An overview of differentiable particle filters for data-adaptive sequential Bayesian inference

Sequential Bayesian inference can be used for continual learning to prevent catastrophic forgetting of past tasks and provide an informative prior when learning new tasks. We revisit sequential Bayesian inference and assess whether using the previous task's posterior as a prior for a new task can prevent catastrophic forgetting in Bayesian neural networks. Our first contribution is to perform sequential Bayesian inference using Hamiltonian Monte Carlo. We propagate the posterior as a prior for new tasks by approximating the posterior via fitting a density estimator on Hamiltonian Monte Carlo samples. We find that this approach fails to prevent catastrophic forgetting, demonstrating the difficulty in performing sequential Bayesian inference in neural networks. From there, we study simple analytical examples of sequential Bayesian inference and CL and highlight the issue of model misspecification, which can lead to sub-optimal continual learning performance despite exact inference. Furthermore, we discuss how task data imbalances can cause forgetting. From these limitations, we argue that we need probabilistic models of the continual learning generative process rather than relying on sequential Bayesian inference over Bayesian neural network weights. Our final contribution is to propose a simple baseline called Prototypical Bayesian Continual Learning, which is competitive with the best performing Bayesian continual learning methods on class incremental continual learning computer vision benchmarks.

The limit order book of an exchange represents an information store of market participants’ future aims and for many traders the information held in this store is of interest. However, information loss occurs between orders being entered into the exchange and limit order book data being sent out. We present an online algorithm which carries out Bayesian inference to replace information lost at the level of the exchange server and apply our proof of concept algorithm to real historical data from some of the world’s most liquid futures contracts as traded on CME GLOBEX, EUREX and NYSE Liffe exchanges.

Real-time Bayesian data assimilation with data selection, correction of model bias, and on-the-fly uncertainty propagation

Bayesian econometric methods have enjoyed an increase in popularity in recent years. Econometricians, empirical economists, and policymakers are increasingly making use of Bayesian methods. The Oxford Handbook of Bayesian Econometrics is a single source about Bayesian methods in specialized fields. It contains articles by leading Bayesians on the latest developments in their specific fields of expertise. The volume provides broad coverage of the application of Bayesian econometrics in the major fields of economics and related disciplines, including macroeconomics, microeconomics, finance, and marketing. It reviews the state of the art in Bayesian econometric methodology, with articles on posterior simulation and Markov chain Monte Carlo methods, Bayesian nonparametric techniques, and the specialized tools used by Bayesian time series econometricians such as state space models and particle filtering. It also includes articles on Bayesian principles and methodology.

With the challenges of the unknown noise and time-varying channel, a new modulation recognition algorithm is proposed in this paper. Firstly, we propose a new dynamic state-space model to describe the sequential modulation recognition problem. And the time-variant fading channel is described by finite state Markov channel. Secondly, based on particle filter technology which can overcome the adverse effects of time-varying noise and sequential Bayesian inference method which can premeditate the dynamic transfer characteristics of the fading channel gain, a new algorithm is proposed to estimate the noise variance, channel state and modulation scheme jointly. The algorithm can raise modulation recognition performance with unknown noise variance in the time-varying channel by simulation.

Radars have been an important tool for mankind since its inception. Several object tracking algorithms have been overlaid over the conventional radar to improve the accuracy of detection. Bayesian methods like Kalman and Particle filters is one such example, where the occurrence of the target at various positions is tracked probabilistically. The objective here is to devise an adaptive target tracking system using a combination of Kalman and particle filters, to improve the accuracy while optimizing the rate of convergence with respect to the basic Bayesian methods. It was observed that Kalman and Particle both have good accuracy of detection at lower accelerations with accuracy decreasing sharply at accelerations above 50 m/s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . This trends of decreasing accuracy at higher accelerations is mitigated with the proposed system where the accuracy reduces below 90% at accelerations above 100 m/s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> .

In road environment, road obstacle tracking is able to extract important information for driving safety. Indeed, kinematic characteristics estimation (relative position, relative speed, ...) provides a clearer road scene comprehension. This estimate is one of the important parameters of driver assistance systems. However, only one sensor generally does not allow to detect quickly (all the potentially dangerous obstacles) in all the directions, under all the atmospheric conditions. Moreover, the degree of obstacle recognition is different according to the sensor used. Multiplication of sensors makes it possible to face these various problems. Amalgamated information will represent entities in further details and with less uncertainty than with only one source. A system of higher level has been thus developed in order to have a robust management of all tracks and measurements coming from various sensors. This system, applied to radar and lidar measurements combination, gives important obstacles characteristics present in the front bumper of our experimental vehicle (VELAC: LASMEA's experimental vehicle for driving assistance). This state estimate is based on the use of various Bayesian methods (Extended Kalman Filter and Particle Filter). Here we will use the fusion of two-obstacle tracking delivered by two independent systems: track-to-track fusion. These two systems propose estimates characterizing obstacles positions and relative speeds. Fusion estimation is based on the use of extended Kalman filter (EKF) or particle filters. A comparison of these two methods is presented in this article.

Beliefs inform the behavior of forward-thinking agents in complex environments. Recently, sequential Bayesian inference has emerged as a mechanism to study belief formation among agents adapting to dynamical conditions. However, we lack critical theory to explain how preferences evolve in cases of simple agent interactions. In this paper, we derive a Gaussian, pairwise agent interaction model to study how preferences converge when driven by observation of each other's behaviors. We show that the dynamics of convergence resemble an Ornstein-Uhlenbeck process, a common model in nonequilibrium stochastic dynamics. Using standard analytical and computational techniques, we find that the hyperprior magnitudes, representing the learning time, determine the convergence value and the asymptotic entropy of the preferences across pairs of agents. We also show that the dynamical variance in preferences is characterized by a relaxation time t^{★} and compute its asymptotic upper bound. This formulation enhances the existing toolkit for modeling stochastic, interactive agents by formalizing leading theories in learning theory, and builds towards more comprehensive models of open problems in principal-agent and market theory.

