Bayesian inference for nonlinear multivariate diffusion models observed with error

Abstract

Diffusion processes governed by stochastic differential equations (SDEs) are a well established tool for modelling continuous time data from a wide range of areas. Consequently, techniques have been developed to estimate diffusion parameters from partial and discrete observations. Likelihood based inference can be problematic as closed form transition densities are rarely available. One widely used solution involves the introduction of latent data points between every pair of observations to allow an Euler-Maruyama approximation of the true transition densities to become accurate. In recent literature, Markov chain Monte Carlo (MCMC) methods have been used to sample the posterior distribution of latent data and model parameters; however, naive schemes suffer from a mixing problem that worsens with the degree of augmentation. In this paper, we explore an MCMC scheme whose performance is not adversely affected by the number of latent values. We illustrate the methodology by estimating parameters governing an auto-regulatory gene network, using partial and discrete data that is subject to measurement error.

Bayesian estimation of incompletely observed diffusions

“…To work around this problem, data-augmentation has been proposed where the latent data are the missing diffusion bridges that connect the discrete time observations. See for instance [3,4,6,[11][12][13]21,24,27]. The resulting algorithm has been shown to be successful provided one is able to draw diffusion bridges between two adjacent discrete time observations efficiently.…”

“…The reason for this is that a continuous sample path fixes the diffusion coefficient by means of its quadratic variation process. This phenomenon was first discussed in [24] and a solution to it was proposed in both [6,12] within the projection-simulation setup. The resulting algorithm is referred to as the innovation scheme, as the innovations of the bridges are used as auxiliary data, instead of the discretised bridges themselves.…”

“…This idea has appeared in several papers. Both [11,12] consider the case where L i = I d with possibly several rows removed (which corresponds to not observing corresponding components of the diffusion). For i < j set X (i:j) = {X t , t ∈ (t i , t j )}.…”

“…The approach taken in [11,12] is to approximate this stochastic process by a finite-dimensional vector and next carry out simulation. [23] call this the projection-simulation strategy and advocate the simulation-projection strategy where an appropriate Monte-Carlo scheme is designed that operates on the infinitely-dimensional space of diffusion bridges.…”

“…For i < j set X (i:j) = {X t , t ∈ (t i , t j )}. [12] discretise the SDE and construct an algorithm according to the steps:…”