Using stochastic gradient search and the optimal filter derivative, it is possible to perform recursive (i.e., online) maximum likelihood estimation in a non-linear state-space model. As the optimal filter and its derivative are analytically intractable for such a model, they need to be approximated numerically. In [Poyiadjis, Doucet and Singh, Biometrika 2018], a recursive maximum likelihood algorithm based on a particle approximation to the optimal filter derivative has been proposed and studied through numerical simulations. Here, this algorithm and its asymptotic behavior are analyzed theoretically. We show that the algorithm accurately estimates maxima to the underlying (average) log-likelihood when the number of particles is sufficiently large. We also derive (relatively) tight bounds on the estimation error. The obtained results hold under (relatively) mild conditions and cover several classes of non-linear state-space models met in practice.

翻译：使用随机梯度搜索和最佳过滤衍生物,可以在非线性状态空间模型中进行循环(即在线)最大概率估计。由于最佳过滤器及其衍生物在分析上难以找到这种模型,因此需要以数字相近。在[Poyiadjis、Doucet和Singh,Biometrika 2018]中,基于粒子近似至最佳过滤衍生物的循环最大可能性算法已经提出,并通过数字模拟进行了研究。在这里,对这个算法及其无线性行为进行了理论分析。我们表明,当粒子数量足够大时,算法准确估计底部(平均)日志相似性的最大值。我们还(相对)得出了估算误差的严格界限。在(相对)温和条件下获得的结果,并涵盖在实践中遇到的若干非线性状态-空间模型类别。