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Exact and computationally efficient likelihoodbased estimation for discretely observed diffusion processes
 Journal of the Royal Statistical Society, Series B: Statistical Methodology
, 2006
"... The objective of this paper is to present a novel methodology for likelihoodbased inference for discretely observed diffusions. We propose Monte Carlo methods, which build on recent advances on the exact simulation of diffusions, for performing maximum likelihood and Bayesian estimation. ..."
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Cited by 58 (12 self)
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The objective of this paper is to present a novel methodology for likelihoodbased inference for discretely observed diffusions. We propose Monte Carlo methods, which build on recent advances on the exact simulation of diffusions, for performing maximum likelihood and Bayesian estimation.
Hierarchical models in the brain
 PLoS Computational Biology
, 2008
"... This paper describes a general model that subsumes many parametric models for continuous data. The model comprises hidden layers of statespace or dynamic causal models, arranged so that the output of one provides input to another. The ensuing hierarchy furnishes a model for many types of data, of a ..."
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Cited by 20 (9 self)
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This paper describes a general model that subsumes many parametric models for continuous data. The model comprises hidden layers of statespace or dynamic causal models, arranged so that the output of one provides input to another. The ensuing hierarchy furnishes a model for many types of data, of arbitrary complexity. Special cases range from the general linear model for static data to generalised convolution models, with system noise, for nonlinear timeseries analysis. Crucially, all of these models can be inverted using exactly the same scheme, namely, dynamic expectation maximization. This means that a single model and optimisation scheme can be used to invert a wide range of models. We present the model and a brief review of its inversion to disclose the relationships among, apparently, diverse generative models of empirical data. We then show that this inversion can be formulated as a simple neural network and may provide a useful metaphor for inference and learning in the brain.
Statistical Aspects of the fractional stochastic calculus
 ANN. STAT
, 2007
"... We apply the techniques of stochastic integration with respect to the fractional Brownian motion and the theory of regularity and supremum estimation for stochastic processes to study the maximum likelihood estimator (MLE) for the drift parameter of stochastic processes satisfying stochastic equati ..."
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Cited by 10 (5 self)
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We apply the techniques of stochastic integration with respect to the fractional Brownian motion and the theory of regularity and supremum estimation for stochastic processes to study the maximum likelihood estimator (MLE) for the drift parameter of stochastic processes satisfying stochastic equations driven by fractional Brownian motion with any level of Holderregularity (any Hurst parameter). We prove existence and strong consistency of the MLE for linear and nonlinear equations. We also prove that a version of the MLE using only discrete observations is still a strongly consistent estimator.
Monte Carlo maximum likelihood estimation for discretely observed diffusion porcesses
 The Annals of Statistics
, 2009
"... This paper introduces a Monte Carlo method for maximum likelihood inference in the context of discretely observed diffusion processes. The method gives unbiased and a.s. continuous estimators of the likelihood function for a family of diffusion models and its performance in numerical examples is com ..."
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Cited by 8 (1 self)
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This paper introduces a Monte Carlo method for maximum likelihood inference in the context of discretely observed diffusion processes. The method gives unbiased and a.s. continuous estimators of the likelihood function for a family of diffusion models and its performance in numerical examples is computationally efficient. It uses a recently developed technique for the exact simulation of diffusions, and involves no discretization error. We show that, under regularity conditions, the Monte Carlo MLE converges a.s. to the true MLE. For datasize n → ∞, we show that the number of Monte Carlo iterations should be tuned as O(n 1/2) and we demonstrate the consistency properties of the Monte Carlo MLE as an estimator of the true parameter value. 1. Introduction. We introduce a Monte Carlo
On simulated likelihood of discretely observed diffusion processes and comparison to closed form approximation
 Journal of Computational and Graphical Statistics
, 2007
"... This article focuses on two methods to approximate the loglikelihood function for univariate diffusions: 1) the simulation approach using a modified Brownian bridge as the importance sampler; and 2) the recent closedform approach. For the case of constant volatility, we give a theoretical justifica ..."
