We consider the problem of inference for nonlinear, multivariate diffusion processes, satisfying It\^o stochastic differential equations (SDEs), using data at discrete times that may be incomplete and subject to measurement error. Our starting point is a state-of-the-art correlated pseudo-marginal Metropolis-Hastings algorithm, that uses correlated particle filters to induce strong and positive correlation between successive likelihood estimates. However, unless the measurement error or the dimension of the SDE is small, correlation can be eroded by the resampling steps in the particle filter. We therefore propose a novel augmentation scheme, that allows for conditioning on values of the latent process at the observation times, completely avoiding the need for resampling steps. We integrate over the uncertainty at the observation times with an additional Gibbs step. Connections between the resulting pseudo-marginal scheme and existing inference schemes for diffusion processes are made, giving a unified inference framework that encompasses Gibbs sampling and pseudo marginal schemes. The methodology is applied in three examples of increasing complexity. We find that our approach offers substantial increases in overall efficiency, compared to competing methods.
翻译:我们考虑了非线性、多变量扩散过程的推论问题,满足了It ⁇ o 随机差分方程(SDEs),在离散时使用的数据,这些数据可能不完整,并容易发生测量错误。我们的出发点是先进的相关伪边际大都会-哈斯廷算法,这种算法使用相关粒子过滤法在连续的概率估计之间产生强烈和积极的相关性。然而,除非SDE的测量错误或维度小,否则相关联性可能会被粒子过滤器中重新标注的步骤侵蚀。因此,我们提出了一个新的增强方案,允许在观察时对潜在过程的值进行调节,完全避免重标步骤的需要。我们将观察时的不确定性与另一个Gibbs步骤结合起来。由此产生的伪边际图与现有的扩散过程推论计划之间的联系,提供了包含Gibbs取样和伪边际计划的统一推论框架。该方法在三个日益复杂的例子中应用。我们发现,我们的方法在总体上提高了效率,与相互竞争的方法相比较。