This paper presents a sensor system for robot localization based on the information obtained from a single camera attached in a fixed place external to the robot. Our approach firstly obtains the 3D geometrical model of the robot based on the projection of its natural appearance in the camera while the robot performs an initialization trajectory. This paper proposes a structure-from-motion solution that uses the odometry sensors inside the robot as a metric reference. Secondly, an online localization method based on a sequential Bayesian inference is proposed, which uses the geometrical model of the robot as a link between image measurements and pose estimation. The online approach is resistant to hard occlusions and the experimental setup proposed in this paper shows its effectiveness in real situations. The proposed approach has many applications in both the industrial and service robot fields.

[1] discovered a mapping formula for Type 1 Vague events, and presented an alternative problem as an example of its application. Since it is well known that the alternative problem results in sequential Bayesian inference, the subsequent research flow is to make the mapping formula multidimensional, to derive the Markov (decision) process by introducing the concept of time, and so on. Furthermore, the stochastic differential equation from which it is derived was formulated. [2] This paper refers to Type 2 Vague events based on the secondary mapping formula. This quadratic mapping formula gives a certain rotation to a non-mapping function by transforming it with a relationship between the two mapping functions. Furthermore, here we refer to the derivation of the Type 2 Vague Markov process and the initial and stop conditions for its rotation.

Uemura [1] discovered a mapping formula that transforms and maps the state of nature into fuzzy events with a membership function that expresses the degree of attribution. In decision theory in no-data problems, sequential Bayesian inference is an example of this mapping formula, and Hori et al. [2] made the mapping formula multidimensional, introduced the concept of time, to Markov (decision) processes in fuzzy events under ergodic conditions, and derived stochastic differential equations in fuzzy events, although in reverse. In this paper, we focus on type 2 fuzzy. First, assuming that Type 2 Fuzzy Events are transformed and mapped onto the state of nature by a quadratic mapping formula that simultaneously considers longitudinal and transverse ambiguity, the joint stochastic differential equation representing these two ambiguities can be applied to possibility principal factor analysis if the weights of the equations are orthogonal. This indicates that the type 2 fuzzy is a two-dimensional possibility multivariate error model with longitudinal and transverse directions. Also, when the weights are oblique, it is a general possibility oblique factor analysis. Therefore, an example of type 2 fuzzy system theory is the possibility factor analysis. Furthermore, we show the initial and stopping condition on possibility factor rotation, on the base of possibility theory.

To minimize the average of a set of log-convex functions, the stochastic Newton method iteratively updates its estimate using subsampled versions of the full objective’s gradient and Hessian. We contextualize this optimization problem as sequential Bayesian inference on a latent state-space model with a discriminatively-specified observation process. Applying Bayesian filtering then yields a novel optimization algorithm that considers the entire history of gradients and Hessians when forming an update. We establish matrix-based conditions under which the effect of older observations diminishes over time, in a manner analogous to Polyak’s heavy ball momentum. We illustrate various aspects of our approach with an example and review other relevant innovations for the stochastic Newton method.

We present a sequential sampling methodology for weakly structural Markov laws, arising naturally in a Bayesian structure learning context for decomposable graphical models. As a key component of our suggested approach, we show that the problem of graph estimation, which in general lacks natural sequential interpretation, can be recast into a sequential setting by proposing a recursive Feynman-Kac model that generates a flow of junction tree distributions over a space of increasing dimensions. We focus on particle McMC methods to provide samples on this space, in particular on particle Gibbs (PG), as it allows for generating McMC chains with global moves on an underlying space of decomposable graphs. To further improve the PG mixing properties, we incorporate a systematic refreshment step implemented through direct sampling from a backward kernel. The theoretical properties of the algorithm are investigated, showing that the proposed refreshment step improves the performance in terms of asymptotic variance of the estimated distribution. The suggested sampling methodology is illustrated through a collection of numerical examples demonstrating high accuracy in Bayesian graph structure learning in both discrete and continuous graphical models.

This chapter deals with the problem of non-linear and non-Gaussian states and parameters estimation using Bayesian methods. The performances of various conventional and state-of-the-art state estimation techniques are compared when they are utilized to achieve this objective. These techniques include the Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), and Particle Filter (PF). In the current work, the authors consider two systems (biological model and power system) to perform evaluation of estimation algorithms. The results of the comparative studies show that the UKF provides a higher accuracy than the EKF due to the limited ability of EKF to accurately estimate the mean and covariance matrix of the estimated states through lineralization of the nonlinear process model. The results also show that the PF provides a significant improvement over the UKF because, unlike UKF, PF is not restricted by linear-Gaussian assumptions which greatly extends the range of problems that can be tackled.

Induction machine is highly nonlinear model with states that change with operating point and temperature. In these cases, estimating these variables from other easily obtained measurements can be extremely useful. This paper deals with the problem of state estimation of induction machine on the basis of a third-order electrical model using Bayesian methods. The performances of Bayesian estimation techniques are compared when they are utilized to achieve this objective. These techniques include the extended Kalman filter (EKF), the unscented Kalman filter (UKF), the particle filter (PF), and the developed improved particle filter (IPF). The estimation results, which are validated using simulations, show that IPF provides improved estimation performance over PF, even with abrupt changes in estimated states, and both of them can provide improved accuracy over UKF and EKF. These advantages of the IPF are due to the fact that it uses a better proposal distribution that takes the latest observation into account.