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Cited by 6 (0 self)
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This article focuses on two methods to approximate the loglikelihood function for univariate diffusions: 1) the simulation approach using a modified Brownian bridge as the importance sampler; and 2) the recent closedform approach. For the case of constant volatility, we give a theoretical justification of the modified Brownian bridge sampler by showing that it is exactly a Brownian bridge. We also discuss computational issues in the simulation approach such as accelerating numerical variance stabilizing transformation, computing derivatives of the simulated loglikelihood, and choosing initial values of parameter estimates. The two approaches are compared in the context of financial applications with annualized parameter values, where the diffusion model has an unknown transition density and has no analytical variance stabilizing transformation. The closedform expansion, particularly the secondorder closedform, is found to be computationally efficient and very accurate when the observation frequency is monthly or higher. It is more accurate in the center than in the tail of the transition density. The simulation approach combined with the variance stabilizing transformation is found to be more reliable than the closedform approach when the observation frequency is lower. Both methods performs better when the volatility level is lower, but the simulation method is more robust to the volatility nature of the diffusion model. When applied to two well known datasets of daily observations, the two methods yield similar parameter estimates in both datasets but slightly different loglikelihood in the case of higher volatility.
Inference for stochastic volatility models using time change transformations
, 2007
"... transformations ..."
Markov chain Monte Carlo algorithms for SDE parameter estimation
, 2008
"... This chapter considers stochastic differential equations for Systems Biology models derived from the Chemical Langevin Equation (CLE). After outlining the derivation of such models, Bayesian inference for the parameters is considered, based on stateoftheart Markov chain Monte Carlo algorithms. St ..."
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Cited by 1 (0 self)
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This chapter considers stochastic differential equations for Systems Biology models derived from the Chemical Langevin Equation (CLE). After outlining the derivation of such models, Bayesian inference for the parameters is considered, based on stateoftheart Markov chain Monte Carlo algorithms. Starting with a basic scheme for models observed perfectly, but discretely in time, problems with standard schemes and their solutions are discussed. Extensions of these schemes to partial observation and observations subject to measurement error are also considered. Finally, the techniques are demonstrated in the context of a simple stochastic kinetic model of a genetic regulatory network. 1
Likelihood based inference for correllated diffusions
, 2007
"... We address the problem of likelihood based inference for correlated diffusion processes using Markov chain Monte Carlo (MCMC) techniques. Such a task presents two interesting problems. First, the construction of the MCMC scheme should ensure that the correlation coefficients are updated subject to t ..."
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Cited by 1 (1 self)
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We address the problem of likelihood based inference for correlated diffusion processes using Markov chain Monte Carlo (MCMC) techniques. Such a task presents two interesting problems. First, the construction of the MCMC scheme should ensure that the correlation coefficients are updated subject to the positive definite constraints of the diffusion matrix. Second, a diffusion may only be observed at a finite set of points and the marginal likelihood for the parameters based on these observations is generally not available. We overcome the first issue by using the Cholesky factorisation on the diffusion matrix. To deal with the likelihood unavailability, we generalise the data augmentation framework of Roberts and Stramer (2001 Biometrika 88(3):603621) to d−dimensional correlated diffusions including multivariate stochastic volatility models. Our methodology is illustrated through simulation based experiments and with daily EUR /USD, GBP/USD rates together with their implied volatilities.
DEM: A variational treatment of dynamic systems
, 2008
"... This paper presents a variational treatment of dynamic models that furnishes timedependent conditional densities on the path or trajectory of a system's states and the timeindependent densities of its parameters. These are obtained by maximising a variational action with respect to conditional den ..."
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This paper presents a variational treatment of dynamic models that furnishes timedependent conditional densities on the path or trajectory of a system's states and the timeindependent densities of its parameters. These are obtained by maximising a variational action with respect to conditional densities, under a fixedform assumption about their form. The action or pathintegral of freeenergy represents a lower bound on the model's logevidence or marginal likelihood required for model selection and averaging. This approach rests on formulating the optimisation dynamically, in generalised coordinates of motion. The resulting scheme can be used for online Bayesian inversion of nonlinear dynamic causal models and is shown to outperform existing approaches, such as Kalman and particle filtering. Furthermore, it provides for dual and triple inferences on a system's states, parameters and hyperparameters using exactly the same principles. We refer to this approach as dynamic expectation maximisation (DEM).