Bayesian filtering deals with computing the posterior distribution of the state of a stochastic dynamic system given noisy observations. In this paper, motivated by applications in counter-adversarial autonomous systems, we consider the following inverse filtering problem: Given a sequence of posterior distributions from a Bayesian filter, what can be inferred about the transition kernel of the state, the observation likelihoods of the sensor and the measured observations? For finite-state Markov chains observed in noise (hidden Markov models), we show that a least-squares fit for estimating the parameters and observations amounts to a combinatorial optimization problem with non-convex objective. Instead, by exploiting the algebraic structure of the corresponding Bayesian filter, we propose an algorithm based on convex optimization for reconstructing the transition kernel, the observation likelihoods and the observations. We discuss and derive conditions for identifiability. As an application of our results, we demonstrate the design of a counter-adversarial autonomous system: By observing the actions of an autonomous enemy, we estimate the accuracy of its sensors and the observations it has received. The proposed algorithms are illustrated via several numerical examples.

中文翻译：

---

隐马尔可夫模型的逆过滤及其在反对抗性自治系统中的应用


贝叶斯过滤处理在给定噪声观测的情况下计算随机动态系统状态的后验分布。在本文中，受反对抗性自治系统中应用的启发，我们考虑以下逆向过滤问题：给定贝叶斯滤波器的后验分布序列，可以推断出状态的转移核、状态的观察可能性传感器和测量的观察结果？对于在噪声中观察到的有限状态马尔可夫链（隐马尔可夫模型），我们表明，用于估计参数和观测值的最小二乘拟合相当于具有非凸目标的组合优化问题。相反，通过利用相应贝叶斯滤波器的代数结构，我们提出了一种基于凸优化的算法，用于重建转移核、观测似然和观测。我们讨论并得出可识别性的条件。作为我们结果的应用，我们展示了反对抗性自主系统的设计：通过观察自主敌人的行为，我们估计其传感器的准确性及其收到的观察结果。通过几个数值例子说明了所提出的算法。