# A Novel Anti-Jamming Technique for INS/GNSS Integration Based on Black Box Variational Inference
## 1 Introduction  
The loosely integrated inertial navigation system (INS) and global navigation satellite system (GNSS) corrects INS errors and helps INS to complete navigation tasks by providing the velocity and position of the GNSS. However, GNSS signals are extremely weak when they reach the ground, hence, the signals are vulnerable to interference [1] such as radio signals that fall into the navigation signal pass band and multipath interference caused by reflection, scattering [2] or obstruction from buildings, tunnels and trees [3]. All of these interferences can lead to unknown noise statistics in GNSS navigation parameters. Thus, GNSS is unable to output accurate navigation information and eventually the INS/GNSS integration is unable to work properly. The results of interference on GNSS parameters can be divided into three categories according to various studies. The first is GNSS outages. Because the GNSS cannot output navigation parameters, it is unable to complete auxiliary tasks [4]. Researchers usually solve such cases of interference by machine learning methods, such as neural network [5], regression algorithm [6], fault detection and isolation [7] or multiple receivers [8]. The second result is outliers in the GNSS navigation parameters [9]. For this type of interference, researchers usually set up the Student's model to solve these issues [10]. The third outcome is that the navigation accuracy of the GNSS is lower than the accuracy without interference, but it can still complete auxiliary tasks, for example, the time-varying noise caused by interference. In this paper, the third type of interference noise will be researched. The adaptive Kalman filter (AKF) is the most common method to solve the problem of time-varying noise [11]. The Sage-Husa AKF estimates the noise statistics recursively based on the maximum a posterior criterion [12, 13]. However, the measurement noise of the actual system may be too small compared with the theoretical value, or the initial state noise setting might be too large, which results in the measurement noise covariance matrix (MNCM) losing its positivity and causes filtering divergence. [PERSON] et al. presented a multiple model AKF (MMAKF), which can deal with the model uncertainty by operating abank of Kalman filters with different models simultaneously [14]. However, the MMAKF suffers from substantial computational complexities, and thus, MMAKF has poor real-time performance. [PERSON] et al. first proposed the variational Bayesian (VB) AKF (VBAKF) model to solve the time-varying noise problem [15]. The VBAKF algorithm iteratively estimates the MNCM by using the variational inference (VI) method. The VBAKF algorithm is currently favored by many scholars because of its low computation requirements and accurate estimation. [PERSON] et al. applied VBAKF in target tracking to achieve accurate estimation of targets [16]. [PERSON] et al. used VBAKF in INS/GNSS integration to estimate unknown MNCM [17]. [PERSON] et al. proposed a series of VB nonlinear filtering for unknown MNCM, such as the VB Extended Kalman Filter (VBEKF), VB Untraced Kalman Filter (VBUKF), etc. [18]. The VB Cubature Kalman Filter (VBCKF) method was proposed to improve the estimation accuracy of nonlinear systems in [19]. VB and Monte Carlo sampling were used to solve the unknown measurement noise and uncertain parameters in [20]. [PERSON] et al. solved the problem of inaccurate measurement noise and process noise with VBAKF [11]. However, [PERSON] et al. proposed that the distribution of the process noise covariance matrix (PNCM) and the distribution of system state are non-conjugate, which cannot be solved directly by VBAKF. Therefore, they solved the inaccurate PNCM by black box variational inference (BBVI) [21]. In the existing anti-jamming methods based on VBAKF, to satisfy the conjugate conditions of the VBAKF algorithm, the prior distribution of MNCM was assumed to be an Inv-Gamma distribution or Inv-Wishart distribution.