Maximum Likelihood Estimation of Continuous-Discrete State-Space Models : Langevin Path Sampling vs. Numerical Integration

Singer, Hermann GND

The likelihood function of a continuous-discrete state space model with state dependent diffusion function is computed by integrating out the latent variables with the help of a Langevin sampler. The continuous time paths are discretized on a time grid in order to obtain a finite dimensional integration. We use importance sampling where the importance density is the conditional density of the latent states given the measurements. We compute analytical derivatives of the log importance density and consider a continuum limit, where the sampler is given by a stochastic partial differential equation. Since the conditional density is only known up to a normalizing constant, we use instead a kernel density estimator. Alternatively, a suboptimal, but explicitly known, importance density is employed. This may be obtained by using a certain reference parameter vector, where the drift function is zero or linear. We compare the Monte Carlo results with numerical methods based on the solution of the Fokker-Planck equation between measurements. We use the repeated multiplication of transition matrices based on a) Euler transition kernels, b) finite differences and c) discretized integral operators. The methods are illustrated for one and two-dimensional Ornstein-Uhlenbeck processes, where analytical results are known.

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Singer, Hermann: Maximum Likelihood Estimation of Continuous-Discrete State-Space Models. Langevin Path Sampling vs. Numerical Integration. Hagen 2016. FernUniversität in Hagen.

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