment: 31 pages, 4 figures. Different pagination from journal version due to incompatible style files but same content; the supplemental file for the journal appears here as appendices B--E

State Space Models in R

We give an overview of some of the software tools available in R, either as built- in functions or contributed packages, for the analysis of state space models. Several illustrative examples are included, covering constant and time-varying models for both univariate and multivariate time series. Maximum likelihood and Bayesian methods to obtain parameter estimates are considered.

Efficient likelihood estimation in state space models

Motivated by studying asymptotic properties of the maximum likelihood estimator (MLE) in stochastic volatility (SV) models, in this paper we investigate likelihood estimation in state space models. We first prove, under some regularity conditions, there is a consistent sequence of roots of the likelihood equation that is asymptotically normal with the inverse of the Fisher information as its variance. With an extra assumption that the likelihood equation has a unique root for each $n$, then there is a consistent sequence of estimators of the unknown parameters. If, in addition, the supremum of the log likelihood function is integrable, the MLE exists and is strongly consistent. Edgeworth expansion of the approximate solution of likelihood equation is also established. Several examples, including Markov switching models, ARMA models, (G)ARCH models and stochastic volatility (SV) models, are given for illustration.Comment: With the comments by Jens Ledet Jensen and reply to the comments. Published at http://dx.doi.org/10.1214/009053606000000614; http://dx.doi.org/10.1214/09-AOS748A; http://dx.doi.org/10.1214/09-AOS748B in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Approximate Bayesian Computation in State Space Models

A new approach to inference in state space models is proposed, based on approximate Bayesian computation (ABC). ABC avoids evaluation of the likelihood function by matching observed summary statistics with statistics computed from data simulated from the true process; exact inference being feasible only if the statistics are sufficient. With finite sample sufficiency unattainable in the state space setting, we seek asymptotic sufficiency via the maximum likelihood estimator (MLE) of the parameters of an auxiliary model. We prove that this auxiliary model-based approach achieves Bayesian consistency, and that - in a precise limiting sense - the proximity to (asymptotic) sufficiency yielded by the MLE is replicated by the score. In multiple parameter settings a separate treatment of scalar parameters, based on integrated likelihood techniques, is advocated as a way of avoiding the curse of dimensionality. Some attention is given to a structure in which the state variable is driven by a continuous time process, with exact inference typically infeasible in this case as a result of intractable transitions. The ABC method is demonstrated using the unscented Kalman filter as a fast and simple way of producing an approximation in this setting, with a stochastic volatility model for financial returns used for illustration