In this paper, we consider static parameter estimation for a class of continuous-time state-space models. Our goal is to obtain an unbiased estimate of the gradient of the log-likelihood (score function), which is an estimate that is unbiased even if the stochastic processes involved in the model must be discretized in time. To achieve this goal, we apply a \emph{doubly randomized scheme} (see, e.g.,~\cite{ub_mcmc, ub_grad}), that involves a novel coupled conditional particle filter (CCPF) on the second level of randomization \cite{jacob2}. Our novel estimate helps facilitate the application of gradient-based estimation algorithms, such as stochastic-gradient Langevin descent. We illustrate our methodology in the context of stochastic gradient descent (SGD) in several numerical examples and compare with the Rhee \& Glynn estimator \cite{rhee,vihola}.
翻译:在本文中, 我们考虑对一组连续时间状态- 空间模型的静态参数估计。 我们的目标是获得对日志相似度( 核心函数) 梯度的不偏倚估计, 即便模型中所涉及的随机过程必须及时分离, 也是公正的。 为了实现这一目标, 我们应用了一个随机的公式( 例如, 参见 { cite{ ub_ mcmc, ub_ grad}), 这涉及到在随机化的第二层上使用一个新型的附加有条件粒子过滤器( CCPF ) 。 我们的新估计有助于应用基于梯度的估计算法, 如随机偏差的兰氏血统 。 我们用数字示例来说明我们的方法, 并与 Rhee ⁇ Glynn 测量器 { cite{ rhee, vhola} 进行比较 。